The Logarithmic Sobolev and Sobolev Inequalities Along the Ricci Flow

Abstract

Based on Perelman’s entropy monotonicity, uniform logarithmic Sobolev inequalities along the Ricci flow are derived. Then uniform Sobolev inequalities along the Ricci flow are derived via harmonic analysis of the integral transform of the relevant heat operator. These inequalities are fundamental analytic properties of the Ricci flow. They are also extended to the volume-normalized Ricci flow and the Kähler–Ricci flow.

1 Introduction

Consider a compact manifold $M$ of dimension $n \ge 3$. Let $g=g(t)$ be a smooth solution of the Ricci flow

$$\begin{aligned} \frac{\partial g}{\partial t}=-2\mathrm{{Ric}} \end{aligned}$$

(1.1)

on $M \times [0, T)$ for some (finite or infinite) $T>0$ with a given initial metric $g(0)=g_0$.

Theorem 1.1

For each $\sigma >0$ and each $t \in [0, T),$ there holds

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}}\le & {} \sigma \int _M \left( |\nabla u|^2 +\frac{R}{4} u^2\right) \mathrm{{dvol}} -\frac{n}{2} \ln \sigma \nonumber \\&+A_1\left( t+\frac{\sigma }{4}\right) +A_2 \end{aligned}$$

(1.2)

for all $u\in W^{1,2}(M)$ with $\int _M u^2 \mathrm{{dvol}}=1,$ where

$$\begin{aligned} A_1= & {} \frac{4}{{\tilde{C}}_S(M, g_0)^2 \mathrm{{vol}}_{g_0}(M)^{\frac{2}{n}}}-\min R_{g_0}, \\ A_2= & {} n\ln {\tilde{C}}_S(M,g_0)+ \frac{n}{2}(\ln n-1), \end{aligned}$$

and all geometric quantities are associated with the metric $g(t)$ (e.g., the volume form $\mathrm{{dvol}}$ and the scalar curvature $R$), except the scalar curvature $R_{g_0},$ the modified Sobolev constant ${\tilde{C}}_S(M,g_0)$ (see Sect. 2 for its definition) and the volume $\mathrm{{vol}}_{g_0}(M)$ which are those of the initial metric $g_0$.

Consequently, there holds for each $t \in [0, T)$

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \frac{n}{2} \ln \left[ \alpha _I\left( \int _M \left( |\nabla u|^2 +\frac{R}{4} u^2\right) \mathrm{{dvol}}+\frac{A_1}{4}\right) \right] \end{aligned}$$

(1.3)

for all $u \in W^{1,2}(M)$ with $\int _M u^2 \mathrm{{dvol}}=1,$ where

$$\begin{aligned} \alpha _I=\frac{2e}{n}e^{\frac{2(A_1t+A_2)}{n}}. \end{aligned}$$

(1.4)

Indeed, a more general result holds true, in which the logarithmic Sobolev inequality along $g(t)$ is derived from a logarithmic Sobolev inequality for $g_0,$ see Theorem 4.2. The exact factor $\frac{n}{2}$ in the term $-\frac{n}{2} \ln \sigma$ in the logarithmic Sobolev inequality (1.2) (also in (1.5) and (1.8) below) is crucial for the purpose of Theorems 1.5 and 1.6. Note that an upper bound for the Sobolev constant $C_S(M,g_0)$ and the modified Sobolev constant ${\tilde{C}}_S(M, g_0)$ can be obtained in terms of a lower bound for the diameter rescaled Ricci curvature and a positive lower bound for the diameter rescaled volume, see Sect. 2. In particular, a lower bound for the Ricci curvature, a positive lower bound for the volume, and an upper bound for the diameter lead to an upper bound for the Sobolev constant and the modified Sobolev constant.

The logarithmic Sobolev inequality in Theorem 1.1 is uniform for all time which lies below a given bound, but deteriorates as time becomes large. This is not a deficiency in the result, however. In general, it is impossible to obtain a uniform logarithmic Sobolev inequality along the Ricci flow which is independent of an upper bound for time. Indeed, by [5], there are smooth solutions of the Ricci flow on torus bundles over the circle which exist for all time, have bounded curvature, and collapse as $t\rightarrow \infty$. In view of the proofs of Theorems 1.5 and 1.7, a uniform logarithmic Sobolev inequality fails to hold along these solutions.

To obtain a uniform logarithmic Sobolev inequality, we employ a natural geometric condition. Let $\lambda _0=\lambda _0(g_0)$ denote the first eigenvalue of the operator $-\Delta +\frac{R}{4}$ for the initial metric $g_0$.

Theorem 1.2

Assume that the first eigenvalue $\lambda _0=\lambda _0(g_0)$ of the operator $-\Delta +\frac{R}{4}$ for the initial metric $g_0$ is positive. Let $\delta _0=\delta _0(g_0)$ be the number defined in (3.12). Let $t \in [0, T)$ and $\sigma >0$ satisfy $t+\sigma \ge \frac{n}{8}C_S(M,g_0)^2\delta _0$. Then there holds

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}}\le & {} \sigma \int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}} -\frac{n}{2}\ln \sigma \nonumber \\&+\frac{n}{2}\ln n + n \ln C_S(M,g_0)+\sigma _0(g_0) \end{aligned}$$

(1.5)

for all $u\in W^{1,2}(M)$ with $\int _M u^2 \mathrm{{dvol}}=1,$ where all geometric quantities are associated with the metric $g(t)$ (e.g., the volume form $\mathrm{{dvol}}$ and the scalar curvature $R$), except the Sobolev constant $C_S(M,g_0)$ and the number $\sigma _0(g_0)$ (defined in (3.13)) which are those of the initial metric $g_0$.

Consequently, there holds for each $t \in [0, T)$

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \frac{n}{2} \ln \left[ \alpha _{II}\int _M \left( |\nabla u|^2 +\frac{R}{4} u^2\right) \mathrm{{dvol}} \right] \end{aligned}$$

(1.6)

for all $u \in W^{1,2}(M)$ with $\int _M u^2 \mathrm{{dvol}}=1,$ where

$$\begin{aligned} \alpha _{II}=2eC_S(M,g_0)^2 e^{\frac{2}{n}\sigma _0(g_0)}. \end{aligned}$$

(1.7)

Combining Theorems 1.1 and 1.2, we obtain a uniform logarithmic Sobolev inequality along the Ricci flow.

Theorem 1.3

Assume that $\lambda _0(g_0) > 0$. For each $t\in [0, T)$ and each $\sigma >0,$ there holds

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \sigma \int _M \left( |\nabla u|^2 +\frac{R}{4}u^2\right) \mathrm{{dvol}} -\frac{n}{2}\ln \sigma +C \end{aligned}$$

(1.8)

for all $u \in W^{1,2}(M)$ with $\int _M u^2 \mathrm{{dvol}}=1,$ where $C$ depends only on the dimension $n,$ a positive lower bound for $\mathrm{{vol}}_{g_0}(M),$ a nonpositive lower bound for $R_{g_0},$ an upper bound for $C_S(M, g_0),$ and a positive lower bound for $\lambda _0(g_0)$.

Consequently, there holds for each $t \in [0, T)$

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \frac{n}{2} \ln \left[ \alpha _{III}\int _M \left( |\nabla u|^2 +\frac{R}{4} u^2\right) \mathrm{{dvol}} \right] \end{aligned}$$

(1.9)

for all $u \in W^{1,2}(M)$ with $\int _M u^2 \mathrm{{dvol}}=1,$ where

$$\begin{aligned} \alpha _{III}=\frac{2e}{n}e^{\frac{2}{n}C}. \end{aligned}$$

(1.10)

The class of Riemannian manifolds $(M, g_0)$ with $\lambda _0(g_0) > 0$ or, more generally, $\lambda _0(g_0)\ge 0$ is a very large one and particularly significant from a geometric point of view. (For example, the condition $\lambda _0(g_0)>0$ holds true when the scalar curvature of $g_0$ is nonnegative and somewhere positive.) On the other hand, we would like to remark that if $\lambda _0>0,$ then there holds $T<\frac{n}{2\lambda _0}$ by the proof of Proposition 1.2 in [6]. Hence, Theorem 1.3 can also be derived as a corollary of Theorem 1.1 alone. Next we note a special consequence of Theorem 1.3.

Corollary 1.4

Assume that $\lambda _0(g_0)> 0$. Then we have at any time $t \in [0, T)$

$$\begin{aligned} \mathrm{{vol}}_{g(t)}(M) \ge e^{-\frac{1}{4}-C} \end{aligned}$$

(1.11)

when ${\hat{R}}(t) \le 0,$ and

$$\begin{aligned} \mathrm{{vol}}_{g(t)}(M) \ge e^{-\frac{1}{4}-C} {\hat{R}}(t)^{-\frac{n}{2}} \end{aligned}$$

(1.12)

when ${\hat{R}}(t)>0$. Here ${\hat{R}}$ denotes the average scalar curvature.

