Abstract
In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estimate, the no-local-collapsing theorem, the pseudo-locality theorem and the strong maximum principle for curvature tensors, are essentially improved for Ricci flows induced by Ricci shrinkers. Our results provide many necessary tools to analyze short time singularities of the Ricci flows of general dimension.
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1 Introduction
A Ricci shrinker is a triple \((M^n, g, f)\) of smooth manifold \(M^n\), Riemannian metric g and a smooth function f satisfying
By a normalization of f, we can assume that
where \(\varvec{\mu }\) is the functional of Perelman. As usual, we define
Lying on the intersection of critical metrics and geometric flows, the study of Ricci shrinkers has already become a very important topic in geometric analysis. Up to dimension 3, all Ricci shrinkers are classified. In dimension 2, the only Ricci shrinkers are \(\mathbb {R}^2, \,S^2\) and \(\mathbb {R}P^2\) with standard metrics, due to the classification of Hamilton [25]. In dimension 3, we know that \(\mathbb {R}^3,\,S^2 \times \mathbb {R},\,S^3\) and their quotients are all possible Ricci shrinkers, based on the work of Perelman [46], Petersen–Wylie [47], Naber [43], Ni–Wallach [45] and Cao–Chen–Zhu [8]. If we assume the curvature operator to be nonnegative, then the Ricci shrinkers are also classified, see Munteanu–Wang [42]. However, an important motivation for the study of the Ricci shrinkers is that the Ricci shrinkers are models for short time singularities of the Ricci flows. In dimension 3, by the Hamilton–Ivey pinch [25, 26, 30], one may naturally assume that the Ricci shrinker has nonnegative curvature operator. If the dimension is strictly greater than 3, the loss of pinch estimate makes the nonnegativity of curvature operator an unsatisfactory condition and should be dropped. Also, it is well known (cf. Haslhofer–Müller [27]) that most interesting singularity models are non-compact. Therefore, to prepare for the singularity analysis of high dimensional Ricci flow, we shall focus only on the study of non-compact Ricci shrinkers without any curvature assumption. Since M is non-compact, the inequality
may fail. The failure of Riemannian curvature bound causes serious consequences. Many fundamental analysis tools, e.g., maximum principle and integration by parts, cannot be applied directly without estimates of the manifold at infinity.
In this paper, we shall provide a solid foundation for many fundamental analysis tools in the Ricci shrinkers. We shall mostly take the point of view that Ricci shrinkers are time slices of self-similar Ricci flow solutions. After a delicate choice of cutoff functions and calculations, we show that most of the fundamental tools, including maximum principle, existence of (conjugate) heat solutions, uniqueness and stochastic completeness, integration by parts, etc., work well on the Ricci shrinker spacetime. Then we use these fundamental tools to study the geometric properties of the Ricci flows induced by the Ricci shrinkers. Therefore, we are able to check that most known important properties of the compact Ricci flows, including monotonicity of Perelman’s functional, no-local-collapsing and pseudo-locality theorem of Perelman, curvature tensor strong maximum principle of Hamilton, do apply on noncompact Ricci shrinkers. Furthermore, since the Ricci flows induced by the Ricci shrinkers are self-similar, we obtain many special properties of the Ricci shrinkers. The first property is the estimate of sharp Logarithmic Sobolev constant, which can be regarded as an improvement of the fact that Perelman’s functional is monotone along each Ricci flow.
Theorem 1
(Optimal Logarithmic Sobolev constant) Let \((M^n,p,g,f)\) be a Ricci shrinker. Then \(\varvec{\mu }(g,\tau )\) is a continuous function for \(\tau >0\) such that \(\varvec{\mu }(g,\tau )\) is decreasing for \(\tau \le 1\) and increasing for \(\tau \ge 1\). In particular, we have
Consequently, the following properties hold.
-
Logarithmic Sobolev inequality. In other words, for each compactly supported locally Lipschitz function u and each \(\tau >0\), we have
$$\begin{aligned}&\int u^2 \log u^2 dV - \left( \int u^2 dV \right) \log \left( \int u^2 dV \right) + \left( \varvec{\mu } +n +\frac{n}{2} \log (4\pi \tau ) \right) \int u^2 dV \nonumber \\&\quad \le \tau \int \left\{ 4|\nabla u|^2 +Ru^2 \right\} dV. \end{aligned}$$(7) -
Sobolev inequality. Namely, for each compactly supported locally Lipschitz function u, we have
$$\begin{aligned} \left( \int u^{\frac{2n}{n-2}}\,dV\right) ^{\frac{n-2}{n}} \le C e^{-\frac{2 \varvec{\mu }}{n}} \int \left\{ 4|\nabla u|^2+Ru^2 \right\} dV \end{aligned}$$(8)for some dimensional constant \(C=C(n)\).
In geometric analysis, it is a fundamental problem to estimate uniform Sobolev constant. When the underlying manifold is noncompact, the uniform Sobolev constant in general does not exist. However, (8) says that there is a uniform (Scalar-)Sobolev constant, depending only on n and \(\varvec{\mu }\). In particular, if the scalar curvature is bounded, i.e., \(\sup _{M} R<\infty \), then there exists a classical Sobolev constant. Namely, for each \(u \in C_{c}^{\infty }(M)\), we have
for some \(C=C(n, \sup _{M} R)\). Note that the term \(e^{-\frac{2 \varvec{\mu }}{n}}\) is almost \(|B(p, 1)|^{-\frac{2}{n}}\) by Lemma 2.5 of [34].
The proof of Theorem 1 follows a similar route as done in Proposition 9.5 of [34], by using the monotonicity of Perelman’s functional along Ricci flow and the invariance of Perelman’s functional under diffeomorphism actions.
Secondly, we can improve the no-local-collapsing theorem of Perelman on the Ricci shrinker Ricci flow. By the fundamental work of Perelman [46], the Ricci flow spacetime can be regarded as a “Ricci-flat” spacetime in terms of reduced volume and reduced distance. Now we can regard Ricci shrinker as a special time slice of the induced Ricci flow. On a Ricci flat manifold, an elementary comparison argument shows that \(\frac{|B(x, r)|}{|B(x,1)|}\) grows at most Euclideanly and at least linearly (cf. [59, 64], and Theorem 2.5 of [35]). This comparison geometry picture has a spacetime version which is used to illustrate the no-local-collapsing (cf. [46, 53]). Although the comparison argument (even the space-time version) does not apply directly in the Ricci shrinker case, we can still show that similar phenomena hold for Ricci shrinkers.
Theorem 2
(Improved no-local-collapsing theorem) Suppose \((M^n,p,g,f)\) is a Ricci shrinker, \(r>1\). Then
Here q is any point on \(\partial B(p, r)\), and C is a dimensional constant.
Although the volume estimate (9a) behaves like the Ricci-flat case, its proof is totally different and much more involved. The proof builds on the the Sobolev inequality (8) and an improvement (cf. Remark 8) of the induction argument due to Munteanu and Wang [41]. The non-collapsing estimate (9b) in general does not hold for Ricci-flat manifold. This indicates that Ricci shrinkers are more rigid than Ricci-flat manifold. See Figure 1 for intuition.
Propagation of non-collapsing on Ricci-shrinkers
The proof of (9b) relies on (6) and an effective volume estimate in [53]. The scale \(\rho \in (0, r^{-1})\) is chosen such that \(R\rho ^2 \le C(n)\) inside B(q, r). If we further assume scalar curvature is uniformly bounded on M, then we shall obtain that every unit ball on the Ricci shrinker M is uniformly non-collapsed. Theorem 2 can be regard as a special case of Theorem 23 and Theorem 23, which are more general versions of the no-local-collapsing. In particular, it indicates that any Ricci shrinker must be \(\kappa \)-noncollpased for some constant \(\kappa >0\), see Remark 7. The proof of Theorems 2, 22 and 23 can be found in Sect. 9. Note that Theorem 2 indicates that the Ricci shrinkers are similar to the Ricci-flat manifolds. Actually, there exist many other similarities between the Ricci-flat manifolds and the Ricci Shrinkers. For example, in [29, 34], it is proved that each sequence of non-collapsed Ricci shrinkers sub-converges to a limit Riemannian conifold Ricci shrinker. Such results are analogue of the weak compactness theorem of non-collapsed Ricci-flat manifolds, by the deep work of Cheeger, Colding and Naber (cf. [12, 14, 20]).
Thirdly, the pseudo-locality theorem of Perelman has an elegant version on the Ricci shrinker Ricci flow. The pseudo-locality theorem of Perelman [46] is a fundamental tool in the study of Ricci flow. It claims that the Ricci flow cannot turn an almost Euclidean domain to a very curved region in a short time period. In the literature, it is known that the pseudo-locality theorem hold for Ricci flow with bounded Riemannian curvature, which condition is clearly not available in the current setting. However, using the existence of special cutoff function, we can show maximum principle and stochastic completeness for conjugate heat kernel. By carefully checking the integration by parts, we obtain that the traditional pseudo-locality theorem holds on the Ricci flow spacetime induced by the Ricci shrinker. Furthermore, the pseudo-locality has the following special version for Ricci shrinkers.
Theorem 3
(Improved pseudo-locality theorem) Suppose that \((M^n,p,g,f)\) is a non-flat Ricci shrinker. Then we have
\(\varvec{\mu }(g,\tau )\) of a Ricci shrinker with bounded geometry
for some small positive constant \(\delta _0=\delta _0(n)\). Furthermore, the following properties are equivalent.
-
(a)
M has bounded geometry. Namely, the norm of Riemannian curvature tensor is bounded from above and the injectivity radius is bounded from below.
-
(b)
The infinitesimal functional satisfies
$$\begin{aligned} \lim _{\tau \rightarrow 0^{+}} \varvec{\mu }(g,\tau )=0. \end{aligned}$$(11) -
(c)
The infinitesimal functional satisfies the gap
$$\begin{aligned} \lim _{\tau \rightarrow 0^{+}} \varvec{\mu }(g,\tau )>-\delta _0. \end{aligned}$$(12)
If one of the above conditions hold, we can define
Then for some positive constant \(C=C(n)\), we have the following explicit estimates
We remark that the gap inequality (10) is not new. It was first proved by Yokota in [57, 58]. However, our proof of (10) is completely different and is the base for the proof of (11), (12) and (14). Theorem 3 also indicates that the bounded geometry for Ricci shrinkers is equivalent to the gap inequality (12). This criterion has divided all Ricci shrinkers into two categories characterized by their graphs of entropies, which are illustrated by Figure 2 and Figure 3. Note that Figure 2 represents the functional behavior of a typical Ricci shrinker, for example, the cylinder \(S^{k} \times \mathbb {R}^{n-k}\) for \(k \ge 2\). Figure 3 represents the functional behavior of a Ricci shrinker with unbounded geometry. However, it is not clear whether such Ricci shrinker exists. For Ricci shrinkers with bounded geometry, it follows from (13) and (14) that the number \(\sqrt{\tau _0}\) can be understood as the regularity scale. Actually, under the scale \(\sqrt{\tau _0}\), all the higher curvature derivatives norm \(|\nabla ^k Rm|\) are bounded by \(C(n,k) \tau _0^{-1-\frac{k}{2}}\), in light of the estimates of Shi [49].
\(\varvec{\mu }(g,\tau )\) of a Ricci shrinker with unbounded geometry
There exist several other special versions and consequences of the pseudo-locality theorems. The proof of all of them, including the proof of Theorem 3, can be found in Sect. 10.
Fourthly, the curvature tensor strong maximum principle, developed by R. Hamilton, works on Ricci shrinker Ricci flows and also has an improved version. Using the curvature tensor maximum principle, Hamilton shows that the nonnegativity of curvature operator is preserved under the Ricci flow and the kernel space is parallel. Therefore, the manifold splits as product when kernel space is nontrivial. Since different time slices of a Ricci shrinker Ricci flow are the same up to scaling and diffeomorphism, the preservation of curvature conditions is automatic. The interesting problem on Ricci shrinker is to show the strong maximum principle, i.e., the splitting of the manifold when eigenvalues of curvature operator satisfy some nonnegativity condition. On this perspective, we can improve the traditional strong maximum principle of curvature operator to the following format.
Theorem 4
(Improved strong maximum principle of curvature tensor) Suppose \((M^n,g,f)\) is a Ricci shrinker and \(\lambda _1 \le \lambda _2 \le \cdots \) are the eigenvalue functions of the curvature operator Rm. Then the following properties hold.
-
If \(\lambda _2 \ge 0\) as a function, then there is a \(k \in \{0,1, 2, \ldots , n\}\) and a closed symmetric space \(N^k\) such that \((M^n, g)\) is isometric to a quotient of \(N^k \times \mathbb {R}^{n-k}\).
-
If \(\lambda _2 \ge 0\) as a function and \(\lambda _2>0\) at one point, then \((M^n,g)\) is isometric to a quotient of round sphere \(S^n\).
The statement in Theorem 4 should be well known to experts in Ricci flow if we replace \(\lambda _2\) by \(\lambda _1\). In fact, by the work of Munteanu–Wang [42] and Petersen–Wylie [47], we know that the same geometry conclusion hold if we replace \(\lambda _2\) in Theorem 4 by \(\lambda _1+\lambda _2\). Their proof builds on the celebrated work of Böhm–Wilking [5] on the closed Ricci flow satisfying \(\lambda _1+\lambda _2>0\) and also relies on a weighted Riemannian curvature integral estimate \(\int |Rm|^2 e^{-f} dV<\infty \). If \(\lambda _1+\lambda _2 \ge 0\), the Riemannian curvature integral estimate can be deduced from the Ricci curvature integral bound \(\int |Rc|^2 e^{-f} dV<\infty \), which follows from a clever integration-by-parts. In Theorem 4, with only condition \(\lambda _2 \ge 0\), Riemannian curvature integral estimate \(\int |Rm|^2 e^{-f} dV<\infty \) becomes nontrivial. As done in [34], we apply local conformal transformations and the classical Cheeger–Colding theory to study the local structure of Ricci shrinkers. Combining the \(L^2\)-curvature estimate of Jiang–Naber [31] with the improved no-local-collapsing Theorem 2, we are able to show that \(\int |Rm|^2 e^{-f} dV<\infty \) always holds true (i.e., Theorem 26). Consequently, the work of Petersen–Wylie [47] applies and the curvature tensor strong maximum principle holds for Ricci shrinkers. Then we are able to obtain \(\lambda _1 \ge 0\) from the condition \(\lambda _2 \ge 0\). Clearly, the condition \(\lambda _2 \ge 0\) is weaker than \(\lambda _1 + \lambda _2 \ge 0\) and Theorem 4 is an improvement of the results of Munteanu–Wang [42] and Petersen–Wylie [47]. Note that \(\lambda _2 \ge 0\) is a novel condition in the Ricci flow literature. It is not clear whether \(\lambda _2 \ge 0\) is preserved by the Ricci flow on a closed manifold. Actually, in Theorem 4, the same conclusion holds if one replace the condition \(\lambda _2 \ge 0\) by an even weak condition
for some \(\epsilon =\epsilon (n)\). The details can be found in Theorem 27. The proof of Theorems 4 and 27 appear in Sect. 11.
The proof of the previous four theorems requires some elementary, but delicate, geometric and analytic facts on Ricci shrinkers.
-
The level sets of f are comparable with geodesic balls.
-
A special cutoff function.
-
Special heat solution and conjugate heat solution on the Ricci shrinker Ricci flow.
-
The existence of heat kernel and stochastic completeness of the backward heat solution.
-
The existence and uniqueness of bounded (conjugate) heat solutions.
After the above estimates are developed, we check that the entropy of Perelman is monotone along the Ricci flow induced by the Ricci shrinker, whose proof needs more delicate integration by parts. Then the proof of Theorem 1 follows a similar route as the one in Proposition 9.5 of [34], with more involved technique. From Theorem 1, we can obtain Theorem 2 by repeatedly choosing proper test function u. When integration by parts are assumed, one can formally follows the routine of Perelman to obtain the differential Harnack inequality (i.e., Theorem 21), and then the traditional pseudo-locality theorem. Combining with a standard localization technique, one can deduce Theorem 3. However, as the functional derivatives contain quadratic Ricci curvature term, many terms concerning high order derivatives need to be carefully handled to verify the integration by parts. This causes many technical difficulties. One key difficulty is the delicate heat kernel estimate to derive the differential Harnack inequality. Therefore, the following heat kernel estimate is in the central position for developing fundamental analytic estimates on Ricci shrinker.
Theorem 5
(Heat Kernel estimate) Let \((M^n,g,f)\) be a Ricci shrinker in \(\mathcal M_n(A)\). Then the following properties hold.
-
(i)
(Heat kernel upper bound)
$$\begin{aligned} H(x,t,y,s) \le \frac{e^{-\varvec{\mu }}}{(4\pi (t-s))^{\frac{n}{2}}}. \end{aligned}$$ -
(ii)
(Heat kernel lower bound) For any \(0<\delta <1\), \(D>1\) and \(0<\epsilon <4\), there exists a constant \(C=C(n,\delta ,D)>0\) such that
$$\begin{aligned} H(x,t,y,s) \ge \frac{C^{\frac{4}{\epsilon }}e^{\varvec{\mu }(\frac{4}{\epsilon }-1)}}{(4\pi (t-s))^{n/2}} \exp {\left( -\frac{d_t^2(x,y)}{(4-\epsilon )(t-s)}\right) } \end{aligned}$$for any \(t \in [-\delta ^{-1},1-\delta ]\) and \(d_t(p,y)+\sqrt{t-s}\le D\).
-
(iii)
(Heat kernel integral bound) For any \(0<\delta <1\), \(D>1\) and \(\epsilon >0\), there exists a constant \(C=C(n,A,\delta ,D,\epsilon )>1\) such that
$$\begin{aligned} \int _{M \backslash B_s(x,r\sqrt{t-s})} H(x,t,y,s)\,dV_s(y) \le C\exp {\left( -\frac{(r-1)^2}{4(1+\epsilon )}\right) } \end{aligned}$$for any \(t \in [-\delta ^{-1},1-\delta ]\), \(d_t(p,x)+\sqrt{t-s}\le D\) and \(r \ge 1\).
We briefly discuss the proof of Theorem 5. Notice that the Logarithmic Sobolev inequality for all scales implies the ultracontractivity of the heat kernel by Davies’ methods (see Chapter 2 of [21]). We prove that the same result (i) holds for Ricci shrinkers. The lower bound of the heat kernel can be estimated by considering the reduced distance (i.e., Theorem 16). We first obtain an on-diagonal lower bound of the heat kernel, in which case the estimate of the reduced distance is straightforward. Then we derive the general off-diagonal lower bound by exploiting a Harnack property (i.e., (200)). To prove the integral upper bound, we consider the probability measure \(v_s(y){:}{=}H(x,t,y,s)\,dV_s(y)\). Following the work of Hein–Naber [28], we show that \(v_s\) satisfies a type of Logarithmic Sobolev inequality (i.e., Theorem 13). The equivalence of the Logarithmic Sobolev inequality and the Gaussian concentration (i.e., Theorem 14) then shows that we can estimate the integral upper bound of the heat kernel by its pointwise lower bound.
Organization of the paper In Sect. 2, we review the definition of the Ricci flows induced by the Ricci shrinkers. We also present the estimates of the potential function and volume upper bound. In Sect. 3, we introduce a family of cutoff functions and prove a maximum principle (i.e., Theorem 6) on Ricci shrinker spacetime. Moreover, we prove the existence and other basic properties of the heat kernel on spacetime. In Sect. 4, we prove the monotonicity of Perelman’s entropy (i.e., Theorem 10). In Sect. 5, we prove Theorem 1. In Sect. 6, we prove the logarithmic Sobolev inequality (i.e., Theorem 13) and the Gaussian concentration (i.e., Theorem 14) of the probability measure induced by the heat kernel. In Sect. 7, Theorem 5 is proved. In Sect. 8, we prove the differential Harnack inequality (i.e., Theorem 21) by using the heat kernel estimates. In Sect. 9, we provide the proof of Theorem 2. In Sect. 10, we prove the pseudo-locality theorem (i.e., Theorems 24) and 3. In the last section, we obtain an \(L^2\)-integral bound of the Riemannian curvature (i.e., Theorem 26). As a consequence, we prove Theorem 4.
2 Preliminaries
For any Ricci shrinker \((M^n,g,f)\), let \({\psi ^t}: M \rightarrow M\) be a 1-parameter family of diffeomorphisms generated by \(X(t)=\dfrac{1}{1-t}\nabla _gf\). That is
By a direct calculation, see [18, Chapter 4], the rescaled pull-back metric \(g(t){:}{=}(1-t) (\psi ^t)^*g\) and the pull-back function \(f(t){:}{=}(\psi ^t)^*f\) satisfy the equation
where \(\left\{ (M,g(t)), -\infty<t<1 \right\} \) is a Ricci flow solution with \(g(0)=g\), that is,
For notational simplicity, we will omit the subscript g(t) if there is no confusion. From (16) and (17), it is easy to show that
Now we define
It follows from (18), (19) and (20) that
Now we define
We have special heat solution and conjugate heat solution:
Note that (27) is equivalent to
Now we have the following estimate of F by using the same method as [9, 27].
Lemma 1
There exists a point \(p \in M\) where F attains its infimum and F satisfies the quadratic growth estimate
for all \(x\in M\) and \(t < 1\), where \(a_+ :=\max \{0,a\}\).
Proof
This originates from the work of Cao–Zhou [9, Theorem 1.1]. We follow the argument of Haslhofer–Müller [27]. It follows from [15] that for any Ricci shrinker \(R \ge 0\) since its corresponding Ricci flow solution is ancient. So from (24), we have
It implies that \(\sqrt{F}\) is \(\frac{1}{2}\)-Lipschitz, since
On the other hand, for any \(x,y \in M\), we choose a minimizing geodesic \(\gamma (s), 0\le s \le d=d_t(x,y)\) joining \(x=\gamma (0)\) and \(y=\gamma (d)\). Assume that \(d >2\), we construct a function
The second variation formula for shortest geodesic implies that
Note that from the Eq. (16),
Therefore from (24) we have
where we used (31) in the last inequality. It is now immediate from (34) that F has a minimum point p. It is clear that \(|\nabla F|=0\) and \(\varDelta F \ge 0\) at the point p by the minimum principle. Hence from (23) and (24) we have
For any \(q \in M\) such that \(d_t(p,q)=d\), it is straightforward from (31) and (34) that
\(\square \)
Note that \(F(\cdot ,t)\) is a pull-back function of \(f(\cdot ,0)\) up to the scale \(\bar{\tau }\), we can choose a base point \(p \in M\) such that p is a minimum point for all \(F(\cdot ,t)\). Now from Lemma 1, \(F(\cdot ,t)\) can be regarded as an approximation of \(\frac{d_t^2}{4}\).
With Lemma 1, we have the following volume estimate whose proof follows from [9, Theorem 1.2].
Lemma 2
There exists a constant \(C=C(n)>0\) such that for any Ricci shrinker \((M^n,g,f)\) with \(p \in M\) a minimum point of f,
Proof
We set \(\rho =2\sqrt{F}\) and \(D(r)=\{x\in M \mid \rho \le r\}\). Moreover we define \(V(r)=\int _{D(r)}\,dV_t\) and \(\chi (r)=\int _{D(r)}R(t)\,dV_t\). It follows from a similar computation as [9, (3.5)], by using (23) and (24), that
If we set \(r_0=\sqrt{2\bar{\tau }(n+2)}\), by integrating (35) we obtain, see [9, (3.6)] for details, that
for any \(r \ge 2\sqrt{\bar{\tau }n}\). Then it follows from Lemma 1 that for any \(r \ge 2\sqrt{\bar{\tau }n}\),
By definition, we have
Moreover, since \(g(t)=\bar{\tau }(\psi ^t)^*g\),
For any x such that \(f(x) \le (n+2)/2\). it follows from Lemma 1 that \(d(p,x) \le c_0(n)\). Therefore for any \(r \ge 2\sqrt{\bar{\tau }n}\),
where the last inequality follows from [34, Lemma 2.3].
Finally, the case \(r \le 2\sqrt{\bar{\tau }n}\) follows from the comparison theorem [55, Theorem 1.2] by using (16). Indeed, for any x with \(d_t(p,x) \le 2\sqrt{\bar{\tau }n}\), it follows from Lemma 1 that \(f(x,t)=\bar{\tau }^{-1}F(x,t) \le C\). Therefore, from (20) we obtain \(|\nabla f|(x,t) \le C{\bar{\tau }}^{-1/2}\). Now it follows from [55, (1.5) of Theorem 1.2] that for any \(s \le r\),
Then the conclusion follows if we let \(s \rightarrow 0\). \(\square \)
3 Cutoff functions, maximum principle and heat kernel
Now we construct a family of cutoff functions which is important when we perform integration by parts throughout the paper.
Fix a function \(\eta \in C^{\infty }([0,\infty ))\) such that \(0\le \eta \le 1\), \(\eta =1\) on [0, 1] and \(\eta =0\) on \([2,\infty )\). Furthermore, \(-C \le \eta '/\eta ^{\frac{1}{2}}\le 0\) and \(|\eta ''|+|\eta '''| \le C\) for a universal constant \(C>0\). For each \(r \ge 1\), we define
Then \(\phi ^r\) is a smooth function on \(M \times (-\infty ,1)\). The following estimates of \(\phi ^{r}\) will be repeatedly used in this paper.
Lemma 3
There exists a constant \(C=C(n)\) such that
Proof
Note that \(F \le 2r\) on the support of \(\phi ^r\), it follows from the assumption of \(\eta \) and (31) that
This finishes the proof of (37). Similarly, by using (22), (24), (29) and (31), we can prove
So (38) and (40) are proved. Then we have
Hence we obtain (39). Finally, using (24) again, we have
which proves (41). \(\square \)
Now we move on to show the maximum principle on general Ricci shrinkers. On a closed manifold, maximum principle holds automatically. If the underlying manifold is noncompact, then some additional assumptions are needed in order the maximum principle to hold. For example, in [35, Theorem 15.2], a condition
is needed for the maximum principle of the static heat equation subsolution u. In our current setting of Ricci shrinker spacetime, the metrics are evolving under Ricci curvature. Then the distance distortion of different time slices is not easy to estimate directly without Ricci curvature bound. Fortunately, we can replace \(d^2\) by f and obtain a maximum principle under a condition similar to (42).
Theorem 6
(Maximum principle on Ricci shrinkers) Let \((M^n,g,f)\) be a Ricci shrinker. Given any closed interval \([a,b] \subset (-\infty ,1)\) and a function u which satisfies \(\square u \le 0\) on \(M \times [a,b]\), suppose that
If \(u(\cdot ,a) \le c\), then \(u(\cdot , b) \le c\).
Proof
From Lemmas 1 and 2, it is easy to see
Therefore, we only need to prove the special case when \(c=0\), by considering \(u-c\).
Multiplying both sides of \(\square u \le 0\) by \(u_+(\phi ^r)^2e^{-2f}\) and integrating on the spacetime \(M \times [a,b]\), then we obtain
For the left side of (44), we have
where we have used \(R \ge 0\), \(f_t=|\nabla f|^2\) and \(u_+(\cdot ,a)=0\). For the right side of (44), we have
Combining (45) and (46), we obtain
where
and
From our construction of \(\phi ^r\), it is easy to see that all functions involved in last three integrals are supported in the spacetime set
Moreover, all the cutoff function terms can be estimated by (37) and (38). For example, we have
Plugging (37), (38) and the above inequality into (49), we arrive at
It follows from (47), (48) and (51) that
Note that the left hand side of the above inequality is independent of r. Letting \(r \rightarrow +\infty \), the finite integral assumption (43) implies that
Therefore, \(u(\cdot ,b) \le 0\) by the continuity of u and positivity of \(e^{-2f(\cdot , b)}\). \(\square \)
The condition (43) is satisfied in many cases. For example, if u is a bounded heat solution. The technique used in the proof of Theorem 6 will be repeatedly used in this paper.
Now we control the spacetime integral of \(|\text {Hess}\,F|^2\).
Lemma 4
For any \(\lambda >0\), \(a<b<1\), there exists a constant \(C=C(a,b,\lambda )\) such that
Proof
From (29) and direct computations,
Multiplying both sides of the above equation by \(\phi ^re^{-\lambda F}\) and integrating on the spacetime \(M \times [a,b]\), we obtain
Now we let \(r \rightarrow \infty \) and the conclusion follows from Lemmas 1 and 2. \(\square \)
Theorem 7
On the Ricci flow spacetime \(M \times (-\infty , 1)\) induced by a Ricci shrinker (M, g, f), there exists a positive heat kernel function H(x, t, y, s) for all \(x, y \in M\) and \(s, t \in (-\infty , 1)\) with \(x \ne y\) and \(s<t\). It satisfies
Furthermore, the heat kernel H satisfies the semigroup property
and the following integral relationships
Proof
We shall divide the proof of Theorem 7 into four steps.
Step 1. Existence of a heat kernel function H solving heat equation and conjugate heat equation.
Fix a compact interval \(I=[a,b] \subset (-\infty ,1)\) and a compact set \(\varOmega \subset M\) with smooth boundary, there exists a Dirichlet heat kernel. The proof can be found in in [19, Chapter 24, Sect. 5]. Regarding \((-\infty ,1)\) as the union of \([-2^{k}, 1-2^{-k}]\), it is easy to see that the Dirichlet heat kernel actually exists on \(\varOmega \times (-\infty , 1)\). Now we let \(\left\{ \varOmega _i \right\} \) be an exhaustion of M by relatively compact domains with smooth boundary such that \(\overline{\varOmega _i}\subset \varOmega _{i+1}\). Let \(H_i(x,t,y,s)\) be the Dirichlet heat kernel of \((\overline{\varOmega _i},g)\). Then the following properties hold.
Let \(\mathbf {n}\) be the outward normal vector of \(\partial \varOmega _i\), then the positivity of \(H_i\) implies that \(\frac{\partial H_i}{\partial \mathbf {n}} \le 0\). Since \(R \ge 0\) on Ricci shrinkers, direct computation shows that
Hence from (61), we have
Similarly, we have
which implies that
As \(H_i>0\) on \(\varOmega _i \times (-\infty ,1)\), it follows from the classical maximum principle that
on \(\varOmega _i \times \varOmega _i \times (-\infty , 1)\). Now we define the heat kernel on \(M \times (-\infty , 1)\) by
From the well-known mean value theorem (cf. Theorem 25.2 in [19]) the interior regularity estimates for the heat equation and conjugate heat equation, it follows from (64) and (66) that \(H_i\) is uniformly bounded when s, t are fixed. Threfore, H exists as a smooth function. Its positivity is guranteed by (67). The regularity estimates also imply that the convergence from \(H_i\) to H is locally smooth. In particular, we can take limit of (59) and (60) to obtain that H solves heat equation and conjugate heat equation on \(M \times (-\infty , 1)\). In other words, (52) and (53) are satisfied.