Similar volume bounds follow from Theorem 1.1 without the condition $\lambda _0(g_0)> 0,$ but they also depend on a (finite) upper bound of $T$.

For a brief account of the logarithmic Sobolev inequalities on the euclidean space, we refer to Appendix 1, which serve as the background for the idea of the logarithmic Sobolev inequality. Both Theorems 1.1 and 1.2 are consequences of Perelman’s entropy monotonicity [6]. We obtained these two results, Theorems 1.3 and 4.2 in 2004 (around the time of the author’s differential geometry seminar talk “An introduction to the logarithmic Sobolev inequality” at UCSB in June 2004). They have also been prepared as part of the notes [13].

Next we apply the theory as presented in Chap. 2 of [2] to derive from Theorem 1.5 a Sobolev inequality along the Ricci flow without any restriction on time. (We came to notice [2] in the paper [19]. Note that the main result presented in [19] (and [20]) is incorrect, as pointed out in [14] (the archive version of the present paper), based on the example in [5]. Subsequently, a correction of this mistake was made in [21].) A particularly nice feature of the theory in Chap. 2 of [2] is that no additional geometric data (such as the volume) are involved in the passage from the logarithmic Sobolev inequality to the Sobolev inequality. Only the non-integral terms in the logarithmic Sobolev inequality and a nonpositive lower bound for the potential function $\Psi$ (see Theorem 5.5) come into play. This leads to the form of the geometric dependence in the following theorem.

Theorem 1.5

Assume $T<\infty$. There are positive constants $A$ and $B$ depending only on the dimension $n,$ a nonpositive lower bound for $R_{g_0},$ a positive lower bound for $\mathrm{{vol}}_{g_0}(M),$ an upper bound for $C_S(M,g_0),$ and an upper bound for $T,$ such that for each $t \in [0, T)$ and all $u \in W^{1,2}(M)$ there holds

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \mathrm{{dvol}} \right) ^{\frac{n-2}{n}} \le A\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}}+B \int _M u^2 \mathrm{{dvol}}, \end{aligned}$$

(1.13)

where all geometric quantities except $A$ and $B$ are associated with $g(t)$.

Under the assumption $\lambda _0(g_0)>0,$ this theorem can be improved as follows.

Theorem 1.6

Assume that $\lambda _0(g_0) > 0$. There is a positive constant $A$ depending only on the dimension $n,$ a nonpositive lower bound for $R_{g_0},$ a positive lower bound for $\mathrm{{vol}}_{g_0}(M),$ an upper bound for $C_S(M,g_0),$ and a positive lower bound for $\lambda _0(g_0),$ such that for each $t \in [0, T)$ and all $u \in W^{1,2}(M)$ there holds

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \mathrm{{dvol}} \right) ^{\frac{n-2}{n}} \le A\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}}, \end{aligned}$$

(1.14)

where all geometric quantities except $A$ are associated with $g(t)$.

We also obtain two results which extend Theorems 1.5 and 1.6 to the set-up of $W^{1,p}(M)$ for all $1<p<n,$ see Theorems 9.6 and 9.7 in Appendix 3. (Theorems 1.5 and 1.6 correspond to the case $p=2$.) These two general results can be thought of as nonlocal versions of Sobolev inequality, because they involve nonlocal pseudo-differential operators. Further results on (conventional) Sobolev inequalities for $2<p<n$ and $1<p<2$ will be presented in [17]. (Part of these results are derived from Theorems 1.5, 1.6, 9.6 and 9.7.) We would like to point out that the $p=2$ case of the Sobolev inequality is the most important for analytic and geometric applications.

The theory in Chap. 2 of [2] is formulated in a general and abstract set-up of symmetric Markov semigroups. By Lemma 5.2, $e^{-tH}$ is a symmetric Markov semigroup, where $H=-\Delta +\frac{R}{4}$ in the case $\lambda _0(g_0)>0$ and $H=-\Delta +\frac{R}{4}-\frac{\min R^-}{4}$ in the general case. Hence, the general theory and results in Chap. 2 of [2] can be applied to our situation. However, to obtain the precise geometric dependence of the Sobolev inequalities in Theorems 1.5 and 1.6, one has to verify the exact geometric nature of the constants which would appear in the many steps of the involved (and tightly formulated) arguments in [2]. Our proofs of Theorems 1.5 and 1.6 would be unclear and non-transparent if we go through a multitude of checking processes. Instead, we adapt the theory in [2] to our geometric set-up and work it out in complete, self-contained details. Another reason for doing so is to obtain some useful extensions of the theory as presented in Sect. 5, Appendix 2 and Appendix 3 (in particular Theorems 9.5–9.7). On the other hand, we think that our presentation makes the theory easily accessible to the general audience of geometric analysis. In particular, our presentation demonstrates in detail how the theory of the Ricci flow interacts with the basic theory of harmonic analysis.

Next we deduce from Theorems 1.5 and 1.6 $\kappa$-noncollapsing estimates for the Ricci flow which are measured relative to upper bounds of the scalar curvature to improve Perelman’s $\kappa$-noncollapsing result [6]. The original $\kappa$-noncollapsing result of Perelman in [6] is formulated relative to bounds for $|Rm|$. Later, a $\kappa$-noncollapsing result for bounded time measured relative to upper bounds of the scalar curvature was obtained independently by Perelman and the present author (see [11]). The $\kappa$-noncollapsing estimates below improve these results in two ways. First, they provide explicit estimates with clear geometric dependence on the initial metric. The estimates below are formulated in terms of more familiar geometric quantities of the initial metric. (If we apply Theorem 4.2, then we obtain $\kappa$-noncollapsing estimates which only depend on the logarithmic Sobolev inequality of the initial metric.) Moreover, the estimates are uniform up to $t=0$ (under a given upper bound for $T$). Secondly, the strategy of deriving the $\kappa$-noncollapsing estimate from the Sobolev inequality is particularly powerful and flexible, and has among others important applications to the Ricci flow with surgeries as constructed by Perelman in his work on the Poincaré conjecture and the geometrization conjecture [7]. It leads to a considerable clarification and simplification of an important step of the main arguments in [7], see Theorem 1.12 below.

Theorem 1.7

Assume that $T<\infty$. Let $L>0$ and $t \in [0, T)$. Consider the Riemannian manifold $(M, g)$ with $g=g(t)$. Assume $R\le \frac{1}{r^2}$ on a geodesic ball $B(x, r)$ with $0<r \le L$. Then there holds

$$\begin{aligned} \mathrm{{vol}}(B(x, r)) \ge \left( \frac{1}{2^{n+3}A+2BL^2}\right) ^{\frac{n}{2}} r^n, \end{aligned}$$

(1.15)

where $A$ and $B$ are from Theorem 1.5.

Theorem 1.8

Assume that $\lambda _0(g_0) > 0$. Let $t \in [0, T)$. Consider the Riemannian manifold $(M, g)$ with $g=g(t)$. Assume $R\le \frac{1}{r^2}$ on a geodesic ball $B(x, r)$ with $r>0$. Then there holds

$$\begin{aligned} \mathrm{{vol}}(B(x, r)) \ge \left( \frac{1}{2^{n+3}A}\right) ^{\frac{n}{2}} r^n, \end{aligned}$$

(1.16)

where $A$ is from Theorem 1.6. In other words, the flow $g=g(t), t\in [0, T)$ is $\kappa$-noncollapsed relative to upper bounds of the scalar curvature on all scales.

As is well-known, a major application of $\kappa$-noncollapsing estimates is to obtain smooth blow-up limits of the Ricci flow at singularities, which is crucial for analysing the structures of singularities of the Ricci flow.

Now we discuss how Theorems 1.5–1.8 lead to uniform Sobolev inequalities and uniform $\kappa$-noncollapsing estimates independent of any upper bound of time for various modified Ricci flows. In particular, they hold both on finite and infinite time intervals.

Consider the modified Ricci flow

$$\begin{aligned} \frac{\partial g}{\partial t}=-2\mathrm{{Ric}}+\lambda (g, t) g \end{aligned}$$

(1.17)

with a smooth scalar function $\lambda (g, t)$ independent of $x \in M$. The volume-normalized Ricci flow

$$\begin{aligned} \frac{\partial g}{\partial t}=-2\mathrm{{Ric}}+\frac{2}{n} {{\hat{R}}} g \end{aligned}$$

(1.18)

on a closed manifold, with ${\hat{R}}$ denoting the average scalar curvature, is an example of the modified Ricci flow. The $\lambda$-normalized Ricci flow

$$\begin{aligned} \frac{\partial g}{\partial t}=-2\mathrm{{Ric}}+\lambda g \end{aligned}$$

(1.19)

for a constant $\lambda$ is another example. (Of course, it reduces to the Ricci flow when $\lambda =0$.) The normalized Kähler–Ricci flow is a special case of it.