Step 2 The heat kernel is a fundamental solution of heat equation and conjugate heat equation.
Let \(\phi \) be a smooth function on M with compact support K. For fixed y and s, we have
where C is independent of \(H_i\). Notice that the last two inequalities hold since we just need to restrict the integral on K, and for a fixed s, when t is close to s, the metrics are uniformly equivalent on \(K \times [s,t]\). Combining (61) with (69), we obtain
Since \(\phi \) has compact support, it is clear that
Plugging the above equation into (70) yields that
which means that
By the arbitrary choice of \(\phi \), we obtain (54). Therefore, H is a fundamental solution of the heat equation. Similary, we can use the limit argument to derive (55) and claim that H is a fundamental solution of the conjugate heat equation.
Step 3 The heat kernel satisfies the semigroup property.
From its construction, \(H_i\) satisfies the semigroup property:
For each compact set \(K \subset M\), it is clear that \(K \subset \varOmega _i\) for large i. By the positivity of each \(H_i\), we have
By the arbitrary choice of \(K \subset M\), the above inequality implies that
By (67), (68) and the positivity of H, we have
whose limit form is
Therefore, the semigroup property (56) follows from the combination of (72) and (73).
Step 4 The integral relationships (57) and (58) are satisfied.
On each compact set \(K \subset M\), since \(K \subset \varOmega _i\) for large i and each \(H_i\) is positive on \(\varOmega _i\), we have
where (64) is applied in the last step. The arbitrary choice of K then yields that
which is nothing but (57). Similar reasoning can pass (66) to obtain
where the inequality will be improved to equality (58) in the following argument. In fact, let \(\phi ^{r}\) be the cutoff function defined in (36). For any fixed x and t, it follows from the cutoff function estimate (40) that
Plugging (74) into the above inequality, we obtain
When r is large, x is covered by the support of \(\phi ^r\) at the time t. Using (55), the above inequality implies that
Since r could be arbitrarily large in the above inequality, we obtain (58) by letting \(r \rightarrow \infty \). \(\square \)
Lemma 5
Suppose \([a,b] \subset (-\infty , 1)\) and \(u_{a}\) is a bounded function on the time slice (M, g(a)). Then
is the unique bounded heat solution with initial value \(u_a\).
Proof
Clearly, u is a well-defined heat solution with the initial value \(u_a\). Suppose \(\tilde{u}\) is another heat solution with initial value \(u_a\). Then \(u-\tilde{u}\) is a bounded heat solution with initial value 0. Therefore, we can apply maximum principle Theorem 6 on \(\pm (u-\tilde{u})\) to obtain that
In other words, \(\tilde{u} \equiv u\) and the uniqueness is proved. \(\square \)
Corollary 1
Suppose \(u_a\) is a smooth, bounded, integrable function on (M, g(a)). Let u be the unique bounded heat solution on \(M \times [a,b]\) starting from \(u_a\). Then we have
Proof
Fix \(r>>1\) and multiply both sides of \(\square u=0\) by \(u (\phi ^r)^2\) and integrating on \(M \times [a,b]\), we obtain
By Lemma 5 we know
Then it follows from (57) that u is bounded and integrable. Consequently, \(u^2\) is integrable. It follows from (37) and (38) that by letting \(r \rightarrow \infty \), we obtain from (77)
Therefore, the assumption of Theorem 6 is satisfied. Since \(\square |\nabla u|^2=-2|\text {Hess}\,u|^2 \le 0\), following the maximum principle, we arrive at (76). \(\square \)
Proposition 1
Suppose u is a heat solution and w is a conjugate heat solution on \(M \times [a,b]\) for \([a,b] \subset (-\infty , 1)\) such that
Then we have
Proof
Fix \(r>>1\). We calculate
Note that \(|\nabla u|\le C\) by Corollary 1. Plugging the cutoff function estimates (37) and (40) into the above inequality, we obtain
Taking \(r \rightarrow \infty \), the right hand side of the above inequality tends to zero, the left hand side converges to
since u is bounded and w is integrable. Consequently, we arrive at (78). \(\square \)
Lemma 6
Suppose \([a,b] \subset (-\infty , 1)\) and \(w_{b}\) is an integrable function on the time slice (M, g(b)). Then
is the unique conjugate heat solution with initial value \(w_{b}\) such that
Proof
Fix a time \(a_0 \in [a,b]\) and let h be an arbitrary smooth function with compact support. Then we solve the heat equation starting from h to obtain a unique bounded function u as
Since w is given by (80), it follows from (58) that w satisfies (81). Suppose \(\tilde{w}\) is another conjugate heat solution starting from \(w_b\) satisfying (81). Then we can apply Lemma 5 to the couple of u and \(\tilde{w}-w\) to obtain that for any \(t \in [a_0,b]\),
In particular,
By the arbitrary choice of h, we obtain \(\tilde{w}(\cdot , a_0)-w(\cdot , a_0) \equiv 0\). Then by the arbitrary choice of \(a_0\), we see that
Therefore, the uniqueness is proved. \(\square \)
Lemma 7
Suppose w is a bounded function on \(M \times [a,b]\) satisfying \(\square ^* w \le 0\) and (81). Then we have
Proof
Without loss of generality, by adding a constant, we may assume that \(\displaystyle \sup _{M} w(\cdot , b)=0\). Then it suffices to show that
At the time slice \(t=a\), we choose an arbitrary nonnegative smooth function h with compact support. Then we solve the forward heat solution starting from h and denote the function by u. It is clear that \(u \ge 0\). Similar to the proof of Proposition 1, we obtain that
since at time \(t=b\) we have \(u \ge 0\) and \(w \le 0\). Therefore, the inequality (83) follows from the arbitrary choice of h. \(\square \)
Theorem 8
(Bounded heat solution) Suppose \(t_0 \in (-\infty , 1)\) and h is a bounded function on the time-slice \((M, g(t_0))\). On \(M \times (t_0, 1)\), starting from h, there is a unique heat solution u which is bounded on each compact time-interval of \([t_0,1)\). The solution is
Similarly, for any bouned integrable function h, starting from h there is a unique conjugate heat solution w which is bounded and integrable uniformly on each compact time interval of \((-\infty , t_0]\). The solution is
Theorem 9
(Maximum principle of bounded functions) Suppose u is a bounded super-heat-solution, i.e., \(\square u \le 0\) on \(M \times [a,b]\). Then
Similarly, if w is a bounded super-conjugate-heat-solution, i.e., \(\square ^* w \le 0\) on \(M \times [a,b]\) satisfying (81). Then
From (27) and (28) from previous section, on the space-time \(M \times (-\infty , 1)\), there are standard heat solution and conjugate heat solutions \(F+\frac{n}{2}t\) and \(\bar{v}=(4\pi (1-t))^{-\frac{n}{2}} e^{-f}\). We can apply Theorems 6 and 9 to compare other supersolutions or subsolutions with \(F+\frac{n}{2}t\) and \(\bar{v}=(4\pi (1-t))^{-\frac{n}{2}} e^{-f}\). In particular, we have the following Lemma.
Lemma 8
Given a smooth function \(\phi \) with compact support on a Ricci shrinker \((M^n,g,f)\). For any \(b<1\), let \(w(x,t)=\int H(y,b,x,t)\phi (y) \,dV_b(y)\) be the bounded solution of conjugate heat equation with \(w(\cdot ,b)=\phi \). Then there exists a constant \(C>0\) such that for \(t \le b\)
Lemma 8 tells us that starting from a compact supported function, the solution of the conjugate heat equation is at least exponentially decaying.
4 Monotonicity of Perelman’s entropy
Recall that on any compact Riemannian manifold \((M^n,g)\), Perelman’s \(\mathcal W\) entropy [46] is defined as
for \(\phi \) a smooth function and \(\tau >0\). Let \(u^2=\frac{e^{-\phi }}{(4\pi \tau )^{n/2}}\), we can rewrite above functional as
For a general Ricci shrinker \((M^n,g,f)\), we define the \(\varvec{\mu }\)-functional as
where
The last integral condition \(\int d^2(p,\cdot )u^2\,dV<\infty \) is imposed for two reasons. First, it follows from Lemma 1 and (20) that
Second, the term \(\int u^2 \log u^2\,dV\) in the definition of \(\overline{\mathcal W}(g,u,\tau )\) is well defined. Indeed, if we consider the rescaled measure \(d\tilde{V} {:}{=}e^{-d^2(p,x)}V\), then it follows from the volume estimate Lemma 2 that \(\tilde{V}(M)\) is finite. Given a \(u \in W_{*}^{1,2}\), we set \(A {:}{=}\{x\in M\mid u(x)<1\}\) and \(\tilde{u} {:}{=}\chi _Au\), where \(\chi _A\) is the characteristic function of the set A. Then it is clear that \(\int d^2(p,x)\tilde{u}^2(x)\,dV<\infty \). By a direct calculation,
where \(\hat{u}^2=\tilde{u}^2 e^{d^2(p,\cdot )}\). By Jensen’s inequality, we obtain
since \(\int \hat{u}^2\,d\tilde{V}=\int \tilde{u}^2\,dV \in [0,1]\). Therefore it follows from (94) that
In other words, it implies that for any \(u \in W_{*}^{1,2}(M)\), the negative part of \(u^2\log u^2\) is integrable and \(\overline{\mathcal W}(g,u,\tau ) \in [-\infty ,+\infty )\). In fact, it will be proved later, see Proposition 15 that \(\overline{\mathcal W}(g,u,\tau )\) cannot be \(-\infty \).
Remark 1
The space \(W_{*}^{1,2}(M)\) can be regarded as a collection of probability measure v such that
-
(i)
\(v=\rho V\), that is, v is absolutely continuous with respect to the volume form V.
-
(ii)
v has finite moment of second order (\(v \in P_2(M)\)), that is, for any point \(q \in M\),
$$\begin{aligned} \int d^2(q,\cdot )\,dv < \infty . \end{aligned}$$ -
(iii)
The Fisher information
$$\begin{aligned} F(\rho ) {:}{=}4\int |\nabla \sqrt{\rho }|^2\,dV<\infty . \end{aligned}$$
Now we show that for any Ricci shrinker, we can always restrict the infimum on all smooth functions with compact support.
Proposition 2
For any Ricci shrinker \((M^n,g,f)\),
Proof
For any function \(u \in W_{*}^{1,2}(M)\) such that \(\overline{\mathcal W}(g,u,\tau )\) is finite, we define a positive constant
It is clear from the definition that \(c_r \le 1\) and \(\lim _{r \rightarrow \infty }c_r=1\). From direct computations,
Now by the definition of \(W_{*}^{1,2}\) and the dominated convergence theorem,
Since \(u^2\log u^2\) is absolutely integrable, by the dominated convergence theorem,
and hence
Similarly, if \(\overline{\mathcal W}(g,u,\tau )=-\infty \), then
For a fixed r, it is not hard to choose a sequence of smooth functions \(u_s\) with compact support by the usual smoothing process such that
\(\square \)
Now we prove the celebrated monotonicity theorem of Perelman on Ricci shrinkers.
Theorem 10
For any Ricci shrinker \((M^n,g,f)\) and \(\tau >0\),
is increasing for \(t < \min \{1,\tau \}\).
Proof
We fix a time \(t_1 < \min \{1,\tau \}\) and an nonnegative smooth function \(\sqrt{\bar{w}}\) with compact support such that \(\int \bar{w}\,dV_{t_1}=1\). By defining
it is straightforward to check that
where we have used stochastic completeness (58) for the last equality.
Lemma 9
For any time \(t_0 < t_1\),
Proof of Lemma 9:
By direct computations,
We estimate I first.
Now it is easy to show that all integrals in I are bounded. Indeed, from Lemma 8, there exists a constant C such that
on \(M \times [t_0,t_1]\), where C depends only on \(t_1\), \(t_2\) and the upper bound of \(w(\cdot , t_1)\).
Therefore for \(t \in [t_0,t_1]\)
Moreover, by using (20),
Now we estimate II in (99).
By our construction of \(\phi ^r\), \(\frac{|\nabla \phi ^r|^2}{\phi ^r}\) is uniformly bounded. Reasoning as before,
Now it is easy to see from (99) and (100) that
where C depends only on \(t_0\), \(t_1\) and the upper bound of \(w(\cdot ,t_1)\). By taking \(r \rightarrow \infty \), we have proved Lemma 9.
Now we define the function \(\phi \) as
By direct computations, see Theorem 9.1 of [46], that if we set
then for \(t<\tau \),
that is, v is a subsolution of the conjugate heat equation.
We set \(\tau _1=\tau _1(t)=\tau -t\) for simplicity. By the definition,
Now we multiply both sides of (101) by \(\phi ^r\) so that
The left side of (103) is
The right side of (103) is
Therefore, we have
Now it is important to use the exact expression of \(\square \phi ^r\), that is,
We consider the first term of v and prove the following lemma. \(\square \)
Lemma 10
Proof of Lemma 10:
From (107), we have
Now
Now the first integral of (109) is bounded by (98) while the second
where the last constant C depends only on \(t_0\), \(t_1\) and the upper bound of \(w(\cdot ,t_1)\).
It is immediate that from (109) by taking \(r \rightarrow \infty \) that \(\lim _{r \rightarrow \infty } I=0\).
We continue to estimate II.
Now we have
since
Therefore \(\lim _{r \rightarrow \infty } III=0\).
Similarly,
Now from Lemma 4 the last integral is bounded since \(w \le Ce^{-f}\), so \(\lim _{r \rightarrow \infty } IV=0\). Therefore, Lemma 10 is proved.
We can estimate the integral of \(v \square \phi ^r\).
From the expression of v in (102), we have
Since we have \(|\square \phi ^r|\le Cr^{-1}\) from (40) and all terms except the first above have bounded integral on spacetime, it is easy to show, by taking into account of the claim, that
Now from (106),
Since we choose \(\sqrt{w}(\cdot ,t_1)\) to be a smooth function with compact support, it is immediate that
\(\square \)
Lemma 11
and
Proof of Lemma 11:
From the definition of v,
All terms except for the first two in the above integral are absolutely integrable, due to \(w \le Ce^{-f}\) and \(R \le \tau ^{-2}F\).
Combining with (112), we conclude that
is bounded above.
Then we have
where we have used
To prove (114), for any \(\epsilon >0\),
By taking \(\epsilon \rightarrow 0\), we conclude that (114) holds. Therefore, the proof of Lemma 11 is complete.
Therefore,
In summary, we have shown from (112) that
Since \(\tau \), \(t_0\), \(t_1\) and \(\sqrt{w}(\cdot ,t_1)\) are arbitrary, the proof of Theorem 10 is complete. \(\square \)
Corollary 2
On a Ricci shrinker \((M^n,g,f)\), the functional \(\varvec{\mu }(g,\tau )\) is decreasing for \( 0<\tau <1\) and increasing for \(\tau >1\).
Proof
The same argument appeared in Step 1, Proposition 9.5 of [34]. We repeat the argument here for the convenience of the readers.
For a fixed constant \(\tau _0>1\), from Theorem 10,
is increasing for \(t <1\). Now as t goes from 0 to 1, \(\frac{\tau _0-t}{1-t}\) goes from \(\tau _0\) to \(\infty \). As \(\tau _0>1\) is arbitrary, we have proved that \(\varvec{\mu }(g,\tau )\) is increasing for all \(\tau >1\). Similarly, for any \(\tau _0<1\), as t goes from 0 to \(\tau _0\), \(\frac{\tau _0-t}{1-t}\) goes from \(\tau _0\) to 0. Therefore, \(\varvec{\mu }(g,\tau )\) is decreasing for all \(\tau <1\).
\(\square \)
5 Optimal logarithmic Sobolev constant—part I
For any Ricci shrinker \((M^n,g,f)\) with the normalization (2), we define
It follows from a direct calculation that \(e^{\varvec{\mu }}\) is comparable to the volume of the unit ball B(p, 1).
Lemma 12
(cf. Lemma 2.5 of [34]) For any Ricci shrinker \((M^n,g,f)\), there exists a constant \(C=C(n)>1\) such that
Next we recall from [1] some standard definitions and properties of the space which satisfies the curvature-dimension estimate.
Definition 1
A Riemannian manifold (M, g, v), equipped with a reference measure \(v=e^{-W}V\) where \(W \in C^2\) and V is the standard volume form, satisfies the \(CD(K,\infty )\) condition if the generalized Ricci tensor
In particular, on a Ricci shrinker \((M^n,g,f)\), if we define
then \(v_0\) is a probability measure and \((M,g,v_0) \in CD(\frac{1}{2},\infty )\). Then the following celebrated theorem of Bakry–Émery can be applied on Ricci shrinkers.
Theorem 11
(Bakry–Émery theorem [2]) For any Riemannian manifold (M, g, v) satisfying the \(CD(K,\infty )\) condition for some \(K>0\), the following logarithmic Sobolev inequality holds
where v and \(\rho \, v\) are probability measures which have finite moments of second order and \(\rho \) is locally Lipschitz.
The original proof by Bakry and Émery is complete for compact manifolds. A proof using the optimal transport by Lott and Villani for the general case can be found in [39, Corollary 6.12], see also [52, Theorem 21.2]. For the self-containedness, we give a proof of the Bakry–Émery theorem for Ricci shrinkers.
Theorem 12
For any Ricci shrinker \((M^n,g,f)\) and any nonnegative function \(\rho \) such that \(\sqrt{\rho }\in W^{1,2}(M,v_0)\) and \(\int d^2(p,\cdot )\rho \,dv_0 <\infty \),
If the equality holds, then either \(\rho \) is a constant or \((M^n,g)\) splits off a \(\mathbb {R}\) factor.
Before we prove Theorem 12, we prove the following two lemmas.
Lemma 13
For any smooth function u(t, x) on \(M \times [0,T]\) such that
and for some constant \(a>0\),
if \(u(\cdot ,0) \le c\), then \(u \le c\) on \(M \times [0,T]\).
Proof
The proof follows from [35, Theorem 15.2] verbatim by using \(\varDelta _f\) and the measure \(v_0\) instead of \(\varDelta \) and the volume form V. \(\square \)
We define a new familiy of cutoff functions by setting
where \(\eta \) is the same function as in (36) and f is the potential function at time 0. A direct calculation shows that
Then it is clear that \(\varDelta _f \overline{\phi }^r\) is supported on \(\{f \ge r\}\) and there exists a constant \(C=C(n)\) such that
Lemma 14
For any smooth bounded function u on M,
Proof
From the integration by parts,
where the last equality holds since u is bounded and \(v_0\) is a probability measure. \(\square \)
Proof of Theorem 12:
We only prove the inequality for \(\rho _0\) such that \(\sqrt{\rho }_0\) is a compactly supported smooth function and the general case follows from approximations as in Proposition 95. In addition, we assume that \(\int \rho _0\,dv_0=1\).
Given such \(\rho _0\), we consider the heat flow with respect to the measure \(v_0\), that is,
It is clear that there exists a constant C such that \(\rho \le C\) on \(M \times [0,\infty )\). Now we set
By direct computations
Therefore, for any \(T>0\),
It follows from Lemma 14 that
and
We compute
From Bochner’s formula,
where we have used the Ricci shrinker equation for the last equality.
Therefore,
A direct calculation shows that
It follows from (124) and Lemma 13 that there exists a constant \(C>0\) such that
Therefore, by (123) and Lemma 14, for any \(T>S>0\),
It follows from (122) that for any \(t \ge 0\),
Moreover, for any \(t >s \ge 0\), it follows from (127) that
Then it is easy to see from (129) that
Now we claim that \(E(t) \rightarrow 0\) if \(t \rightarrow \infty \). Since E(t) is decreasing by (128), we only need to prove the claim by considering a sequence \(t_i \rightarrow \infty \). We define \(u_i=\sqrt{\rho (t_i,\cdot )}\), then
and by (130),
Then by taking a subsequence, we claim that \(u_i\) converges to \(u_{\infty }\) weakly in \(W^{1,2}(M,v_0)\). It is clear from (131) and (132) that \(u_{\infty } \equiv 1\). Since we can assume that \(u_i\) converges to 1 almost everywhere,
by the dominated convergence theorem. Therefore, \(E(t) \rightarrow 0\) if \(t \rightarrow \infty \).
It follows from (122) and (130) that
If the equality holds and \(\rho _0\) is not a constant, it follows from (127) that
Therefore, \((M^n,g)\) splits off a \(\mathbb {R}\) factor.
In summary, the proof of Theorem 12 is complete.
Using the Bakry–Émery theorem, Carrillo and Ni have proved in [10] the following result. \(\square \)
Proposition 3
(Carrillo–Ni [10]) For any Ricci shrinker \((M^n,g,f)\), we have
where \(f_0\) is the normalization of f defined in (117).
Proof
We shall follow the argument of Carrillo and Ni. The proof is given for the self-containedness.
For any Ricci shrinker \((M^n,g,f)\) and any smooth function u on M with compact support such that \(\int u^2 \,dV=1\), we define \(w=u^2e^{f_0}\). Then it is clear that both \(v_0\) and \(wv_0\) belong to \(P_2(M)\) from the estimates of f and dV.
It follows from Theorem 12 that
By rewriting (135) in terms of u, we have
It follows from the integration by parts for the last term that (136) becomes
It follows from the \(|\nabla f|^2+R=f\) and \(\varDelta f+R=\frac{n}{2}\) that \(|\nabla f|^2-2\varDelta f-f=R-f\). Therefore, by (137) that
By the arbitrary choice of u, the above inequality means that
On the other hand, if we set \(u_1=e^{-\frac{f_0}{2}}\), it follows from direct calculation that
Recall that \(R+|\nabla f|^2=f\) and \(R+\varDelta f=\frac{n}{2}\) on a Ricci shrinker. So the above equation can be simplified as
Then it follows from definition that
Therefore, (134) follows from the combination of (138) and (139). \(\square \)
Corollary 3
For any Ricci shrinker \((M^n,g,f)\), if there exist more than one minimizer \(u \in W^{1,2}_{*}\) for \(\overline{\mathcal {W}} (g,u,1)\), then (M, g) must split off a \(\mathbb {R}\) factor.
Proof
If u is a minimizer other than \(e^{-\frac{f_0}{2}}\), then the same proof as Proposition 3 shows that
where \(w=u^2e^{f_0}\). Then the conclusion follows from the equality case of Theorem 12. \(\square \)
Proposition 3 indicates that \(\varvec{\mu }\) is the optimal log-Sobolev constant for \((M^n,g,f)\) on scale 1. We shall improve (134) by showing that \(\varvec{\mu }\) is in fact the optimal log-Sobolev constant for all scales. Note that the same result has already been proved for compact Ricci shrinkers in Proposition 9.5 of [34].
Proposition 4
For any Ricci shrinker \((M^n,g,f)\), we have
We first show two important intermediate steps before we prove Proposition 4.
Lemma 15
For each \(\tau \in (0,1)\), we have
Proof
Fix \(\eta _0 \in (0,1)\). Let w be a nonnegative, compactly supported smooth function satisfying the normalization condition \(\int w \,dV=1\). We now regard w as a smooth function on the time slice \(t=0\) and solve the conjugate heat equation \(\square ^* w=0\). Then w is a smooth function on the space-time \(M \times (-\infty , 0)\). It follows from Lemma 8 that there exists a constant \(C>0\) such that
By the diffeomorphism invariance of the \(\overline{\mathcal {W}}\)-functional, it is easy to see that
where we have used the notation
Notice that \(\int u^2 \,dV \equiv 1\) according to our construction. It follows from definition and direct calculations that
By (144), the inequality (142) can be understood as
for some constant C indepenent of t. Consequently, as \(f \ge 0\), we obtain
Note that \(\theta (t)<1\) when \(t<0\). Plugging the above inequality into (146), and noting that
we can use (143) to obtain
From (145), it is clear that \(\displaystyle \lim _{t \rightarrow -\infty } \theta (t)=1\). On the right hand side of the above inequlaity, letting \(t \rightarrow -\infty \), we arrive at
Since \(w(\cdot , 0)\) could be arbitrary smooth nonnegative function satisfying the normalization condition, and \(g=g(0)\), in light of (95), it is clear that (141) follows from the above inequality. \(\square \)
Lemma 16
For each \(\tau \in (1, \infty )\), we have
Proof
For any \(u \in W_{*}^{1,2}\) and \(\tau >1\),
By the arbitrary choice of \(u \in W_{*}^{1,2}\), it follows that
Let \(\tau \rightarrow 1^{+}\), we obtain that
By Corollary 2, we know that \(\varvec{\mu }(g, \tau )\) is an increasing function of \(\tau \) for \(\tau \in (1, \infty )\). Then it is clear that (147) follows directly from the above inequality. \(\square \)
Proof of Proposition 4:
It follows from the combination of Lemmas 15 and 16. \(\square \)
Lemma 17
Suppose (M, g) is a complete Riemannian manifold with Sobolev constant \(C_{RS}\). Namely, for each smooth function u with compact support, we have
Then for each positive \(\tau \), the following estimates hold for any \(u \in W_{*}^{1,2}\),
where
Proof
By Jensen’s inequality, we know that
Plugging the Sobolev inequality (148) into the above inequality yields that
It follows that
Let \(x=\int \left( 4|\nabla u|^2+Ru^2 \right) dV\). The above inequality can be rewritten as
Since \(\tau x>0\), it follows from (152) that
On the other hand, it is clear that
Suppose \(\tau x \ge n^2\), then the combination of (152) and (154) implies that \(\tau x \le 2E\). Consequently, we always have
Clearly, (149) follows from the combination of (153) and (155). \(\square \)
Corollary 4
(Sobolev inequality) Let \(\Big \{(M^n,g(t)),\, t\in (-\infty ,1)\Big \}\) be the Ricci flow solution of a Ricci shrinker \((M^n,p,g,f)\), there exists a constant \(C=C(n)\) such that at any time \(t<1\),
for any smooth function u with compact support.
Proof
We consider the Schrödinger operator \(H=-2\varDelta +\frac{R}{2}\) and the quadratic forms \(Q(u){:}{=}\int (Hu)u\,dV_t\) with its corresponding Markov semigroup \(\{e^{-Hs},\,s\ge 0\}\). Since \(\varvec{\mu }(g(t),\tau ) =\varvec{\mu }(g,\frac{\tau }{1-t})\ge \varvec{\mu }\), we have
for any \(\int u^2\,dV_t=1\), where \(\beta (\tau )=-\frac{n}{2}-\frac{n}{4}\log (4\pi \tau )-\varvec{\mu }\). Then it follows from [21, Corollary 2.2.8] that for any \(s>0\),
where \(M(s){:}{=}\frac{1}{s}\int _0^s \beta (\tau )\,d\tau \). Now we use the same argument as in [21, Theorem 2.4.2] to derive the Sobolev inequaltiy. It follows from (157) that for any \(u \in L^2\),
Since \(e^{-Hs}\) is self-adjoint, by taking the conjugation of (158) we obtain
Therefore, for any \(s>0\),
Combining (160) with the fact that \(e^{-Hs}\) is a contraction on \(L^{\infty }\), it follows from the Riesz-Thorin interpolation that for any \(q \in [1,\infty ]\).
We now write
where
It follows from (161) that
for some constant \(c=c(n)\). Given \(\lambda >0\), we define \(T>0\) by \(\frac{\lambda }{2}=ce^{-\frac{\varvec{\mu }}{q}}\Vert u\Vert _qT^{\frac{1}{2}-\frac{n}{2q}}\). It is clear that
since \(e^{-Hs}\) is a contraction on \(L^q\). For any \(1<q<n\), we set \(\frac{1}{r}=\frac{1}{q}-\frac{1}{n}\), then it follows from our choice of \(\lambda \) that
In other words,
where \(\Vert \cdot \Vert _{r,w}\) denotes the weak \(L^r\) space. Therefore, it follows from the Marcinkiewicz interpolation theorem that
where \(\frac{1}{p}=\frac{1}{2}-\frac{1}{n}\). Therefore, (156) is a direct consequence. \(\square \)
Remark 2
It follows from the above corollary that the Yamabe invariant of \((M^n,g,f) \)
Here Y depends only on the conformal class of g. Hence it implies some connections between a Ricci shrinker and its conformal class. Note that it is shown in [63] that each Ricci shrinker has a conformal metric such that its Ricci curvature has local bound depending only on the dimension. This fact plays a key role in [34].
Proposition 5
On a Ricci shrinker \((M^n,g,f)\), the functional \(\varvec{\mu }(g,\tau )\) is a continuous function of \(\tau \in (0, \infty )\).
Proof
Fix \(\tau _0 \in (0, \infty )\). We need to show both the upper semi-continuity and the lower semi-continuity as \(\tau _0\).
The upper-semicontinuity is more or less standard. Fix \(u \in W_{*}^{1,2}\), we have
By taking the infimum among all qualified u’s, we have
Hence \(\varvec{\mu }(g,\tau )\) is upper semicontinuous.
The lower semicontinuity relies on the estimate (149) in Lemma 17. Actually, for arbitrary \(u \in W_{*}^{1,2}\) satisfying the normalization condition, direct calculation shows that
For any \(\tau _i \rightarrow \tau _0\), we choose \(u_i \in W_{*}^{1,2}\) such that
Together with (165), this implies that
By Corollary 4, the Sobolev constant on each Ricci shrinker is finite. It follows from (149) and (150) that \(\int (4|\nabla u_i|^2+Ru_i^2)\,dV\) is uniformly bounded. In (166), replacing u by \(u_i\) and letting \(i \rightarrow \infty \), we obtain
Combining the above inequality with (167), we obtain that
which is the lower semi-continuity at \(\tau _0\). The continuity of \(\varvec{\mu }(g,\tau )\) with respect to \(\tau \) at \(\tau _0\) follows from the combination of the above inequality and (165). \(\square \)
6 Optimal logarithmic Sobolev constant—part II
We first prove the log-Sobolev inequality for the conjugate heat kernel following [28]. The proof in [28] is for spacetime with bounded geometry. Since we do not impose any curvature restriction here, more should be done due to the integration by parts.
Theorem 13
For any Ricci shrinker \((M^n,g,f)\) with its heat kernel H(x, t, y, s),
Here \(dv_s(y)=H(x,t,y,s)dV_s(y)\) for any \(x\in M\) and \(s<t<1\) and \(\rho \) is any nonnegative function such that \(\sqrt{\rho }\in W^{1,2}(M,v_s)\) and \(\int d^2(p,\cdot )\rho \,dv_s <\infty \). If the equality holds, then either \(\rho \) is a constant or \((M^n,g)\) splits off a \(\mathbb {R}\) factor.
Proof
By a similar approximation process as in Sect. 4, we only need to prove the inequality for any \(\rho \) such that \(\sqrt{\rho }\) is a compactly supported smooth function. Without loss of generality, we assume \(s=0\) and fix \(T>0\) and \(q \in M\). Moreover, we set \(w(x,t)=H(q,T,x,t)\), \(dv=w(y,0)\,dV_0(y)\) and \(\rho (x,t)\) is the bounded solution of the heat equation starting from \(\rho (x)\). In the proof, we denote \(\rho (x,t)\) by \(\rho \) with the time t implicitly understood. We also assume that \(\rho \) is uniformly bounded by 1 on \(M \times [0,T]\).