Let $g=g(t)$ be a smooth solution of the modified Ricci flow (1.17) on $M \times [0 ,T)$ for some $T>0,$ which is allowed to be finite or infinite. Let $g_0=g(0)$ denote the initial metric. First we have the following results.

Theorem 1.9

Theorem 1.6 and 1.8 extend to the above $g=g(t)$.

This result simply follows from scaling invariance of the estimates in Theorems 1.6 and 1.8. Here no additional condition is required. In the general case without the assumption $\lambda _0(g_0)>0,$ we need an additional condition. Set

$$\begin{aligned} T^*=\int _0^T e^{-\int _0^t \lambda (g(s), s)ds} \mathrm{{d}}t. \end{aligned}$$

(1.20)

Theorem 1.10

Assume that $T^*<\infty$.

• (1) There are positive constants $A$ and $B$ depending only on the dimension $n,$ a nonpositive lower bound for $R_{g_0},$ a positive lower bound for $\mathrm{{vol}}_{g_0}(M),$ an upper bound for $C_S(M,g_0),$ and an upper bound for $T^*,$ such that for each $t \in [0, T)$ and all $u \in W^{1,2}(M)$ there holds

  $$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \mathrm{{dvol}} \right) ^{\frac{n-2}{n}}\le & {} A\int _M \left( |\nabla u|^2 +\frac{R}{4}u^2\right) \mathrm{{dvol}}\nonumber \\&+Be^{-\int _0^t \lambda (g(s), s)ds} \int _M u^2 \mathrm{{dvol}}. \end{aligned}$$

  (1.21)

• (2) Let $L>0$ and $t \in [0, T)$. Consider the Riemannian manifold $(M, g)$ with $g=g(t)$. Assume $R\le \frac{1}{r^2}$ on a geodesic ball $B(x, r)$ with $0<r \le L$. Then there holds

  $$\begin{aligned} \mathrm{{vol}}(B(x, r)) \ge \left( \frac{1}{2^{n+3}A+2Be^{-\int _0^t \lambda (g(s), s)ds}L^2}\right) ^{\frac{n}{2}} r^n. \end{aligned}$$

  (1.22)

This theorem is a simple consequence of the scaling behavior of the estimates in Theorems 1.5 and 1.7. Combining Theorems 1.9 and  1.10 with Perelman’s scalar curvature estimate [10], we obtain the following corollary.

Theorem 1.11

Let $g=g(t)$ be a smooth solution of the normalized Kähler–Ricci flow

$$\begin{aligned} \frac{\partial g}{\partial t}=-2\mathrm{{Ric}}+2\gamma g \end{aligned}$$

(1.23)

on $M \times [0, \infty )$ with a positive first Chern class, where $\gamma$ is the positive constant such that the Ricci class equals $\gamma$ times the Kähler class. (We assume that $M$ carries such a Kähler structure.) Then the Sobolev inequality (1.24) holds true with $\lambda (g(s), s)=2\lambda$. Moreover, there is a positive constant $L$ depending only on the initial metric $g_0=g(0)$ and the dimension $n$ such that the inequality (1.22) holds true for all $t \in [0, T)$ and $0<r \le L$.

If $\lambda _0(g_0)>0,$ then the Sobolev inequality (1.14) holds true for $g$. Moreover, there is a positive constant depending only on the initial metric $g_0$ and the dimension $n$ such that the inequality (1.16) holds true for all $t \in [0, T)$ and $0<r \le L$. Consequently, blow-up limits of $g$ at the time infinity satisfy (1.16) for all $r>0$ and the Sobolev inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \mathrm{{dvol}} \right) ^{\frac{n-2}{n}} \le A\int _M |\nabla u|^2 \mathrm{{dvol}} \end{aligned}$$

(1.24)

for all $u$. (In particular, they must be noncompact.)

Finally, we would like to mention that Theorems 1.5 and 1.6 hold true for the Ricci flow with surgeries of Perelman [7], with suitable modifications as stated below.

Theorem 1.12

Let $n=3$ and $g=g(t)$ be a Ricci flow with surgeries as constructed in [7] on its maximal time interval $[0, T_{max}),$ with suitably chosen surgery parameters. Let $g_0=g(0)$. Then there holds at each $t \in [0, T_{max})$

$$\begin{aligned} \left( \int _M |u|^{6} \mathrm{{dvol}} \right) ^{\frac{1}{3}} \le A(t)\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}}+B(t) \int _M u^2 \mathrm{{dvol}} \end{aligned}$$

(1.25)

for all $u \in W^{1,2}(M),$ where $A(t)$ and $B(t)$ are bounded from above in terms of a nonpositive lower bound for $R_{g_0},$ a positive lower bound for $\mathrm{{vol}}_{g_0}(M),$ an upper bound for $C_S(M,g_0),$ and an upper bound for $t$.

If $\lambda _0(g_0)>0,$ then there holds at each $t \in [0, T_{max})$

$$\begin{aligned} \left( \int _M |u|^{6} \mathrm{{dvol}} \right) ^{\frac{1}{3}} \le A(t)\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}} \end{aligned}$$

(1.26)

for all $u \in W^{1,2}(M),$ where $A(t)$ is bounded from above in terms of a nonpositive lower bound for $R_{g_0},$ a positive lower bound for $\mathrm{{vol}}_{g_0}(M),$ an upper bound for $C_S(M,g_0),$ a positive lower bound for $\lambda _0(g_0),$ and an upper bound for $m(t)$.

$\kappa$-noncollapsing estimates follow as before, which lead to a considerable simplification of the arguments in [7] about preserving the $\kappa$-noncollapsing property after surgeries. Similar results hold true in higher dimensions whenever similar surgeries are performed. (The constants also depend on the dimension $n$.)

This result follows from Theorems 1.5 and 1.6, and a general result on Sobolev inequalities under surgeries. The details can be found in [18] and its sequel. In [7], the surgery parameters are chosen such that several key properties of the Ricci flow are preserved after surgery. One is the $\kappa$-noncollapsing property. Since the Sobolev inequalities (1.25) and (1.26) are derived without using the $\kappa$-noncollapsing property, the choice of the surgery parameters is also simplified. The $\kappa$-noncollapsing property follows as a consequence of (1.25) and (1.26).

The results in this paper (except Theorem 1.12) extend to the dimension $n=2,$ see [15].

This paper first appeared on the archive in 2007 as [14].

2 The Sobolev Inequality

Consider a compact Riemannian manifold $(M, g)$ of dimension $n\ge 3$. Its Poincaré-Sobolev constant (for the exponent 2) is defined to be

$$\begin{aligned} C_{P,S}(M,g)=\sup \{\Vert u-u_M\Vert _{\frac{2n}{n-2}}: u \in C^1(M), \Vert \nabla u\Vert _2=1\}, \end{aligned}$$

(2.1)

where $\Vert u\Vert _p$ denotes the $L^p$ norm of $u$ with respect to $g,$ i.e., $\Vert u\Vert _p=(\int _M |u|^p\mathrm{{dvol}})^{1/p}$ ($\mathrm{{dvol}}=d\mathrm{{vol}}_g$). In other words, $C_{P,S}(M,g)$ is the smallest number such that the Poincare–Sobolev inequality

$$\begin{aligned} \Vert u-u_M\Vert _{\frac{2n}{n-2}} \le C_{P,S}(M,g) \Vert \nabla u\Vert _2 \end{aligned}$$

(2.2)

holds true for all $u \in C^1(M)$ (or all $u \in W^{1,2}(M)$). The Sobolev constant of $(M,g)$ (for the exponent 2) is defined to be

$$\begin{aligned} C_S(M,g)=\sup \{\Vert u\Vert _{\frac{2n}{n-2}}-\frac{1}{\mathrm{{vol}}(M)^{\frac{1}{n}}}\Vert u\Vert _2: u \in C^1(M), \Vert \nabla u\Vert _2=1. \end{aligned}$$

(2.3)

In other words, $C_S(M,g)$ is the smallest number such that the inequality

$$\begin{aligned} \Vert u\Vert _{\frac{2n}{n-2}} \le C_{S}(M,g)\Vert \nabla u\Vert _2+\frac{1}{\mathrm{{vol}}(M)^{\frac{1}{n}}}\Vert u\Vert _2 \end{aligned}$$

(2.4)

holds true for all $u\in W^{1,2}(M)$.

Definition

We define the modified Sobolev constant ${\tilde{C}}_S(M,g)$ to be $\max \{C_S(M,g), 1\}$.

The Hölder inequality leads to the following basic fact.