It is clear from the definition of w that
and
Therefore, we have
By direct computations
Similarly,
Since
we have for any \(s \in [0,T]\),
With (169) and (171), we have proved so far that if r is sufficiently large,
where
It remains to show that when \(r \rightarrow \infty \) the sum is less or equal to 0.
We first notice that as \(\rho \) is smooth with compact support, by using (170) and the maximum principle,
Here the assumption in Theorem 6 can be checked as (98).
Now we have for the first term I
For the second term II,
Similarly for the third term III,
The fourth term IV is more involved, by computation we have
From (172), we have
where we denote \(\text {Hess}\,\rho -\rho ^{-1}d\rho \otimes d\rho \) by h and \(\epsilon \in (0,1)\).
To deal with the last integral, we notice from Lemma 18 that
and hence
Therefore, \(\lim _{r \rightarrow \infty }V\) is finite and \(\lim _{r \rightarrow \infty }|IV| \le -\epsilon \left( \lim _{r \rightarrow \infty }V\right) \). By taking \(\epsilon \rightarrow 0\), we obtain that \(\lim _{r \rightarrow \infty }|IV|=0\) and hence
If the equality holds and \(\rho \) is not a constant, it follows from (173) that
Therefore, \((M^n,g)\) splits off a \(\mathbb {R}\) factor. \(\square \)
For fixed x, t and s, Theorem 13 implies that the probability measure \(dv_s(y)=H(x,t,y,s)dV_s(y)\) satisfies the log-Sobolev inequality with the constant \(\frac{1}{2(t-s)}\). It is a standard fact that log-Sobolev condition implies the Talagrand’s inequality and equivalently, the Gaussian concentration, see [52, Theorems 22.17, 22.10]. In particular we have the following theorem, see also [28, Theorem 1.13].
Theorem 14
(Gaussian concentration) For any Ricci shrinker \((M^n,g,f)\) with its heat kernel H(x, t, y, s) and reference measure \(dv_s(y)=H(x,t,y,s)dV_s(y)\) and any \(\sigma >0\)
where A and B are two sets on M such that \(d_s(A,B) \ge r>0\).
Proof
From Theorem (13), we have for any probability measure \(\rho dv_s\),
By a further approximation, we can assume (175) holds for any locally Lipschitz \(\rho \). Now it follows from [52, Theorem 22.17] that \(dv_s\) satisfies the \(T_2\) Talagrand inequality, that is,
for any measure \(\eta \in P_2(M)\), where \(W_2\) is the Wasserstein distance of second order. For any two sets A and B on M such that \(d_s(A,B) \ge r>0\). We set \(\eta =\frac{1_A}{v_s(A)}v_s\) and \(v=\frac{1_B}{v_s(B)}v_s\). Then on the one hand,
and hence
On the other hand, it follows from the definition of \(W_2\) that
where \(\pi \) is the optimal transport between \(\eta \) and v.
Therefore by computation
\(\square \)
In fact, with the Gaussian concentration, we can prove that \(v_s\) has finite square-exponential moment.
Corollary 5
For any Ricci shrinker \((M^n,g,f)\) with its heat kernel H(x, t, y, s) and reference measure \(dv_s(y)=H(x,t,y,s)dV_s(y)\), if \(a< \frac{1}{4(t-s)}\), then
Proof
We choose a constant \(\sigma >0\) such that \(a < \frac{1}{4(1+\sigma )(t-s)} \). It follows from Theorem 14 that for any integer \(k \ge 2\),
Hence
where we have used Lemma 2. Since \(a< \frac{1}{4(1+\sigma )(t-s)}\), it is easy to show that the last sum is finite. \(\square \)
7 Heat kernel estimates
We first prove a pointwise upper bound for the heat kernel H. The idea of the proof is from [21, Chapter 2], see also [61].
Theorem 15
(Ultracontractivity) For any Ricci shrinker \((M^n,g,f)\),
Proof
We fix \(x \in M\) and two constants \(s<T<1\). For notational simplicity, we assume that \(\tau =T-t\) and \(\partial _{\tau }=-\partial _t\). We also fix a function \(p(\tau )=\frac{T-s}{T-s-\tau }\) for \(\tau \in [0,T-s)\). For any nonnegative smooth function h with compact support we define
then \(\square ^* w=0\).
Now we compute,
If we multiply both sides above by \(p^2({\tau })||w\phi ^r||^{p({\tau })}_{p({\tau })}\) and use the fact
then we have
where
Now we divide both sides of (179) by \(||w\phi ^r||_{p({\tau })}\), then
where
We denote \(v=(w\phi ^r)^{\frac{p({\tau })}{2}}/||(w\phi ^r)^{\frac{p({\tau })}{2}}||_2\) so that \(||v||_2=1\). Now by direct computations,
So (180) becomes
where
Now we obtain
Since \(\frac{p({\tau })-1}{p'({\tau })}=\frac{\tau (T-s-\tau )}{T-s}>0\), we have from (181)
Now we divide both sides by \(p^2(\tau )\), we have
where
Now we integrate both sides of (183) and estimate the two terms of right side separately.
For a number \(L < T-s\), we integrate (183) from 0 to L so that
By direct computations,
Now we consider the term \(U(\tau )\).
Since we construct w through a smooth function with compact support,
for a constant C uniformly on \(M \times [T-s-L,T-s]\). On the other hand, by Lemma 11\(\sqrt{w} \in W_{*}^{1,2}\) for any \(\tau >0\), in particular
Now the second term in (186) can be estimated as
For any fixed L, it is easy to say \(U(\tau )\) is uniformly bounded for any \(\tau \in [T-s-L,T-s]\) and \(r \ge 1\). By taking \(r \rightarrow \infty \) in (184), from the dominated convergence theorem,
Now by taking \(L \rightarrow T-s\) we have
Therefore,
Since h(x) can be any smooth function with compact support, we derive that
\(\square \)
Now we derive the lower bound of H. Recall that the reduced distance between (x, t) and (y, s) are defined as
where
Now we have the following important estimate, see Corollary 9.5 of [46]. The proof is motivated by [16, Proposition 1].
Theorem 16
For any Ricci shrinker \((M^n,g,f)\),
Proof
We set
It follows from the definition of \(l_{(x,t)}(y,s)\), see [46] and [56], that
and
For any \(x,y \in M\), \(s<T\) and small \(\epsilon >0\) we have
Here and after we omit all z, t for notational simplicity.
By the integration by parts, we have
Therefore,
Now we multiply both sides of \(\square H=0\) by \((\phi ^r)^2H\) and do the integration.
It is immediate by taking \(r \rightarrow \infty \) that
For fixed \(\epsilon \), we have
For the first term,
since L is uniformly bounded on \(M \times [s+\epsilon ,T-\epsilon ]\) and H is integrable.
For the second term,
Now we claim
Indeed, it follows from [58, Eq. (3.3)] that for any \(t \in [s+\epsilon ,T-\epsilon ]\),
and hence
where
Therefore, it is clear from Lemmas 1 and 2 that the claim (197) holds.
It is immediate from (195) that
Now it follows from (194) that by taking \(r \rightarrow \infty \),
As \(\epsilon \rightarrow 0\), both \(H(z,T-\epsilon ,y,s)\) and \(L(x,T,z,s+\epsilon )\) are uniformly bounded (in terms of z). We conclude from the definition of \(\delta \) function that by taking \(\epsilon \rightarrow 0\)
\(\square \)
We also need the following gradient estimate from [60].
Lemma 18
For any Ricci shrinker \((M^n,g,f)\), suppose u is a positive bounded solution of the heat equation \(\square u=0\) on \(M \times [0,T]\), then
where \(\varLambda =\max _{M \times [0,T]}u\).
Proof
From a direction computation
Now the theorem follows from Theorem 6 if
Notice that this follows the same proof as Lemma 9. \(\square \)
Now we have the following corollary of Lemma 18, see [60, Eq. (3.44)].
Corollary 6
With the same conditions as Lemma 18, for any \(\sigma >0\),
Proof
We rewrite Lemma 18 as
and hence
By squaring both sides above, we have
Then the conclusion follows immediately. \(\square \)
We now prove the pointwise lower bound of the heat kernel H.
Theorem 17
For any Ricci shrinker \((M^n,g,f)\), \(0<\delta <1\), \(D>1\) and \(0<\epsilon <4\), there exists a constant \(C=C(n,\delta ,D)>0\) such that
for any \(t \in [-\delta ^{-1},1-\delta ]\) and \(d_t(p,y)+\sqrt{t-s}\le D\).
Proof
From Theorem 16,
By the definition of l and \(\partial _z f(y,z)=|\nabla f|^2 \ge 0\),
and hence
for some constant \(C=C(n,\delta ,D)>0\).
By using (200) for the heat kernel on \(M \times [\frac{t+s}{2},t]\), we obtain
where we have used the result in Theorem 15 for the upper bound.
Therefore,
The conclusion follows by choosing \(\sigma =4/\epsilon -1\). \(\square \)
Remark 3
From the proof a more precise bound is, for any \(0<\epsilon <4\),
In order to further estimate the upper bound of H, it is crucial to compare distance functions from different time slices. We first prove the second order estimate of the heat equation soluton on Ricci shrinkers, see [3, Lemma 3.1].
Lemma 19
Let \((M^n,g(t)),\, t\in [0,1)\) be the Ricci flow solution of a Ricci shrinker and let u be a postive solution to the heat equation \(\square u=0\) and \(u \le \varLambda \) on \(M \times [0,T]\). Then there exists a constant \(C =C(n)\) such that
Proof
By rescaling, we assume that \(\varLambda =1\). Let \(L_1=-\varDelta u+\frac{|\nabla u|^2}{u}-R\), then it follows from [3, Eqs. (3.3), (3.4)] that
From (205) we have
Now at the maximum point of \(L_1\phi ^r\), we have
so we obtain
By taking \(r \rightarrow \infty \), we have \(L_1 \le C(n)t^{-1}\). Now if we set \(L_2=\varDelta u+\frac{|\nabla u|^2}{u}-R\), then similarly
Therefore by the same method, we prove that \(L_2 \le C(n)t^{-1}\).
Now the proof is complete. \(\square \)
By applying the above lemma to the heat kernel, we immediately have from Theorem 15 that
Lemma 20
For any Ricci shrinker \((M^n,g,f)\), there exists a constant \(C =C(n)\) such that
for any \(s<t<1\).
Now we can prove the local distance distorsion on Ricci shrinkers. Notice that a similar estimate has been obtained on compact manifolds, see [3, Theorem 1.1].
Theorem 18
(Local distance distorsion estimate) For any Ricci shrinker \((M^n,p,g,f) \in \mathcal M_n(A)\), \(0<\delta <1\) and \(D>1\), there exists a constant \(Y=Y(n,A,\delta ,D)>1\) such that for any two points q and z in M with \(d_t(p,q) \le D\) and \(d_t(q,z)=r \le D\),
for any \(t \in [-\delta ^{-1},1-\delta -r^2]\) and \(s \in [t-Y^{-1} r^2,t+Y^{-1}r^2]\).
Proof
In the proof, all constants \(C_i\) and \(c_i\) depend on \(n,A,\delta \) and D. Fix a time \(T \in [-\delta ^{-1},1-\delta -r^2]\), a point q with \(d_T(p,q) \le D\) and \(r \le D\), we set \(w(x,t)=H(x,t,q,T-r^2)\). It follows from Theorem 17 that \(w(y,T) \ge C_1 r^{-n}\) for any y with \(d_T(q,y) \le r\). For any \(y \in B_T(q,r)\), we have from Lemma 20 that
for \(t \in [T-r^2/2,T+r^2]\). Since \(d_T(p,y) \le d_T(p,q)+d_T(q,y) \le 2D\), it is clear from Lemma 1 that \(F(y,T) \le c_1\). Moreover, it follows from (22) and (24) that
Therefore, it is clear that for any \(t \in [T-r^2/2,T+r^2]\), \(F(y,t) \le c_3\) and hence \(R(y,t) \le c_4\) from (24). Since \(r \le D\), we have from (211)
Now we set \(c_5=C_1(2C_3)^{-1}\), it follows from \(w(q,T) \ge C_1r^{-n}\) and (212) that \(w(y,t) \ge \frac{C_1}{2}r^{-n}\) on \(B_T(q,r) \times [T-c_5r^2,T+c_5r^2]\). On the one hand, it follows from Corollary 6 that \(w \ge C_4r^{-n}\) on \(B_t(y,r) \times \{t\}\). On the other hand, by Lemma 1, F and hence R is bounded on \(B_t(y,r) \times \{t\}\), we conclude from Theorem 23 that
For any point z with \(d_T(q,z)=r\), we consider a geodesic \(\gamma \) connecting q and z. We claim that for any \(t \in [T-c_5r^2,T+c_5r^2]\), \(d_t(q,z) \le C_6 r\), where \(C_6=8(C_4C_5)^{-1}\). Otherwise, we take a maximal set \(\{y_i\}_{i=1}^N \subset \gamma \) such that \(B_t(y_i,r)\) are mutually disjoint. In particular, it implies that \(\{B_t(y_i,2r)\}\) covers \(\gamma \). Then it is easy to see \(C_6 r \le 4Nr\) and hence \(N \ge \frac{C_6}{4}\). However, it follows from (57) that
which is a contradiction. Now we set \(c_6=c_5(2C_6)^{-2}\) and claim that \(d_t(y,z) \ge (2C_6)^{-1}r\) for any \(t \in [T-c_6r^2,T+c_6r^2]\). Otherwise, we can find a time \(t \in [T-c_6r^2,T+c_6r^2]\) such that \(d_t(y,z)=(2C_6)^{-1}r\). Since \(c_6r^2 = c_5(2C_6)^{-2}r^2\), the argument before shows that \(r=d_T(q,z) \le C_6 d_t(q,z)=r/2\) and this is impossible.
Therefore, by choosing \(Y=\max \{c_6^{-1},2C_6\}\), the conclusion follows. \(\square \)
Now we prove that H has the exponential decay in the integral sense.
Theorem 19
For any Ricci shrinker \((M^n,p,g,f)\in \mathcal M_n(A)\), \(0<\delta <1\), \(D>1\) and \(\epsilon >0\), there exists a constant \(C=C(n,A,\delta ,D,\epsilon )>1\) such that
for any point \(x \in M\), \(t \in [-\delta ^{-1},1-\delta ]\), \(d_t(p,x)+\sqrt{t-s}\le D\) and \(r \ge 1\).
Proof
It follows from Theorem (14) with \(\sigma =\epsilon \) that
for any \(r \ge 1\). So we only need to prove the first integral to be bounded below.
Theorem 18 shows that there exists a constant \(Y=Y(n,A,\delta ,D)>1\) such that for any y with \(d_s(x,y) \le \sqrt{t-s}\), we have \(d_t(x,y) \le Y\sqrt{t-s}\). Therefore, it follows from Theorem 17 that
for any y with \(d_s(x,y) \le \sqrt{t-s}\).
It implies that
where we have used the fact that R is locally bounded. \(\square \)
As we have proved that all distance functions to the base point p are comparable, we prove the following weaker upper bound.
Theorem 20
For any Ricci shrinker \((M^n,p,g,f)\in \mathcal M_n(A)\), \(x \in M\) and \(s<t<1\), there exist constants \(C=C(n,A,x,t,s)>1\) and \(c=c(n,A,x,t,s)>0\) such that
Proof
Fix \(s<t<1\) and x and we require that all constants in the proof depend on n, x, s, t and A. Notice that since s and t are fixed, f is comparable to \(d_0^2(p,\cdot )\) by Lemma 1.
For an \(\epsilon >0\) to be chosen later, we have from the semigroup property
where \(l=\frac{s+t}{2}\).
Now from Theorem 19
Note that here we can always assume that \(\epsilon d_0(p,y)\) is large.
We choose \(\phi \) which is identical 1 on \(B_l(p,c_2\epsilon d_0(p,y))\) and supported on \(B_l(p,2c_2\epsilon d_0(p,y))\) where we choose \(c_2\) that \(B_0(p,\epsilon d_0(p,y)) \subset B_l(p,c_2\epsilon d_0(p,y))\).
If we set \(w=\frac{e^{-f}}{(4\pi \tau )^{n/2}}\), there are \(c_3\) and \(c_4\) that for any \(z \in M\)
Now, we have
where \(c_6=c_5c_3\) and the last inequality follows from Lemma 8. Indeed, if we consider \(m(u,s):=\int H(z,l,u,s) \phi (z)\,dV_l(z)\), then it follows from (215) and Lemma 8 that
for any \(u \in M\) and \(s \le l\). In particular, (216) holds if \(u=y\).
By the definition of w,
Hence,
If we choose \(\epsilon =\sqrt{\frac{c_8}{2c_4}}\), it follows from (214) and (217) that
\(\square \)
8 Differential Harnack inequality on Ricci shrinkers
In this subsection, we prove that Perelman’s differential Harnack inequality holds on Ricci shrinkers.
For any Ricci shrinker \((M^n,g,f)\), we fix a point \(q \in M\) and a time \(T <1\). Moreover, we set
and \(\tau =T-t\).
We first prove
Lemma 21
For any r such that \(\phi ^r=1\) on an open neighborhood of (q, T),
Proof
We set \(K_r=\text {supp}\, \phi ^r \bigcap M \times [T-1,T]\). Since we only care about the integral on the compact set \(K_r\) when t is sufficiently close to T, we can assume that the distances on different time slices from t to T are uniformly comparable. Now all constants C’s in the rest of the proof depend on \(q,T,\varvec{\mu }\) and the geometry on \(K_r\). In particular, they are independent of \(\tau \).
Now we have from Theorem 19 that
Moreover, from Theorem 17,
for \((x,t) \in K_r\).
Now we set \(d_t=d_t(q,x)\), then for any \(A \ge 1\), we have
Now we have
Therefore, we conclude that
where \(\eta (A) \rightarrow 0\) if \(A \rightarrow +\infty \).
In addition, it follows from Theorem 15 that \( b(x,t) \ge \varvec{\mu }\) and hence
where the last inequality is from (220).
The inequalities (222) and (223) indicates that the integral \(\int bw\phi ^r\,dV_t\) is concentrated on the scale \(\sqrt{\tau }\).
We take a sequence \(\tau _i \rightarrow 0\) and set \(g_i(t)=\tau _i^{-1}g(T-\tau _it)\) and \(w_i(\cdot ,t)=\tau _i^{n/2}w(\cdot ,T-\tau _it)\). Then we have
where \(\varDelta _i\) and \(R_i\) are with respect to \(g_i\).
Since \(g_i\) is a blow-up sequence for the metric g and \(K_r\) has bounded geometry, it is easy to show that \((M,g_i,q)\) subconverges to \((\mathbb {R}^n,g_E,0)\) and \(w_i\) converges a positive smooth function \(w_{\infty }\) on \(\mathbb {R}^n \times (0,\infty )\) such that
Now we can show as (55) that \(w_{\infty }\) is in fact a fundamental solution of the heat equation on the Euclidean space. Moreover it is easy to see by Fatou’s inequality that
for any time \(t>1\). Now it follows from [22, Corollary 9.6] that \(w_{\infty }\) is the heat kernel based at 0, that is,
From the smooth convergence,
By direct computations,
Therefore, it is straightforward from (222), (223) and the fact that \(\phi ^r\) is equal to 1 on a neighborhood of (q, T) that
\(\square \)
Remark 4
The same proof of Lemma 21 shows that if u is a bounded smooth function on \(M \times [T-1,T]\), then
Now we set \(d=d_T(q,\cdot )\), it follows from (203) that
In terms of b, we have
We denote \(K^r_t=\{r \le F(\cdot , t) \le 2r\}\), then we have
Lemma 22
There exist constants \(C_0\) and \(C_1\) which depend only on \(\varvec{\mu },q\) and T such that
for any \(r \ge C_1\).
Proof
From Lemma 1, there exists \(C_1>0\) such that for any \(x \in K_t^r\) where \(t \in [T-1,T]\),
if \(r \ge C_1\).
It follows from (227) that \(|b|\le -\varvec{\mu }+c_1\frac{d^2}{\tau }+c_2\). So we only need to estimate
Now it follows from the definition of \(\phi ^r\) that \(K_t^r \subset \{c_4r \le d^2 \le c_5r\}\) if \(C_1\) is sufficiently large, therefore
Note that here we have used (220). \(\square \)
Now we have the following spacetime integral estimate.
Lemma 23
where C depends only on \(\varvec{\mu },n,q\) and T.
Proof
From the evolution equation
we immediately have
From an elementary computation,
where we have used \(\nabla w=-w\nabla b\).
On the one hand we have,
On the other hand
Now (234) becomes
where
Integrate (237) from \(T-1\) to \(T-\epsilon \), we have
where
At the time \(T-1\), since \(b=-\log w-\frac{n}{2}\log {4\pi }\), we have
where the last inequality can be seen from Theorem 20.
Moverover,
Now it follows from Theorem 20 and Lemma 2 that
So if we let \(r \rightarrow \infty \) in (238), the proof is complete. \(\square \)
From Lemma 23, we have
Lemma 24
There exist a sequence \(\tau _i \rightarrow 0\) and a constant \(C>0\) such that
Proof
If the conclusion does not hold, we can find a function \(C(\tau )\) such that \(\lim _{\tau \rightarrow 0}C(\tau )=+\infty \) and
But it obviously contradicts Lemma 23 if \(\epsilon \) is sufficiently small. \(\square \)
Lemma 25
For any \(\theta >0\),
Proof
It follows from Lemma 23 that
\(\square \)
Now we fix a nonnegative function u on the time slice \(T-1\) such that \(\sqrt{u}\) smooth and compactly supported. We denote by the same u as its heat equation solution.
Then we have
Lemma 26
There exists a constant \(C>0\) such that
on \(M \times [T-1,T]\).
Proof
The conclusion follows directly from
and Theorem 6. Note that the assumption in Theorem 6 can be checked similarly as Lemma 9\(\square \)
We also need the following lemma, whose proof is similar to Lemma 4.
Lemma 27
There exists a constant \(C>0\) such that
Proof
From the evolution equation \(\square |\nabla F|^2=-2|\text {Hess}\,F|^2\), we have
Integrate above from \(T-1\) to T, we get
From (37) and (40), there exists a constant C independent of r such that
Therefore,
Now the lemma follows by taking \(r \rightarrow \infty \). \(\square \)
With the same proof, we have
Lemma 28
There exists a constant \(C>0\) such that
As before, we set
and therefore
Now we prove
Lemma 29
There exist a sequence \(\tau _i \rightarrow 0\) and a constant \(C>0\) independent of r and i such that
Proof
From integration by parts, we have
In addition,
Now the conclusion follows immediately from Lemmas 21, 24 and 26. \(\square \)
We are now ready to estimate the squared term in (246).
Lemma 30
Proof
We denote \(A= 2\tau \left| Rc+\text {Hess}\,b-\frac{g}{2\tau }\right| ^2w\). By computations,
Now we have
For the last integral,
and
By the explicit expression \(\square \phi ^r=-nr^{-1}\eta '/2-r^{-2}\eta ''|\nabla F|^2\), we have
In addition,
To estimate the last two terms, since \(|\nabla u|\) is uniformly bounded,
Note that we have
and
Now we integrate (247) from \(T-1\) to \(T-\tau _i\),
Therefore,
For a fixed i, from Theorem 20, Lemmas 4, 25, 26 and 27, we have by taking \(r \rightarrow \infty \) that
where the last inequality follows from Lemma 29.
Now the lemma follows from (249) by taking \(i \rightarrow \infty \). \(\square \)
A consequence of Lemma 30 is
Lemma 31
There exists a sequence \(\tau _j \rightarrow 0\) such that
Proof
It follows from Lemmas 30 and 25 that
Now the conclusion is obvious. \(\square \)
Note that the sequence \(\tau _j\) may not be the same sequence \(\tau _i\) in Lemma 24.
Finally, we can prove Perelman’s differential Harnack inequality.
Theorem 21
Proof
As \(T-1\) can be any time \(S<T\), we just need to prove \(v \le 0\) on \(T-1\).
For the chosen \(\tau _j\) obtained in Lemma 31, we have
On the one hand,
On the other hand,
In addition, it follows from Lemma 21 and Remark 4 that
Combining (251), (252) and (253), it follows immediately from Lemmas 31 and 21 that
Now we consider (248), with \(\tau _i\) replaced by \(\tau _j\), and let \(j \rightarrow \infty \).
It is easy to see all integrals above converge to zero if \(r \rightarrow \infty \), by Lemmas 22, 26, 27 and 28. Therefore,
By the arbitrary choice of u at \(T-1\), we have proved that \(v \le 0\). \(\square \)
Remark 5
Note that as in Perelman’s paper [46], Theorem 16 is a corollary of Theorem 21. Our proof of Theorem 21 is different from most literature, for instance [11, 44], in that we do not need a pointwise gradient estimate of the conjugate heat kernel, see [44, Lemma 2.2].
Remark 6
The proof of Theorem 21 shows the following identity. For any \(S<T<1\),
9 The no-local-collapsing theorems
We need to use the local entropy in [53]. Let us first recall some notations. Let \(\varOmega \) be a domain in M. Then we define (cf. (91) and (92) and Sect. 2 of [53]):
When the meaning is clear in the context, the metric g may be dropped. Note that if \(\varOmega \) does not appear, it means the default set is M. We shall exploit the argument in [53] to obtain volume ratio estimate.
Theorem 22
Suppose \((M^n, g, f)\) is a Ricci shrinker and \(B=B(x,r) \subset M\) is a geodesic ball with \(R \le \varLambda \), then we have
for some \(c=c(n)>0\). If \(r \in (0, 1)\), then (256) can be improved to
Proof
We first show (256). By Theorem 3.3 of [53], we know that
where \(\varvec{\nu }(B,r^2)\) is the local \(\varvec{\nu }\)-functional of B on the scale \(r^2\). Since (M, g) is a Ricci shrinker, it follows from (6) in Theorem 1 that
If \(r \in (0, 1)\), then \(r^2 \in (0, 1)\). By the monotonicity in Theorem 1, the above inequality can be written as
Therefore, we obtain (256) and (257), after we plugging (259) and (260) into (258) respectively. \(\square \)
Theorem 23
Suppose \((M^n, g, f)\) is a Ricci shrinker and \(B=B(q,r) \subset M\) is a geodesic ball with \(R \le \varLambda \), then we have
Proof
Choose \(\rho _0 \in [0,r]\) such that \(\displaystyle \inf _{s \in [0, r]}s^{-n}|B(q,s)|\) is achieved at \(\rho _0\). There are two cases \(\rho _0=0\) and \(\rho _0>0\), which we shall discuss separately.
Case 1 \(\rho _0=0\).
In this case, we have
where \(\omega _n\) is the volume of the unit Euclidean ball. Actually, it is not hard to observe that
Let \(\tau \rightarrow 0^{+}\), it is clear that \((M^n,p,\tau ^{-1}g)\) converges to \((\mathbb {R}^n,0,g_E)\) in the Cheeger–Gromov sense. By Lemma 3.2 of [36], we have
As \(\varvec{\mu }(g,\tau )\) is decreasing on (0, 1) by Lemma 15, then (263) follows from the above inequality. Consequently, (261) follows from the combination of (262) and (263).
Case 2 \(\rho _0>0\).
We choose a nonincreasing smooth function \(\eta \) on \(\mathbb {R}\) such that \(\eta =1\) on \((-\infty ,1/2]\) and 0 on \([1,\infty )\). We also define \(u(x)=\eta (\frac{d(q,x)}{\rho _0})\). From (156) in Corollary 4, we obtain
where the last inequality follows from \(R \le \varLambda \le \varLambda r^2 \rho _0^{-2}\). According to the choice of \(\rho _0\), we obtain
Combining the previous two steps yields that
Recall that \(r^{-n}|B(q,r)| \ge \rho _0^{-n} |B(q,\rho _0)|\) by our choice of \(\rho _0\). Therefore, (261) follows directly from the above inequality. \(\square \)
Remark 7
Theorem 23 indicates that any Ricci shrinker is \(\kappa \)-noncollapsed for some positive constant \(\kappa \) which depends only on the dimension n and the lower bound of \(\varvec{\mu }\).
Note that Theorem 22 is based on the Logarithmic Sobolev inequality, and Theorem 23 relies on the Sobolev inequality. Each of Theorems 22 and 23 has its own advantage and will be used in the remainder of the section. Bascially, Theorem 22 is sharper when r is very small and Theorem 23 is more accurate in the situation when \(\varLambda r^2\) is large.
Using the Sobolev constant estimate in Corollary 4, we can further improve Theorem 6.1 of [41] stating that for any noncompact Ricci shrinker, the volume increases at least linearly.
Proposition 6
For any noncompact Ricci shrinker \((M^n,p,g,f)\), there exist big positive constant \(r_0=r_0(n)\) and small positive constant \(\epsilon _0=\epsilon _0(n)\) such that
Proof
Similar to the proof of Lemma 2, we follow the notation of [41] to denote
From Lemma 1, V(r) is almost the volume of geodesic ball B(p, r), with the advantage that the estimate of V(r) is relatively easier than the estimate of |B(p, r)|. Actually, by Eqs. (6.24) and (6.25) of [41], we know that
whenever \(t \ge C_1\) for some dimensional constant \(C_1=C_1(n)\). Now we define
Therefore, in order to prove (265), it suffices to show that
where \(\epsilon _0=\epsilon _0(n)\) will be determined later.
We shall prove (269) by a contradiction argument. If (269) were wrong, then there exists an \(r \ge 2r_0\) such that \(V(r) \le \epsilon _0 e^{\varvec{\mu }}r\) for \(\epsilon _0\) to be determined later, we claim that
Indeed, by our assumption the case \(m=0\) is true. We assume that the conclusion is true for all \(m=0,1,2,\ldots ,k\) and proceed to show it holds for \(m=k+1\).
For any \(t \ge r_0\), we define
Let \(t=t_m\) and plug the above u into the Sobolev inequality (156). We obtain
for some \(C_3=C_3(n)\). For \(0\le m\le k\), it follows from our induction assumption and (267) that
Summing (271) from \(m=0\) to \(m=k\), we have
Recall that \(\chi (t) \le \frac{n}{2}V(t)\) by (3.4) of [9]. Plugging this fact into the above inequality yields that
where \(C_4=(6+3n)C_3=C_4(n)\). Now we choose
Clearly, \(\epsilon _0=\epsilon _0(n)\). Then it follows from (272) that
It is clear from (273) that
and hence
Therefore, the induction is complete and (270) is proved. By the arbitrary choice of m, the total volume of the Ricci shrinker is finite, which contradicts Lemma 6.2 of [41](See also Theorem 3.1 of [7] by Cao–Zhu). Therefore, the proof of (269) is established by this contradiction. Consequently, (265) holds by Lemma 1. Note that \(r_0\) and \(\epsilon _0\) are defined in (268) and (274). Both of them can be calculated explicitly. \(\square \)
Remark 8
In [41, Theorem 6.1], the authors have obtained a weaker lower bound
for two constants \(C>0\) and \(c>1\) depending only on n.