Lemma 2.1

There holds for all $u\in W^{1,2}(M)$

$$\begin{aligned} \Vert u\Vert _{\frac{2n}{n-2}} \le C_{P,S}(M,g)\Vert \nabla u\Vert _2+\frac{1}{\mathrm{{vol}}(M)^{\frac{1}{n}}}\Vert u\Vert _2. \end{aligned}$$

(2.5)

In other words, there holds $C_S(M,g) \le C_{P,S}(M,g)$.

Another basic constant, the Neumann isoperimetric constant of $(M, g),$ is defined to be

$$\begin{aligned} C_{N, I}(M,g)=\sup \left\{ \frac{\mathrm{{vol}}(\Omega )^{\frac{n-1}{n}}}{A(\partial \Omega )}: \Omega \subset M \text{ is } \text{ a } C^1 \text{ domain }, \mathrm{{vol}}(\Omega ) \le \frac{1}{2} \mathrm{{vol}}(M)\right\} ,\nonumber \\ \end{aligned}$$

(2.6)

where $A(\partial \Omega )$ denotes the $n-1$-dimensional volume of $\partial \Omega$.

Lemma 2.2

There holds for all $u \in W^{1,2}(M)$

$$\begin{aligned} \Vert u-u_M\Vert _{\frac{2n}{n-2}} \le 2(1+\sqrt{2}) \frac{n-1}{n-2} C_{N,I}(M, g) \Vert \nabla u\Vert _2. \end{aligned}$$

(2.7)

In other words, there holds $C_{P,S}(M,g) \le 2(1+\sqrt{2})\frac{n-1}{n-2} C_{N,I}(M, g)$.

For the proof, see [12]. The following estimate of the Neumann isoperimetric constant follows from S. Gallot’s estimate in [4]. We define the diameter rescaled Ricci curvature ${\hat{R}}ic(v,v)$ of a unit tangent vector $v$ to be $\mathrm{{diam}}(M)^2 {\mathrm{Ric}}(v,v),$ and set $\kappa _{\mathrm{{{{R}}ic}}}=\min _v\{{\hat{R}}ic(v,v)\}$. Then we set ${\hat{\kappa }}_{{\hat{R}}ic}=|\min \{\kappa _\mathrm{Ric}, -1\}|$. We also define the diameter rescaled volume ${\hat{v}}ol(M)$ to be $\mathrm{{vol}}(M)\mathrm{{diam}}(M)^{-n}$.

Theorem 2.3

There holds

$$\begin{aligned} C_{N,I}(g, M) \le C(n, {\hat{\kappa }}_{{\hat{R}}ic}){\hat{v}}ol(M)^{-\frac{1}{n}}, \end{aligned}$$

(2.8)

where $C(n, {\hat{\kappa }}_{{\hat{R}}ic})$ is a positive constant depending only on $n$ and ${\hat{\kappa }}_{{\hat{R}}ic}$.

Note that ${\hat{\kappa }}_{{\hat{R}}ic}$ can be replaced by a certain integral lower bound of the Ricci curvature, see [3].

3 The Logarithmic Sobolev Inequalities on a Riemannian Manifold

The various versions of the logarithmic Sobolev inequality on the Euclidean space as presented in Appendix 1 allow suitable extensions to Riemannian manifolds. We formulate a log gradient version and a straight version, cf. Appendix 1. As in the last section, let $(M, g)$ be a compact Riemannian manifold of dimension $n$.

Theorem 3.1

There holds

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le n \ln \left( C_S(M,g) \Vert \nabla u\Vert _2+ \frac{1}{\mathrm{{vol}}_g(M)^{\frac{1}{n}}}\right) , \end{aligned}$$

(3.1)

provided that $u \in W^{1,2}(M)$ and $\Vert u\Vert _2=1$.

Proof

Set $q=\frac{2n}{n-2}$. Since $\ln$ is concave and $\int _M u^2 \mathrm{{dvol}} =1,$ we have by Jensen’s inequality

$$\begin{aligned} \ln \int _M u^q \mathrm{{dvol}} = \ln \int _M u^2 \cdot u^{q-2} \mathrm{{dvol}} \ge \int _M u^2 \ln u^{q-2}. \end{aligned}$$

(3.2)

It follows that

$$\begin{aligned} \int _M u^2 \ln u\le & {} \frac{1}{q-2} \ln \int _M u^q \mathrm{{dvol}} \nonumber \\= & {} \frac{q}{q-2} \ln \Vert u\Vert _q \nonumber \\\le & {} \frac{n}{2} \ln \left( C_S(M,g) \Vert \nabla u\Vert _2+ \frac{1}{\mathrm{{vol}}_g(M)^{\frac{1}{n}}}\Vert u\Vert _2\right) . \end{aligned}$$

(3.3)

$\square$

Lemma 3.2

There holds

$$\begin{aligned} \ln (x+B) \le \alpha x+\alpha B-1-\ln \alpha \end{aligned}$$

(3.4)

for all $B \ge 0, \alpha >0$ and $x >-B$.

Proof

Consider the function $y=\ln (x+B)-\alpha x$ for $x>-B$. Since $y \rightarrow -\infty$ as $x \rightarrow -B$ or $x \rightarrow \infty ,$ it achieves its maximum somewhere. We have

$$\begin{aligned} y'=\frac{1}{x+B}-\alpha . \end{aligned}$$

(3.5)

Hence, the maximum point is $x_0=\frac{1}{\alpha }-B$. It follows that the maximum of $y$ is $y(x_0)=\alpha B-1-\ln \alpha$.$\square$

Theorem 3.3

For each $\alpha >0$ and all $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1,$ there holds

$$\begin{aligned} \int _M u^2 \ln u^2 \le \frac{n\alpha C_S(M,g)^2}{2} \int _M |\nabla u|^2 -\frac{n}{2}\ln \alpha + \frac{n}{2}(\ln 2+\alpha \mathrm{{vol}}_g(M)^{-\frac{2}{n}}-1)\nonumber \\ \end{aligned}$$

(3.6)

and

$$\begin{aligned} \int _M u^2 \ln u^2\le & {} \frac{n\alpha C_S(M,g)^2}{2} \int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) -\frac{n}{2}\ln \alpha \nonumber \\&+\frac{n\alpha }{2}\left( \mathrm{{vol}}_g(M)^{-\frac{2}{n}}-\frac{\min R^-}{4}C_S(M,g)^2\right) +\frac{n}{2}(\ln 2-1).\qquad \quad \end{aligned}$$

(3.7)

(The notation of the volume is omitted.)

Proof

By (3.1), we have for $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1$

$$\begin{aligned} \int _M u^2 \ln u^2\le & {} \frac{n}{2} \ln \left( C_S(M,g) \Vert \nabla u\Vert _2+ \frac{1}{\mathrm{{vol}}_g(M)^{\frac{1}{n}}}\right) ^2 \nonumber \\\le & {} \frac{n}{2} \ln 2 + \frac{n}{2} \ln \left( C_S(M,g)^2 \int _M |\nabla u|^2+ \frac{1}{\mathrm{{vol}}_g(M)^{\frac{2}{n}}}\right) . \end{aligned}$$

(3.8)

Applying Lemma 3.2 with $x=C_S(M,g)^2 \int _M |\nabla u|^2$ and $B=1,$ we then arrive at (3.6). The inequality (3.7) follows from (3.6).$\square$

Lemma 3.4

Let $A>0, B>0$ and $\gamma >0$ such that $A\ge \frac{1}{\gamma +B}$. Then we have

$$\begin{aligned} \ln (x+B)\le Ax-\ln A+\ln (\gamma +B)-\ln \gamma -1 \end{aligned}$$

(3.9)

for all $x \ge \gamma$.

Proof

First consider the function $y=\ln t-\gamma t$ for $t>0$. Since $y\rightarrow -\infty$ as $t \rightarrow 0$ or $t \rightarrow \infty ,$ $y$ achieves its maximum somewhere. We have $y'=\frac{1}{t}-\gamma$. Hence, the maximum is achieved at $\frac{1}{\gamma }$. It follows that the maximum is $y(\frac{1}{\gamma })=-\ln \gamma -1$. We infer

$$\begin{aligned} \ln A-\gamma A \le -\ln \gamma -1. \end{aligned}$$

(3.10)

Next we consider the function $y=\ln (x+B)-Ax+\ln A$ for $x \ge \gamma$. By (3.10), we have $y(\gamma )=\ln (\gamma +B)-A\gamma +\ln A \le \ln (\gamma +B)-\ln \gamma -1$. On the other hand, we have $y'=\frac{1}{x+B}-A \le \frac{1}{\gamma +B}-A \le 0$. We arrive at (3.9).$\square$

Theorem 3.5

Assume that the first eigenvalue $\lambda _0=\lambda _0(g)$ of the operator $-\Delta +\frac{R}{4}$ is positive. For each $A \ge \delta _0$ and all $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1,$ there holds

$$\begin{aligned} \int _M u^2 \ln u^2 \le \frac{nAC_S^2}{2}\int _M\left( |\nabla u|^2+\frac{R}{4}u^2\right) -\frac{n}{2}\ln A+\frac{n}{2}\ln 2+\sigma _0, \end{aligned}$$

(3.11)

where

$$\begin{aligned}&\delta _0=\delta _0(g)= \left( \lambda _0 C_S^2+\frac{1}{\mathrm{{vol}}_g(M)^{\frac{2}{n}}} -C_S^2 \frac{\min R^-}{4}\right) ^{-1},\end{aligned}$$

(3.12)

$$\begin{aligned}&\sigma _0=\sigma _0(g)==\frac{n}{2}\left[ \ln \left( \lambda _0 C_S^2+\frac{1}{\mathrm{{vol}}_g(M)^{\frac{2}{n}}} -C_S^2 \frac{\min R^-}{4}\right) -\ln (\lambda _0 C_S^2)-1\right] ,\nonumber \\ \end{aligned}$$

(3.13)

and $C_S=C_S(M,g)$.