We are now ready to prove the improved no-local-collapsing, i.e., Theorem 2.
Proof of Theorem 2
It follows from Lemma 2 and Proposition 6 that
By Lemma 12, we know \(e^{\varvec{\mu }}|B(p,1)|^{-1}\) is uniformly bounded from above and from below. Multiplying each term of the above inequality by \(e^{\varvec{\mu }}|B(p,1)|^{-1}\) and adjusting C if necessary, we arrive at
which is nothing but (9a). We proceed to prove (9b). Recall that \(q \in \partial B(p,r)\) and \(\rho \in (0, r^{-1})\) for some \(r>1\). Triangle inequality implies that
It follows from Lemma 1 that \(f \le C r^2\) for some \(C=C(n)\) on B(p, 2r). Since \(R+|\nabla f|^2=f\) and \(R \ge 0\), it follows that \(R \le Cr^2\) on B(p, 2r). In particular, we have \(R\rho ^2 \le Rr^{-2} \le C(n)\) on \(B(q,\rho )\). Consequently, we can apply Theorem 23 on the ball \(B(q,\rho )\) to obtain (9b). \(\square \)
10 The pseudolocality theorems
In this section, we prove the pseudo-locality theorems on Ricci shrinker and discuss their applications.
Based on the Harnack estimate, following a classical point-picking, or maximum principle argument, we are able to obtain the following pseudo-locality theorem.
Theorem 24
There exist positive numbers \(\epsilon _0=\epsilon _0(n)\) and \(\delta _0=\delta _0(n)\) with the following properties.
Let \(\{(M^n, g(t)), -\infty< t < 1\}\) be the Ricci flow induced from a Ricci shrinker \((M^n, p, g)\). Suppose \(t_0 \in (-\infty , 1)\) and \(B_{g(t_0)}(x,r) \subset M\) is a geodesic ball satisfying
Then for each \(t \in (t_0, \min \{t_0+\epsilon _0^2 r^2, 1\})\) and \(y \in B_{g(t)}(x, 0.5r)\), we have
The statement in Theorem 24 is a slight improvement of Theorem 10.1 of [46]. The basic idea of the proof is already contained in Propositions 3.1 and 3.2 of Tian–Wang [51]. Note that the isoperimetric constant estimate in Peleman’s statement is only used to (cf. Lemma 3.5 of [53]) estimate the local entropy (i.e., (254) and (255)) \(\varvec{\nu } (B_{g(t_0)}(x,r), g(t_0), r^2)\). The statement (275) seems to be more straightforward. The conclusion (276) follows from a standard point-picking argument, whenever the differential Harnack estimate, i.e., Theorem 21 holds. More details can be found in [32, Sect. 30], [11, Sect. 8], [19, Chapter 21], or [54].
As Ricci shrinker Ricci flows are self-similar, we can improve the estimate (276) by the following property.
Theorem 25
Suppose \((M^n, p, g, f)\) is a Ricci shrinker, \(B=B(q,r) \subset M\) is a geodesic ball satisfying
Then we have
where \(D=d(p,q)+\sqrt{2n}\).
Proof of Theorem 25
We fix \(\xi \le \epsilon _0\) a small positive number, whose value will be determined later (i.e., (281)). We set
where \(\psi ^{s}\) is the diffeormorphism (i.e., (15)) generated by \(\frac{\nabla f}{1-s}\).
Claim
By choosing \(\xi \) properly, we have
By (15) and (2), along the flow line \(\psi ^s(\tilde{q})\) where s goes from t to 0, we compute
From the definition of \(\psi ^s\), we have
For each \(s \ge t=-(\xi r)^2\), the integration of the above inequality yields that
where we applied (30) in the last step. Therefore, it follows from (279) that
Plugging the fact that \(t=-(\xi r)^2\) into the above inequality, we arrive at
Now we define \(\xi \) as follows.
Therefore, if \(Dr \le \epsilon _0^{-1}\), it follows from (280) that
If \(Dr >\epsilon _0^{-1}\), it also follows from (280) that
Therefore, no matter what the value of r is, we always have (278). The proof of the Claim is complete.
We proceed to prove (276). Since \(g(t)=(1-t)(\psi ^t)^*g\), it is clear that
It follows from the scaling property of \(\varvec{\nu }\) that
Therefore, we can apply Theorem 24. For each \(s \in (t, \min \{t+(\epsilon _0r)^2, 1\}]\) and \(x \in B_{g(s)}(\tilde{q}, 0.5 r)\), we have
In particular, we can choose \(s=0\). Since \(g=g(0)\), for each \(x \in B(\tilde{q}, 0.5r)\), we obtain
Note that \(B(q, 0.5 \epsilon _0 r) \in B(\tilde{q}, \epsilon _0 r) \subset B(\tilde{q}, 0.5r)\) by (278). Plugging (281) into (285a), we obtain (276). \(\square \)
Now we apply Theorems 24 and 25 to study the geometric properties of (M, g) in terms of \(\varvec{\mu }\). In particular, we are ready to finish the proof of Theorem 3.
Proof of Theorem 3
We divide the proof into several steps.
Step 1 The gap property (10) holds.
It suffices to show that \(\varvec{\mu } \ge -\delta _0\) implies that (M, g) is isometric to the Euclidean space.
Following directly from its definition, as \(B(x,r) \subset M\), it is clear that
Combining the above inequality with the optimal Logarithmic Sobolev inequality, we obtain
Therefore, if \(\varvec{\mu } \ge -\delta _0\), then each ball B(x, r) will satisfy the condition (275). By choosing \(r>>D\), we can apply (276) to obtain that
Let \(r \rightarrow \infty \), we obtain that \(|Rm|(x) \equiv 0\). By the arbitrary choice of x, we obtain that \(|Rm| \equiv 0\). In particular, \(Rc \equiv 0\). Then the Ricci shrinker equation implies that \(f_{ij}=\frac{g_{ij}}{2}\). Therefore, (M, g) is isometric to a metric cone which is also a smooth manifold. This forces that (M, g) is isometric to the standard Euclidean space \((\mathbb {R}^n, g_{E})\). Thus, the proof of (10) is complete.
Step 2 The inequality (12) and (13) imply the curvature and injectivity radius bound (14).
Recall that (10) means \(\varvec{\mu }(g, 1)<-\delta _0\). If (12) holds, by continuity and monotonicity of \(\varvec{\mu }(g, \tau )\), it is clear that there exists some \(\tau \in (0,1)\) such that
Then the \(\tau _0\) in (14) is well defined. Namely, \(\tau _0\) is the largest \(\tau \in (0, 1)\) such that the above equality holds. It follows from the definition of \(\tau _0\) and \(\varvec{\nu }\) that
For each ball \(B_{g(0)}(x,r) \subset M\), we know \( \varvec{\nu }(B_{g(0)}(x,r), g, \tau _0) \ge -\delta _0\). In particular, we can choose \(r=\sqrt{\tau _0}\). Now we apply Theorem 24 on the time slice \(t_0=0\), with scale \(\sqrt{\tau _0}\), to obtain that
In particular, we have
Up to rescaling, since \(g(0)=g\), we arrive atl
which is nothing but (14a). Plugging (287) into (257) of Theorem 22, we obtain that each geodesic ball \(B(\cdot , \sqrt{\tau _0})\) has volume bounded below by \(c(n) \tau _0^{\frac{n}{2}}\). Therefore, the injectivity radius estimate of Cheeger–Gromov–Taylor [13] applies and we arrive at (14b). The proof of (14) is complete.
Step 3 The bounded geometry estimate (14) implies the equality (11), i.e., \(\displaystyle \lim _{\tau \rightarrow 0^+}\varvec{\mu }(g,\tau ) =0\).
We shall argue in the way similar to that in Theorem 1.1 of [62], with more details on the regularity estimate.
Assume otherwise that there exists a sequence \(\tau _i \rightarrow 0^+\) such that
If we set \(g_i=\tau _i^{-1}g\), then all metrics \(g_i\) have uniformly bounded geometry. More precisely, there exist positive constants K and \(v_0\) such that
Notice that for any i, there exists a large domain
for some large \(r_i>>1\) such that
The geometry bound (289) actually implies higher order derivatives of curvatures and \(\sqrt{f}\) are also uniformly bounded (cf. Sect. 4 of [34]). Therefore, it is not hard to see that \(\partial B_i\) is smooth. All the covariant derivatives of second fundamental forms of \(\partial B_i\) are bounded independent of i.
It follows from [48] that a minimizer \(u_i\) of \(\varvec{\mu }(B_i, g_i,1)\) exists. More precisely, \(u_i \in W_0^{1,2}(B_i)\) is a positive smooth function on \(B_i\) satisfying the normalization condition
and solve the Dirichlet problem
Here \(dV_i\), \(\varDelta _i\) and \(R_i\) denote the volume form, Laplacian operator and scalar curvature with respect to \(g_i\) respectively. The number \(\lambda _i\) is defined by
Recall that \(\displaystyle \lim _{\tau \rightarrow 0^{+}} \varvec{\mu }(g,\tau ) \le 0\) by (264). Then it follows from (291) that \(\lambda _i\) is uniformly bounded. Since curvature is uniformly bounded, the classical \(L^2\)-Sobolev constant of \((B_i, g_i)\) is uniformly bounded. In light of (293), the Moser iteration then implies \({ \left\| u_i \right\| }_{C^0}\) is uniformly bounded, see [62, Lemma 2.1(a)] or the proof of Proposition 3.1 of [51]. Then it follows from [23, Corollary 8.36] that \(\Vert u_i\Vert _{C^{1,\frac{1}{2}}(\bar{B}_i)}\) are uniformly bounded. Since all \(\partial B_i\) have uniformly higher regularities, the bootstrapping, see [23, Theorem 6.19], shows that \(\Vert u_i\Vert _{C^{k,\frac{1}{2}}(\bar{B}_i)}\) are uniformly bounded for any \(k \ge 2\).
Let \(q_i\) be a point where \(u_i\) achieves maximum value in \(B_i\). By (293), at \(q_i\) we have
whence we derive
for some uniform constant \(c_0\).
In light of (289) and the discussion below (291), we know that \((M^n, q_i, g_i)\) subconverges to Euclidean space \((\mathbb {R}^n, 0, g_{E})\) in \(C^{\infty }\)-Cheeger–Gromov topology. The set \(B_i\) converges to a limit set \(B_{\infty }\). If \(d(q_i, \partial B_i) \rightarrow \infty \), then \(B_{\infty }=\mathbb {R}^n\). Otherwise, by the estimate of second fundamental form and its covariant derivatives, \(\partial B_i\) converge to a smooth \((n-1)\)-dimensional set \(\partial B_{\infty }\). In light of the uniform bound of \({ \left\| u_i \right\| }_{C^{k,\frac{1}{2}}}\) and the uniform regularity of \(\partial B_i\), by taking subsequence if necessary, we can assume that \(u_i\) converges in smooth topology to a smooth function \(u_{\infty } \in C^{\infty }(\bar{B}_{\infty })\). Furthermore, \(u_{\infty } \equiv 0\) on \(\partial B_{\infty }\).
In view of (294), the convergence process implies that
Furthermore, we have on \(B_{\infty }\) that
where \(\lambda _{\infty }=n+\frac{n}{2}\log (4\pi )+\varvec{\mu }_{\infty }\). Let \(\tilde{u}=c^{-1}u_{\infty }\). Then \(\int _{B_{\infty }} \tilde{u}^2 dV_{\infty }=1\). The above equation becomes
Since \(c \in (0, 1)\) by (295) and \(\varvec{\mu }_{\infty }<0\) by (288), then an integration by parts shows that
which is a contradiction. So we finish the proof of Step 3.
Step 4 The three properties are equivalent.
By Step 2, it is clear that \((c) \Rightarrow (a)\). Then Step 3 means that \((a) \Rightarrow (b)\). It is obvious that \((b) \Rightarrow (c)\). Therefore, we obtained the equivalence of properties (a), (b) and (c) in Theorem 3. The proof of the Theorem is complete. \(\square \)
Corollary 7
There exists a small positive number \(\epsilon =\epsilon (n)>0\) such that for any nonflat Ricci shrinker \((M^n, p, g, f)\), we have
Proof
We argue by contradiction.
If (297) were wrong, then we can have a sequence of nonflat Ricci shrinkers \((M_i, p_i, g_i)\) such that
By Proposition 5.8 of [34], it is clear that \(\varvec{\mu }_i = \varvec{\mu } (M_i, p_i, g_i)\) is uniformly bounded from below. Using Theorem 1.1 of [34], the above convergence can be improved to be in the \(C^{\infty }\)-Cheeger–Gromov sense
It is not hard to see that \(\varvec{\mu }\) is continuous with respect to the above convergence (cf. Theorem 1.2(c) of [34]). Therefore, we have
It follows that \(\varvec{\mu }_i >-\delta _0\) for large i. Therefore, each \((M_i, g_i)\) is isometric to Euclidean space by Theorem 3. This contradicts our choice of \((M_i, g_i)\). The proof of (297) is established by this contradiction. \(\square \)
Corollary 8
Let \((M^n,g,f)\) be a Ricci shrinker and let \(q \in M\) be a point such that
Then there exist a positive constant \(C=C(n)\) such that
for any \(x \in B(q,\frac{1}{2}e^{-CD}D^{-\frac{1}{2}})\) and \(t \in [0,1)\), where \(D=d(p,q)+\sqrt{2n}\).
Proof
By the assumption, it follows from Theorem 24 by choosing \(r=\epsilon _0^{-1}\) that
for any \(t \in (0,1)\) and \(d_{g(t)}(q,x) \le \frac{1}{2}\epsilon _0^{-1}\). In addition, from Theorem 25 we have
for any \(x \in B(q,\frac{1}{2})\). From (298), (299) and [15, Theorem 3.1] that there exist a positive constant \(C=C(n)\) such that for any \(x \in B_t(q,\frac{1}{2}D^{-\frac{1}{2}})\),
From (300), it is easy to see by comparing the distances that
for any \(t \in [0,1)\).
Therefore, for any \(x \in B(q,\frac{1}{2}e^{-CD}D^{-\frac{1}{2}})\),
Along the flow line of \(\psi ^t(x)\),
and hence by solving the corresponding ODE,
Combining (302) and (304), the conclusion follows. \(\square \)
Since f is almost \(\frac{d^2}{4}\) by Lemma 1, Corollary 8 shows that the curvature is quadratically decaying along the flow line. Next we prove that if there exists a tubular neighborhoold of some level set of f whose isoperimetric constant is almost Euclidean, then globally the curvature is quadratically decaying.
Corollary 9
For any Ricc shrinker \((M^n,g,f)\), if there exists an \(a>0\) such that for any \(x \in f^{-1}(a)\),
then the curvature is quadratically decaying and each end has a unique smooth tangent cone at infinity.
Proof
We can assume that (M, f) is nonflat, otherwise there is nothing to prove. Now we reparametrize \(\psi ^t\) by defining for any \(s \in (-\infty ,\infty )\)
It is clear from the definition of \(\psi ^t\) that
In other words, \(\tilde{\psi }^s\) is the one-parameter group of diffeomorphisms generated by \(\nabla f\). Now we set
where \(D_1=2\sqrt{a}+5n+\sqrt{2n}+4\).