Proof

Arguing as in the proof of Theorem 3.3, we deduce for $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1$

$$\begin{aligned}&\int _M u^2 \ln u^2 \le \frac{n}{2}\ln 2+ \frac{n}{2} \ln \left( C_S^2\int _M |\nabla u|^2 +\frac{1}{\mathrm{{vol}}_g(M)^{\frac{2}{n}}}\right) \nonumber \\&\quad \le \frac{n}{2}\ln 2+ \frac{n}{2} \ln \left[ C_S^2\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) +\frac{1}{\mathrm{{vol}}_g(M)^{\frac{2}{n}}} -C_S^2 \frac{\min R^-}{4}\right] \!\!.\qquad \qquad \end{aligned}$$

(3.14)

Applying (3.9) with $\gamma =\lambda _0 C_S^2,$ $B=\frac{1}{\mathrm{{vol}}_g(M)^{\frac{2}{n}}} -C_S^2 \frac{\min R^-}{4}$ and $x=C_S^2\int _M (|\nabla u|^2+\frac{R}{4}u^2),$ we then arrive at (3.11) for each $A \ge (\gamma +B)^{-1}$.$\square$

4 The logarithmic Sobolev Inequality Along the Ricci Flow

Let $M$ be a compact manifold of dimension $n$. Consider Perelman’s entropy functional

$$\begin{aligned} {\mathcal {W}}(g, f, \tau )=\int _M \left[ \tau (R+|\nabla f|^2)+f-n\right] \frac{e^{-f}}{(4\pi \tau )^{\frac{n}{2}}} \mathrm{{dvol}}, \end{aligned}$$

(4.1)

where $\tau$ is a positive number, $g$ is a Riemannian metric on $M,$ and $f\in C^{\infty }(M)$ satisfies

$$\begin{aligned} \int _M \frac{e^{-f}}{(4\pi \tau )^{\frac{n}{2}}} \mathrm{{dvol}}=1. \end{aligned}$$

(4.2)

All geometric quantities in (4.1) and (4.2) are associated with $g$. To relate to the idea of logarithmic Sobolev inequalities, we make a change of variable

$$\begin{aligned} u=\frac{e^{-\frac{f}{2}}}{(4\pi \tau )^{\frac{n}{4}}}. \end{aligned}$$

(4.3)

Then (4.2) leads to

$$\begin{aligned} \int _M u^2 \mathrm{{dvol}}=1 \end{aligned}$$

(4.4)

and we have

$$\begin{aligned} {\mathcal {W}}(g, f, \tau )={\mathcal {W}}^*(g, u, \tau ) -\frac{n}{2}\ln \tau -\frac{n}{2}\ln (4\pi )-n, \end{aligned}$$

(4.5)

where

$$\begin{aligned} {\mathcal {W}}^*(g, u,\tau )=\int _M \left[ \tau (4|\nabla u|^2+R u^2)-u^2 \ln u^2 \right] \mathrm{{dvol}}. \end{aligned}$$

(4.6)

We define $\mu ^*(g, \tau )$ to be the infimum of ${\mathcal {W}}^*(g, u, \tau )$ over all $u$ satisfying (4.4).

Next let $g=g(t)$ be a smooth solution of the Ricci flow

$$\begin{aligned} \frac{\partial g}{\partial t}=-2\mathrm{{Ric}} \end{aligned}$$

(4.7)

on $M\times [0, T)$ for some (finite or infinite) $T>0$. Let $0<t^*<T$ and $\sigma >0$. We set $T^*=t^*+\sigma$ and $\tau =\tau (t)=T^*-t$ for $0\le t \le t^*$. Consider a solution $f=f(t)$ of the equation

$$\begin{aligned} \frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R+\frac{n}{2\tau } \end{aligned}$$

(4.8)

on $[0, t^*]$ with a given terminal value at $t=t^*$ (i.e., $\tau =\sigma$) satisfying (4.2) with $g=g(t^*)$. Then (4.2) holds true for $f=f(t), g=g(t),$ and all $t \in [0, t^*]$. Perelman’s monotonicity formula says

$$\begin{aligned} \frac{\mathrm{{d}} {\mathcal {W}}}{\mathrm{{d}} t}=2\tau \int _M |Ric+\nabla ^2 f-\frac{1}{2\tau } g|^2 \frac{e^{-f}}{(4\pi \tau )^{\frac{n}{2}}} \mathrm{{dvol}} \ge 0, \end{aligned}$$

(4.9)

where ${\mathcal {W}}={\mathcal {W}}(g(t), f(t), \tau (t))$. Consequently,

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t} {\mathcal {W}}^*(g,u,\tau ) \ge \frac{n}{2}\frac{\mathrm{{d}}}{\mathrm{{d}}t} \ln \tau , \end{aligned}$$

(4.10)

where $g=g(t), \tau =\tau (t),$ and

$$\begin{aligned} u=u(t)= \frac{e^{-f(t)/2}}{(4\pi \tau (t))^{\frac{n}{4}}}, \end{aligned}$$

(4.11)

which satisfies the equation

$$\begin{aligned} \frac{\partial u}{\partial t}=-\Delta u+\frac{|\nabla u|^2}{u}+\frac{R}{2}u. \end{aligned}$$

(4.12)

It follows that

$$\begin{aligned} \mu ^*(g(t_1), \tau (t_1)) \le \mu ^*(g(t_2, \tau (t_2))+\frac{n}{2} \ln \frac{\tau _1}{\tau _2}, \end{aligned}$$

(4.13)

for $t_1<t_2,$ where $\tau _1=\tau (t_1)$ and $\tau _2=\tau (t_2)$. Choosing $t_1=0$ and $t_2=t^*,$ we then arrive at

$$\begin{aligned} \mu ^*(g(0), t^*+\sigma ) \le \mu ^*(g(t^*), \sigma )+\frac{n}{2} \ln \frac{t^*+\sigma }{\sigma }. \end{aligned}$$

(4.14)

Since $0<t^*<T$ is arbitrary, we can rewrite (4.14) as follows

$$\begin{aligned} \mu ^*(g(t), \sigma ) \ge \mu ^*(g(0), t+\sigma )+ \frac{n}{2}\ln \frac{\sigma }{t+\sigma } \end{aligned}$$

(4.15)

for all $t \in [0, T)$ and $\sigma >0$ (the case $t=0$ is trivial).

We will also need the following elementary lemma.

Lemma 4.1

Let $a>0$ and $b$ be constants. Then the minimum of the function $y=a \sigma -\frac{n}{2} \ln \sigma +b$ for $\sigma >0$ is $\frac{n}{2} \ln (\alpha a),$ where

$$\begin{aligned} \alpha =\frac{2e}{n} e^{\frac{2b}{n}}. \end{aligned}$$

(4.16)

Proof

Since $y \rightarrow \infty$ as $t\rightarrow 0$ or $t \rightarrow \infty ,$ it achieves its minimum somewhere. We have $y'=a-\frac{n}{2\sigma },$ whence the minimum is achieved at $\sigma =\frac{n}{2a}$. Then the minimum equals $y(\frac{n}{2a}),$ which leads to the desired conclusion.$\square$

Proof of Theorem 1.1

We apply Theorem 3.3 with $g=g_0$ to estimate $\mu ^*(g_0, t+\sigma )$. Consider $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1$. We choose

$$\begin{aligned} \alpha =\frac{8(t+\sigma )}{n{\tilde{C}}_S(M,g_0)^2} \end{aligned}$$