We claim that any \(x \in T_{\epsilon _1}(f^{-1}(a)){:}{=}\bigcup _{q\in f^{-1}(a)}B(q,\epsilon _1)\) is not a stationary point of \(\tilde{\psi }^s\). Otherwise, it follows from Corollary 8 that
for any \(s \ge 0\). However, when \(s \rightarrow \infty \), \(|Rm|(x)=0\) and this contradicts our nonflatness assumption.
Now we choose \(c<a<d\) such that for any \(x \in \partial T_{\frac{\epsilon _1}{2}}(f^{-1}(a))\), either \(f(x) \le c\) or \(f(x) \ge d\). By continuity, there exists a positive constant \(\epsilon \ll \epsilon _1\) such that for any \(x \in T_{\epsilon }(f^{-1}(a))\), \(f(x) \in (c+\epsilon ,d-\epsilon )\). We set \(U {:}{=}T_{\epsilon }(f^{-1}(a))\) and claim that for any \(y \in U\), there exists an \(x \in f^{-1}(a)\) such that \(\tilde{\psi }^{s}(x)=y\) for some s. If \(f(y)=a\), then the claim is obvious. If \(f(y) <a\), we consider the flow line \(\tilde{\psi }^{s}(y)\) for \(s \ge 0\). Notice that by the definition of \(\tilde{\psi }^{s}\),
Therefore, by the local compactness and our previous no stationary argument, the flow will continue and along the flow f is strictly increasing as long as \(\tilde{\psi }^{s}(y)\) stays in \(T_{\epsilon _1}(f^{-1}(a))\). We set \(s_0\) to be the first time such that \(\tilde{\psi }^{s}(y)\) reaches \(\partial T_{\frac{\epsilon _1}{2}}(f^{-1}(a))\). In particular, \(f(\tilde{\psi }^{s}(y)) \le c\) or \(f(\tilde{\psi }^{s}(y)) \ge d\). Since \(f(y) \in (c+\epsilon ,d-\epsilon )\), it must be \(f(\tilde{\psi }^{s}(y)) \ge d\). As \(f(y)<a<f(\tilde{\psi }^{s_0}(y))\), there exists an \(s \in (0,s_0)\) such that \(f(\tilde{\psi }^{s}(y))=a\) by continuity. Therefore, if we set \(x=\tilde{\psi }^{s}(y) \in f^{-1}(a)\), then \(\tilde{\psi }^{-s}(x)=y\) and the claim follows. Similarly, for the case \(f(y)>a\), the claim is also true.
Next we prove that for any y such that \(f(y)>a\), there exists an \(x \in U\) such that \(\tilde{\psi }^{s}(x)=y\) for some s. Fix such y, we choose any curve \(\{\gamma (z):\,z\in [0,1]\}\) such that \(\gamma (0)=p\) and \(\gamma (1)=y\). In particular, since p is the minimum point of f, there exists a \(z_0 \in [0,1)\) such that \(\gamma (z_0) \in f^{-1}(a)\) and for all \(z \in (z_0,1]\), \(f(\gamma (z))>a\). Now we define \(I \subset [z_0,1]\) such that \(z \in I\) if and only if there exists an \(x \in U\) such that \(\tilde{\psi }^{s}(x)=\gamma (z)\) for some s. In particular, I is not empty as \(z_0 \in I\). It is clear that I is open, since U is open. Now we prove the closedness of I. For a sequence \(z_i \in I\) such that \(z_i \rightarrow z_{\infty } \in [z_0,1]\), \(f(z_i)>a\) if i is sufficiently large. By our definition of I and the claim with its proof, there exists \(x_i \in f^{-1}(a)\) and \(s_i >0\) such that \(\tilde{\psi }^{s_i}(x_i)=\gamma (z_i)\). Note that \(s_i\) must be bounded. Indeed, by Corollary 8,
If \(s_i \rightarrow \infty \), then it forces \(|Rm|(\gamma (z_{\infty }))=0\) and this is a contradiction. By compactness and taking the subsequence, there exist \(x_{\infty } \in f^{-1}(a)\) and \(s_{\infty } \ge 0\) such that \(x_i \rightarrow x_{\infty }\) and \(s_i \rightarrow s_{\infty }\). By continuity, \(\tilde{\psi }^{s_{\infty }}(x_i)=\gamma (z_{\infty })\). To summarize, \(I=[z_0,1]\) and in particular, \(\tilde{\psi }^{s}(x)=\gamma (1)=y\) for some \(x \in U\) and \(s \in \mathbb {R}\). By the claim again, we have proved that for any y with \(f(y) \ge a\), there exists an \(x \in f^{-1}(a)\) such that \(\psi ^s(x)=y\) for some \(s \ge 0\).
Therefore, for any point y outside the compact set \(\{f \le a\}\), it follows from Corollary 8 that
See Figure 4 for intuition in the case \(a>1\).
The quadratic decay of curvature
In other words, the curvature is quadratically decaying. Since a Ricci shrinker can be regarded as an ancient Ricci flow, it follows from Shi’s local estimates [50] that
for all \(k=1,2,\ldots \). It follows immediately that any tangent cone at infinity must be smooth. Finally, the uniqueness follows from [17, Theorem 2], see also [33, Lemma A.3]. \(\square \)
Remark 9
The proof of Corollary 9 shows that the manifold \(\{x\in M \mid \,f(x) \ge a\}\) is diffeomorphic to \(f^{-1}(a) \times [0,1)\).
11 Strong maximum principle for curvature operator
The purpose of this section is to prove Theorem 4. We remind the readers that all constants C’s in this section depend only on the dimension n.
We first show an \(L^2\)-integral estimate of Riemannian curvature.
Theorem 26
Suppose \((M^n,p,g,f)\) is a Ricci shrinker satisfying \(\varvec{\mu } \ge -A\), and \(\lambda \) is a positive number. Then we have
for some \(I=I(n,A,\lambda )<\infty \).
Theorem 26 is the consequence of the improved no-local-collapsing theorem (i.e., Theorem 2), the local conformal transformation technique (cf. Sect. 3 of [34]), and the curvature estimate of Jiang–Naber (i.e., [31]).
Lemma 32
For any Ricci shrinker \((M^n,p,g,f)\) and any constant \(D >100n\), we have
where A(D, 2D) is the annulus \(B(p,2D) \backslash B(p, D)\).
Proof
Fix a cutoff function \(\psi \) on \(\mathbb {R}\) such that \(\psi =1\) on [1, 2] and \(\psi =0\) outside \([\frac{1}{2},3]\). By defining \(\eta (x)=\psi (\frac{d(p,x)}{D})\), we compute
where for the second line we have used \(\text {div}(Rc\,e^{-f})=0\). Consequently, by Lemmas 1 and 2, we have
Plugging the estimates in Lemmas 1 and 2 into the above inequality, we arrive at (307). \(\square \)
In the proof of Lemma 32, if we choose \(\psi \) such that \(\psi =1\) on \((-\infty ,1]\) and \(\psi =0\) on \([2,\infty )\), then a similar argument shows the following Lemma.
Lemma 33
For any Ricci shrinker \((M^n,p,g,f)\), we have
The details of the proof of Lemma 33 is almost identical to that of Lemma 32. So we leave it to interested readers. Note that Lemma 33 provides an explicit upper bound of [40, Theorem 1.1]. Starting from Lemmas 32 and 33, we are ready to prove Theorem 26.
Proof of Theorem 26:
We only prove the case when \(\lambda =1\). The general case is similar and is left to interested readers.
For any point \(q \in M\) such that \(d(p,q)=D>100n\), we set \(r=\frac{1}{D}\), \(\bar{f}=f-f(q)\), then under the conformal transformation \(\bar{g}{:}{=}e^{-\frac{2\bar{f}}{n-2}}g\), we have
where the proof and the definition of the Kulkarni–Nomizu product 
can be found in [4, Theorem 1.165]. It follows from [34, Lemma 3.5] that
Therefore, by the same proof as in [34, Lemma 3.7], we have
Since \(R \le CD^2\) on B(q, r), it follows from Theorem 23 that \(|B(q,r)| \ge Ce^{\varvec{\mu }}r^n\) and hence
One can also use Theorem 2 to obtain the above estimate directly.
By defining \(\tilde{g}{:}{=}r^{-2}\bar{g}\), we have \(|\widetilde{Rc}|_{\tilde{g}} \le C\) on \(B_{\tilde{g}}(q,e^{\frac{1}{n-2}})\) and \(|B_{\tilde{g}}(q,e^{\frac{1}{n-2}})|_{\tilde{g}} \ge Ce^{\varvec{\mu }}\). By shrinking balls to its half size if necessary, it follows from [31, Theorem 1.6] that
for some constant \(I_0=I_0(n,A)\).
From (310), we have on B(q, r),
Therefore, we have
where we have used Lemma 32 and (314). Consequently, there exists \(I_1=I_1(n,A)\) such that
For any constant \(D >100n\), we apply Vitali’s lemma for the covering \(\{B(q,\frac{1}{4D})\}_{q\in A(D,2D)}\). If we assume that \(\{B(q_i,\frac{1}{4D})\}_{1 \le i \le k}\) is a maximal collection of mutually disjoint sets, then \(\{B(q_i,\frac{1}{2D})\}_{1 \le i \le k}\) cover A(D, 2D). It is clear from definition that
By Lemma 2 and (313), we obtain \(k \le CD^{2n}\). Combining (315) with the above inequality implies that
Similarly, by exploiting Lemma 33, we have
where \(D_0=100n\) and \(I_2=I_2(n,A)\).
Now we set \(D_i=2^i D_0\) and decompose the integral as
Plugging (316) and (317) into the above equation, we arrive at
Since both \(I_1\) and \(I_2\) depend only on n and A, it is clear that I relies only on n and A and we arrive at (306). The proof of Theorem 26 is complete. \(\square \)
From (306) and [40, Theorem 1.2], a direct corollary of Theorem 26 is the following estimate.
Corollary 10
For any Ricci shrinker \((M^n, g, f)\in \mathcal M_n(A)\), there exists a constant \(I=I(n,A)<\infty \) such that
Theorem 26 is an important step for verifying maximum principle on curvature operators. The curvature operator on two-forms are defined as \(\mathcal {R}: \varLambda ^2 \rightarrow \varLambda ^2: \mathcal R(e^i \wedge e^j,e^k \wedge e^l)=R_{ijkl}\). The two-form \(e^i \wedge e^j {:}{=}e^i \otimes e^j-e^j\otimes e^i\) and the inner product on \(\varLambda ^2\) is defined as \(\langle A,B\rangle {:}{=}-\frac{1}{2}tr(AB)\) for \(A,B\in \varLambda ^2=\mathfrak {so}(n)\). In other words, for \(w=\frac{1}{2}\sum _{i,j}w_{ij}e^i \wedge e^j\), we have
In the setting of Ricci shrinker \((M^n,g,f)\), the following equation (see [24]) holds:
Here \(Q(\mathcal R) {:}{=}\mathcal R^2+\mathcal R^{\#}\) and \(\mathcal R^{\#}\) is defined as
for any \(u,v \in \varLambda ^2\). If we choose an orthonormal basis \(\{\phi _i\}\) of \(\varLambda ^2\), then
If we assume \(\lambda _1 \le \lambda _2 \le \cdots \) are all eigenvalues of \(\mathcal R\) on \(\varLambda ^2\), then we have the following rigidity theorem.
Theorem 27
There exists a constant \(\epsilon =\epsilon (n)>0\) such that for any Ricci shrinkers \((M^n,g,f)\), if \(\lambda _2 \ge -\epsilon \dfrac{\lambda _1^2}{|R-2\lambda _1|}\), then \(\lambda _1 \ge 0\). Consequently, \((M^n,g)\) is isometric to a quotient of \(N^k \times \mathbb {R}^{n-k}\) for some \(0 \le k \le n\), where \(N^k\) is a closed symmetric space.
Proof
It suffices to prove \(\lambda _1 \ge 0\). Namely, \((M^n,g)\) has nonnegative curvature operator. The further conclusion follows from [42, Corollary 4].
We fix a point q and assume that \(\phi _1\) is an eigenvector of \(\lambda _1\). Extending \(\phi _1\) by parallel transport on a small neighborhood of q, we have
Therefore if we assume that \(\phi _i\) are eigenvectors of \(\lambda _i\), then in the barrier sense,
where \(C_{i,j}=\langle [\phi _i,\phi _j],\phi _1 \rangle \). Notice that \(C_{i,j}=0\) if \(i=1\) or \(j=1\).
We claim that \(|C_{i,j}| \le 2\). Indeed, if we assume that \(\phi _i,\phi _j\) and \(\phi _1\) are represented by the antisymmetric matrices A, B and C respectively, then \(C_{i,j}=-\frac{1}{2}tr((AB-BA)C)=-tr(ABC)\). By choosing a basis such that \(A_{2k-1,2k}=a_k=-A_{2k,2k-1}\) for \(k \le [n/2]\) and 0 otherwise, we have
Here we have used the fact that \(|A|^2=|B|^2=|C|^2=2\).
Next we prove that if \(\epsilon \) is properly chosen, then we have
From the definition of \(\lambda _i\), we notice that \(\sum \lambda _i=R/2\). Therefore, we fix \(\lambda _1\) and \(\lambda _2\) and minimize P under the restriction \(\sum \lambda _i=R/2\). We can assume that \(\lambda _2<0\), otherwise \(P \ge 0\) from its definition. We also set \(c_n=n(n-1)/2\) and assume that \(\lambda _1 \le \lambda _2 \le \cdots \le \lambda _{s+1}\) are all eigenvalues smaller than 0. Therefore,
It is easy to show that \(P_1\) is minimized when \(\lambda _2=\lambda _3=\cdots =\lambda _{s+1}\) and \(\lambda _{s+2}=\cdots =\lambda _{c_n}\). It follows that
By solving the above quadratic inequality, we obtain that \(P_1\) and hence P are nonnegative if
If we choose \(\epsilon =\frac{1}{(1+\sqrt{2})(c_n-2)}\), then it is clear that for any \(1\le s \le c_n-2\),
Therefore, from (318) we obtain \(\varDelta _f \lambda _1 \le \lambda _1\). Since \(\lambda _1 \in L^2(e^{-f}\,dV)\) by (306), then it follows from [47, Theorem 4.4] that \(\lambda _1 \ge 0\). \(\square \)
We conclude this section by the proof of Theorem 4.
Proof of Theorem 4:
Since \(\lambda _2 \ge 0\), we can apply Theorem 27 to obtain \(\lambda _1 \ge 0\). Therefore, \(M^n\) is a finite quotient of \(N^k \times \mathbb {R}^{n-k}\). Note that only the case \(k=n\) is possible. For otherwise the second smallest eigenvalue must be 0. Since \(N^n\) is a compact Einstein manifold such that the curvature operator is 2-positive, it follows from [5] that its universal covering must be \(S^n\). \(\square \)
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Acknowledgements
Yu Li would like to thank Jiyuan Han and Shaosai Huang for helpful comments. Bing Wang would like to thank Haozhao Li and Lu Wang for their interests in this work. Part of this work was done while both authors were visiting IMS (Institute of Mathematical Sciences) at ShanghaiTech University during the summer of 2018. They wish to thank IMS for its hospitality. Last but not least, the authors would like to thank the anonymous referees for several valuable comments that help improve the exposition of the paper.
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Yu Li is partially supported by research fund from SUNY Stony Brook and Bing Wang is partially supported by NSF Grant DMS-1510401 and research funds from USTC and UW-Madison.