(4.17)

in (3.6) and deduce

$$\begin{aligned} \int _M u^2 \ln u^2\le & {} 4(t+\sigma )\int _M |\nabla u|^2 -\frac{n}{2}\ln \frac{8(t+\sigma )}{n{\tilde{C}}_S^2}\nonumber \\&+\frac{n}{2} \cdot \frac{8(t+\sigma )}{n{\tilde{C}}_S^2 \mathrm{{vol}}_{g_0}(M)^{\frac{2}{n}}} +\frac{n}{2}(\ln 2-1) \nonumber \\\le & {} (t+\sigma ) \int _M (4|\nabla u|^2+Ru^2) +(t+\sigma )\left( \frac{4}{n{\tilde{C}}_S^2 \mathrm{{vol}}_{g_0}(M)^{\frac{2}{n}}}-\min _{t=0} R\right) \nonumber \\&-\frac{n}{2} \ln (t+\sigma )+\frac{n}{2}(2\ln {\tilde{C}}_S+\ln n -2\ln 2 -1), \end{aligned}$$

(4.18)

where ${\tilde{C}}_S={\tilde{C}}_S(M,g_0)$. It follows that

$$\begin{aligned} \mu ^*(g(0), t+\sigma )\ge & {} \frac{n}{2} \ln (t+\sigma ) -(t+\sigma )\left( \frac{4}{n{\tilde{C}}_S^2 \mathrm{{vol}}_{g_0}(M)^{\frac{2}{n}}}-\min _{t=0} R\right) \nonumber \\&-\frac{n}{2}(2\ln {\tilde{C}}_S+\ln n -2\ln 2 -1). \end{aligned}$$

(4.19)

Combining this with (4.15) leads to

$$\begin{aligned} \mu ^*(g(t), \sigma )\ge & {} \frac{n}{2} \ln \sigma -(t+\sigma )\left( \frac{4}{n{\tilde{C}}_S^2 \mathrm{{vol}}_g(M)^{\frac{2}{n}}}-\min _{t=0} R\right) \nonumber \\&-\frac{n}{2}(2\ln {\tilde{C}}_S+\ln n -2\ln 2 -1), \end{aligned}$$

(4.20)

or

$$\begin{aligned} \mu ^*\left( g(t), \frac{\sigma }{4}\right)\ge & {} \frac{n}{2} \ln \sigma -\left( t+\frac{\sigma }{4}\right) \left( \frac{4}{n{\tilde{C}}_S^2 \mathrm{{vol}}_g(M)^{\frac{2}{n}}}-\min _{t=0} R\right) \nonumber \\&-\frac{n}{2}(2\ln {\tilde{C}}_S+\ln n-1), \end{aligned}$$

(4.21)

which is equivalent to (1.2).

To see (1.3), we apply Lemma 4.1 to (1.2) with $a=\int _M(|\nabla u|^2+\frac{R}{4}u^2)\mathrm{{dvol}}+\frac{A_1}{4}$ and $b=A_1t+A_2$.$\square$

Proof of Theorem 1.2

This is similar to the proof of Theorem 1.1. We apply Theorem 3.5 with $g=g_0$ to estimate $\mu ^*(g_0, t+\sigma )$. Assume $t+\sigma \ge \frac{n}{8}C_S(M,g_0)^2\delta _0(g_0)$. We set

$$\begin{aligned} A=\frac{8(t+\sigma )}{nC_S(M,g_0)^2}. \end{aligned}$$

(4.22)

Then there holds $A \ge \delta _0(g_0)$. Using this $A$ in (3.11), we deduce for $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1$

$$\begin{aligned} \int _M u^2 \ln u^2\le & {} 4(t+\sigma )\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) -\frac{n}{2}\ln (t+\sigma )\nonumber \\&+\frac{n}{2}(2\ln C_S(M,g_0)+\ln n-2\ln 2)+\sigma _0(g_0). \end{aligned}$$

(4.23)

It follows that

$$\begin{aligned} \mu ^*(g_0, t+\sigma ) \ge \frac{n}{2}\ln (t+\sigma )-\frac{n}{2}(2\ln C_S(M,g_0)+\ln n-2\ln 2)-\sigma _0(g_0).\nonumber \\ \end{aligned}$$

(4.24)

Combining this with (4.15) yields

$$\begin{aligned} \mu ^*(g(t), \sigma )\ge \frac{n}{2} \ln \sigma -\frac{n}{2}(2\ln C_S(M,g_0)+\ln n-2\ln 2)-\sigma _0(g_0). \end{aligned}$$

(4.25)

Replacing $\sigma$ by $\frac{\sigma }{4},$ we then arrive at (1.5).

To see (1.6), we apply Lemma 4.1 to (1.5) with $a=\int _M (|\nabla u|^2+\frac{R}{4}u^2)$ and $b=\frac{n}{2}\ln n+n \ln C_S(M, g_0)+\sigma _0(g_0)$. Note that by the maximum principle and the evolution equation of the scalar curvature associated with the Ricci flow, $\min R$ is nondecreasing, which implies that $a>0$.$\square$

Note that the proofs of Theorems 1.1 and 1.2 lead to the following general result. Indeed, Theorems 1.1 and 1.2 follows from it.

Theorem 4.2

Let $g=g(t)$ be a smooth solution of the Ricci flow on $M \times [0, T)$ for some (finite or infinite) $T>0$. Let $h(\sigma )$ be a scalar function for $\sigma >0$. Assume that the initial metric $g_0=g(0)$ satisfies the logarithmic Sobolev inequality

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \sigma \int _M \left( |\nabla u|^2 + \frac{R}{4}u^2\right) \mathrm{{dvol}} +h(\sigma ) \end{aligned}$$

(4.26)

for each $\sigma >0$ and all $u\in W^{1,2}(M)$ with $\int _M u^2 \mathrm{{dvol}}=1$. Then there holds at each $t \in [0, T)$

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \sigma \int _M \left( |\nabla u|^2 + \frac{R}{4}u^2\right) \mathrm{{dvol}} +h(4(t+\sigma )) \end{aligned}$$

(4.27)

for each $\sigma >0$ and all $u\in W^{1,2}(M)$ with $\int _M u^2 \mathrm{{dvol}}=1$.

Proof of Theorem 1.3

Let $t\in [0, T)$ and $\sigma >0$. If $\sigma < \frac{n}{8}C_S(M, g_0)^2 \delta _0(g_0),$ we apply Theorem 1.1. Otherwise, we apply Theorem 1.2. Then we arrive at (1.8). To see (1.9), we note that by [6] the eigenvalue $\lambda _0(g(t))$ is nondecreasing. Hence, $\lambda _0(g(t))>0$ for all $t,$ which implies that $\int _M (|\nabla u|^2+\frac{R}{4}u^2)>0$ for all $t$. Hence, we can apply Lemma 4.1 to (1.8) with $a=\int _M (|\nabla u|^2+\frac{R}{4}u^2)$ and $b=C$ to arrive at the desired inequality. $\square$

Proof of Corollary 1.4 to Theorem 1.3

Choosing $u=\mathrm{{vol}}_{g(t)}(M)^{-\frac{1}{2}}$ in (1.8), we infer

$$\begin{aligned} \ln \frac{1}{\mathrm{{vol}}_{g(t)}(M)}\le \frac{\sigma }{4}{\hat{R}}(t)-\frac{n}{2}\ln \sigma +C. \end{aligned}$$

(4.28)

If ${\hat{R}}(t) \le 0,$ we choose $\sigma =1$ to arrive at (1.11). If ${\hat{R}}(t)>0,$ we choose $\sigma ={\hat{R}}(t)^{-1}$ to arrive at (1.12).$\square$

5 The Sobolev Inequality Along the Ricci Flow

We first present a general result which converts a logarithmic Sobolev inequality to a Sobolev inequality. It follows straightforwardly from more general results in [2]. Consider a compact Riemannian manifold $(M,g)$ of dimension $n \ge 1$. Let $\Psi \in L^{\infty }(M),$ which we call a potential function. We set $H=-\Delta +\Psi$. Its associated quadratic form is

$$\begin{aligned} Q(u)=\int _M (|\nabla u|^2 +\Psi u^2) \mathrm{{dvol}}, \end{aligned}$$

(5.1)

where $u \in W^{1,2}(M)$. We also use $Q$ to denote the corresponding bilinear form, i.e.,

$$\begin{aligned} Q(u, v)=\int _M (\nabla u \cdot \nabla v +\Psi u v)\mathrm{{dvol}}. \end{aligned}$$

(5.2)

Consider the operator $e^{-tH}$ associated with $H$. It is characterized by the property that for $u_0 \in L^2(M),$ $u=e^{-tH}u_0$ satisfies the heat equation

$$\begin{aligned} \frac{\partial u}{\partial t}=-Hu \end{aligned}$$

(5.3)

for $t>0$ and the initial condition $u(0)=u_0$. We have the spectral formula

$$\begin{aligned} e^{-tH}u=\sum e^{-\lambda _i t} \phi _i <u, \phi _i>_2, \end{aligned}$$

(5.4)

for $u \in L^2(M),$ where $\{\phi _i\}$ is a complete set of $L^2$-orthonormal eigenfunctions of $H$ and $\lambda _1\le \lambda _2 \le \cdot \cdot \cdot$ are the corresponding eigenvalues. Since $\lambda _i \rightarrow \infty ,$ $e^{-tH}: L^2(M) \rightarrow L^2(M)$ is a bounded operator. On the other hand, there holds

$$\begin{aligned} e^{-tH}u=\int _M K(\cdot , y, t)u \mathrm{{dvol}}_y, \end{aligned}$$

(5.5)

where $K(x,y,t)$ denotes the heat kernel of $H$.

Lemma 5.1

The extension of $e^{-tH}$ for $t>0$ to $L^1(M)$ by the spectral formula (5.4) defines a bounded linear operator $e^{-tH}: L^1(M) \rightarrow W^{2, p}(M)$ for each $0<p<\infty$.

Proof

By elliptic regularity, we have $\phi _i \in W^{2, p}(M)$ for each $i$ and $0<p<\infty$. The elliptic $W^{2,p}$ estimates and Sobolev embedding lead to $\Vert \phi _i\Vert _{2, p} \le c_p(|\lambda _i|+1)^{m_n}$ for some $c_p>0$ independent of $i$ and a natural number $m_n$ depending only on $n$. The Sobolev embedding then implies $\Vert \phi _i\Vert _{\infty } \le c(\lambda _i|+1)^{m_n}$ for some $c>0$ independent of $i$. Now we have for $u \in L^1(M)$

$$\begin{aligned} \sum _{i\ge 1} e^{-\lambda _i t} |<u, \phi _i>_2 | \cdot \Vert \phi _i\Vert _{2, p} \le \left( \sum _{i \ge 1} e^{-\lambda _i t} \Vert \phi _i\Vert _{\infty } \Vert \phi _i\Vert _{2, p}\right) \Vert u\Vert _1. \end{aligned}$$

(5.6)

By the above estimates, the last series converges. The desired conclusion follows.$\square$

Lemma 5.2

Assume $\Psi \ge 0$. Then $e^{-tH}$ for $t>0$ is a contraction on $L^p(M)$ for each $1\le p \le \infty ,$ i.e.,

$$\begin{aligned} \Vert e^{-tH}u\Vert _p \le \Vert u\Vert _p \end{aligned}$$

(5.7)

for all $u \in L^p(M)$. It is also a contraction on $W^{1,2}(M)$ with respect to the norm $Q(u)^{\frac{1}{2}}$ (if $Q>0,$ i.e., $\lambda _1>0$) or the norm $(Q(u)+\int _M u^2 \mathrm{{dvol}})^{\frac{1}{2}}$ (if $\lambda _1=0$). Moreover, it is positivity preserving, i.e., $e^{-tH}u \ge 0$ if $u \ge 0$ and $u \in L^2(M)$.

Proof

The maximum principle implies that $e^{-tH}$ is a contraction on $L^{\infty }(M)$ for $t>0$. For $t>0$ and $u \in L^1(M),$ we set $\phi =sgn(e^{-tH}u),$ i.e., $\phi =1$ where $e^{-tH}u \ge 0$ and $\phi =-1$ where $e^{-tH}u<0$. There holds

$$\begin{aligned} \Vert e^{-tH}u\Vert _1= & {} \int _M \phi e^{-tH}u \mathrm{{dvol}} = \int _M u e^{-tH} \phi \mathrm{{dvol}} \nonumber \\\le & {} \Vert e^{-tH}\phi \Vert _{\infty } \Vert u\Vert _1 \le \Vert \phi \Vert _{\infty } \Vert u\Vert _1=\Vert u\Vert _1. \end{aligned}$$

(5.8)

Hence, $e^{-tH}$ is a contraction on $L^1(M)$. By the Riesz–Thorin interpolation theorem (see Appendix 3), $e^{-tH}$ is a contraction on $L^p(M)$ for each $1< p < \infty$.

The contraction property of $e^{-tH}$ on $W^{1,2}(M)$ follows from the spectral formula (5.4) because $\lambda _1 \ge 0$. (The contraction property of $e^{-tH}$ on $L^2(M)$ also follows from (5.4).) Finally, the positivity preserving property of $e^{-tH}$ is a consequence of the maximum principle.$\square$

Theorem 5.3

Let $0<\sigma ^*\le \infty$. Assume that for each $0<\sigma <\sigma ^*$ the logarithmic Sobolev inequality

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \sigma Q(u)+ \beta (\sigma ) \end{aligned}$$

(5.9)

holds true for all $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1,$ where $\beta$ is a nonincreasing continuous function. Assume that

$$\begin{aligned} \tau (t)=\frac{1}{2t}\int ^t_0 \beta (\sigma )d\sigma \end{aligned}$$

(5.10)

is finite for all $0<t < \sigma ^*$. Then there holds

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le e^{\tau (t)-\frac{3t}{4}\inf \Psi ^-} \Vert u\Vert _2 \end{aligned}$$

(5.11)

for each $0<t< \frac{1}{4}\sigma ^*$ and all $u \in L^2(M)$. There also holds

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le e^{2\tau (\frac{t}{2})-\frac{3t}{4} \inf \Psi ^-} \Vert u\Vert _1 \end{aligned}$$

(5.12)

for each $0<t< \frac{1}{4}\sigma ^*$ and all $u \in L^1(M)$.

The proof of this theorem is presented in Appendix 2. Note that (5.12) is equivalent to an upper bound for the heat kernel. The nonincreasing condition on $\beta$ can easily be removed (the function $\tau (t)$ needs to be slightly modified).

Theorem 5.4

• (1) Assume $\Psi \ge 0$. Let $\mu >2$ and $c>0$. Assume that the inequality

  $$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le c t^{-\frac{\mu }{4}} \Vert u\Vert _2 \end{aligned}$$

  (5.13)

  holds true for each $t>0$ and all $u\in L^2(M)$. Then the Sobolev inequality

  $$\begin{aligned} \Vert u\Vert ^2_{\frac{2\mu }{\mu -2}} \le C(\mu , c) Q(u) \end{aligned}$$

  (5.14)

  holds true for all $u \in W^{1,2}(M),$ where the positive constant $C(\mu , c)$ can be bounded from above in terms of upper bounds for $c, \mu$ and $\frac{1}{\mu -2}$.

• (2) Let $\mu >2$ and $c>0$. Assume that the inequality

  $$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le c_1 t^{-\frac{\mu }{4}} \Vert u\Vert _2 \end{aligned}$$

  (5.15)

  holds true for each $0<t<1$ and all $u\in L^2(M)$. Then the Sobolev inequality

  $$\begin{aligned} \Vert u\Vert ^2_{\frac{2\mu }{\mu -2}} \le C(\mu , c) (Q(u)+(1-\inf \Psi ^-)\Vert u\Vert _2^2) \end{aligned}$$

  (5.16)

  holds true for all $u \in W^{1,2}(M),$ where $C(\mu , c)$ has the same property as the above $C(\mu , c)$.

The proof of this theorem is presented in Appendix 3. Combining Theorems 5.3 and 5.4, we arrive at the following result.

Theorem 5.5

Let $0<\sigma ^*<\infty$. Assume that for each $0<\sigma < \sigma ^*$ the logarithmic Sobolev inequality

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \sigma Q(u)-\frac{\mu }{2} \ln \sigma +C \end{aligned}$$

(5.17)

holds true for all $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1,$ where $\mu$ and $c$ are constants such that $\mu >2$. Then we have the Sobolev inequality

$$\begin{aligned} \Vert u\Vert ^2_{\frac{2\mu }{\mu -2}} \le \left( \frac{\sigma ^*}{4}\right) ^{1-\frac{n}{\mu }} C({\bar{C}}, \mu ) \left( Q(u)+\frac{4-\sigma ^* \min \Psi ^-}{\sigma ^*} \Vert u\Vert _2^2\right) \end{aligned}$$

(5.18)

for all $u \in W^{1,2}(M),$ where $C({\bar{C}}, \mu )$ is from Theorem 5.4 and ${\bar{C}}$ is defined in (5.22) below.

Proof

For $\lambda >0,$ we consider the metric ${\bar{g}}=\lambda ^{-2}g$ and the potential function ${\bar{\Psi }}=\lambda ^2 \Psi$. Let ${\bar{H}}=-\Delta _{{\bar{g}}}+ {\bar{\Psi }}$ and ${\bar{Q}}$ be the associated quadratic form. It follows from (5.17) that

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}}_{{\bar{g}}} \le \sigma {\bar{Q}}(u)-\frac{\mu }{2} \ln \sigma +(n-\mu )\ln \lambda +C \end{aligned}$$

(5.19)

for $0<\sigma < \lambda ^{-2} \sigma ^*$ and $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1$. Choosing $\lambda =\frac{1}{2}\sqrt{\sigma ^*},$ we obtain

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}}_{{\bar{g}}} \le \sigma {\bar{Q}}(u)-\frac{\mu }{2} \ln \sigma +\frac{n-\mu }{2}(\ln \sigma ^* -2\ln 2) +C \end{aligned}$$

(5.20)

for each $0<\sigma <4$. By Theorem 5.3, we have for each $0<t<1$ and $u \in L^2(M)$

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le {\bar{C}} t^{-\frac{\mu }{4}} \Vert u\Vert _{2, {\bar{g}}}, \end{aligned}$$

(5.21)

where

$$\begin{aligned} {\bar{C}}=2^{\frac{\mu -n}{2}} (\sigma ^*)^{\frac{n-\mu }{4}}e^{\frac{\mu }{4}-\frac{3\sigma ^*}{16}\min \Psi ^-+\frac{1}{2}C}. \end{aligned}$$

(5.22)

Applying Theorem 5.4 and converting back to $g,$ we then arrive at (5.18).$\square$

Proof of Theorem 1.6 Applying Theorems 1.3 and 5.5 with $\Psi =\frac{R}{4},$ $\mu =n,$ and $\sigma ^*=4,$ we deduce

$$\begin{aligned} \Vert u\Vert ^2_{\frac{2n}{n-2}} \le c \left( \int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}} +\left( 1-\frac{\min _{t} R^-}{4}\right) \int _M u^2 \mathrm{{dvol}} \right) ,\qquad \end{aligned}$$

(5.23)

where $c=c(C, -\min _t R^-)$. By the maximum principle, we have $\min _t R^- \ge \min _{t=0} R^-$. Hence, we arrive at

$$\begin{aligned} \Vert u\Vert ^2_{\frac{2n}{n-2}} \le c \left( \int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}} +\left( 1-\frac{\min _{0} R^-}{4}\right) \int _M u^2 \mathrm{{dvol}} \right) \qquad \end{aligned}$$

(5.24)

with $c=c(C, -\min _0 R^-)$. Since $\lambda _0$ is nondecreasing along the Ricci flow [6], we obtain

$$\begin{aligned} \Vert u\Vert ^2_{\frac{2n}{n-2}} \le c\left( 1+\frac{1}{\lambda _0(g_0)} \left( 1-\frac{\min _{0} R^-}{4}\right) \right) \left( \int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}} \right) \nonumber \\ \end{aligned}$$

(5.25)

which leads to (1.14).$\square$

Proof of Theorem 1.5

This is similar to the above proof.$\square$

6 The $\kappa$-Noncollapsing Estimate

It is obvious that Theorems 1.7 and 1.8 follow from Theorem 1.5, Theorem 1.6, and the following result.

Theorem 6.1

Consider the Riemannian manifold $(M,g)$ for a given metric $g,$ such that for some $A>0$ and $B>0$ the Sobolev inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \mathrm{{dvol}} \right) ^{\frac{n-2}{n}} \le A\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{{dvol}} +B\int _M u^2 \mathrm{{dvol}} \end{aligned}$$

(6.1)

holds true for all $u \in W^{1,2}(M)$. Let $L>0$. Assume $R\le \frac{1}{r^2}$ on a geodesic ball $B(x, r)$ with $0<r\le L$. Then there holds

$$\begin{aligned} \mathrm{{vol}}(B(x, r)) \ge \left( \frac{1}{2^{n+3}A+2BL^2}\right) ^{\frac{n}{2}} r^n. \end{aligned}$$

(6.2)

Proof

Let $L>0$. Assume that $R\le \frac{1}{r^2}$ on a closed geodesic ball $B(x_0, r)$ with $0<r\le L,$ but the estimate (6.2) does not hold, i.e.,

$$\begin{aligned} \mathrm{{vol}}(B(x_0, r)) < \delta r^n, \end{aligned}$$

(6.3)

where

$$\begin{aligned} \delta =\left( \frac{1}{2^{n+3}A+2BL^2}\right) ^{\frac{n}{2}}. \end{aligned}$$

(6.4)

We derive a contradiction. Set ${\bar{g}}=\frac{1}{r^2} g$. Then we have for ${\bar{g}}$

$$\begin{aligned} \mathrm{{vol}}(B(x_0, 1)) < \delta \end{aligned}$$

(6.5)

and $R\le 1$ on $B(x_0, 1)$. Moreover, (6.1) leads to the following Sobolev inequality for ${\bar{g}}$

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \right) ^{\frac{n-2}{n}} \le A\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) +BL^2\int _M u^2, \end{aligned}$$

(6.6)

where the notation of the volume form is omitted. For $u \in C^{\infty }(M)$ with support contained in $B(x_0, 1),$ we then have

$$\begin{aligned} \left( \int _{B(x_0, 1)} |u|^{\frac{2n}{n-2}} \right) ^{\frac{n-2}{n}} \le A\int _{B(x_0, 1)} \left( |\nabla u|^2+\frac{1}{4}u^2\right) +BL^2\int _{B(x_0,1)} u^2. \end{aligned}$$

(6.7)

By Hölder’s inequality and (6.5), we have

$$\begin{aligned} \int _{B(x_0, 1)} u^2 \le \delta ^{\frac{2}{n}} \left( \int _{B(x_0, 1)} |u|^{\frac{2n}{n-2}}\right) ^{\frac{n-2}{n}}. \end{aligned}$$

(6.8)

Hence, we deduce

$$\begin{aligned} \left( \int _{B(x_0, 1)} |u|^{\frac{2n}{n-2}} \right) ^{\frac{n-2}{n}}\le & {} A\int _{B(x_0, 1)} |\nabla u|^2 +\left( \frac{A}{4}+BL^2\right) \delta ^{\frac{2}{n}} \left( \int _{B(x_0, 1)} |u|^{\frac{2n}{n-2}}\right) ^{\frac{n-2}{n}} \nonumber \\\le & {} A\int _{B(x_0, 1)} |\nabla u|^2 +\frac{1}{2}\left( \int _{B(x_0, 1)} |u|^{\frac{2n}{n-2}}\right) ^{\frac{n-2}{n}}. \end{aligned}$$

(6.9)

It follows that

$$\begin{aligned} \left( \int _{B(x_0, 1)} |u|^{\frac{2n}{n-2}} \right) ^{\frac{n-2}{n}}\le & {} 2A\int _{B(x_0, 1)} |\nabla u|^2. \end{aligned}$$

(6.10)

Next consider an arbitrary domain $\Omega \subset B(x_0, 1)$. For $u \in C^{\infty }(\Omega )$ with support contained in $\Omega ,$ we deduce from (6.10) via Hölder’s inequality

$$\begin{aligned} \int _{B(x_0, 1)} |u|^2 \le 2A\mathrm{{vol}}(\Omega )^{\frac{2}{n}}\int _{\Omega } |\nabla u|^2. \end{aligned}$$

(6.11)

Hence, we arrive at the following Faber-Krahn inequality:

$$\begin{aligned} \lambda _1(\Omega ) \mathrm{{vol}}(\Omega )^{\frac{2}{n}} \ge \frac{1}{2A}, \end{aligned}$$

(6.12)

where $\lambda _1(\Omega )$ denotes the first Dirichlet eigenvalue of $-\Delta$ on $\Omega$. By the proof of Proposition 2.4 in [1], we then infer

$$\begin{aligned} \mathrm{{vol}}(B(x,\rho )) \ge \left( \frac{1}{2^{n+3}A}\right) ^{\frac{n}{2}} \rho ^n \end{aligned}$$

(6.13)

for all $B(x, \rho ) \subset B(x_0, 1)$. Consequently, we have

$$\begin{aligned} \mathrm{{vol}}(B(x_0, 1)) \ge \left( \frac{1}{2^{n+3}A}\right) ^{\frac{n}{2}}, \end{aligned}$$

(6.14)

contradicting (6.5).

For the convenience of the reader, we reproduce here the arguments in the proof of Proposition 2.4 in [1]. Consider $B(x, \rho ) \subset B(x_0, 1)$. Set $u(y)=\rho -d(x, y)$. Then we obtain

$$\begin{aligned} \lambda _1(B(x, \rho )) \equiv \lambda _1( int \, B(x, \rho ))\le \frac{\mathrm{{vol}}(B(x, r))}{\int _{B(x, \rho /2)} u^2} \le \frac{4\mathrm{{vol}}(B(x, \rho ))}{\rho ^2 \mathrm{{vol}}(B(x, \rho /2))}. \end{aligned}$$

(6.15)

By (6.12), we then infer

$$\begin{aligned} \mathrm{{vol}}(B(x, \rho )) \ge \left( \frac{\rho ^2}{2A}\right) ^{\frac{n}{n+2}} 4^{-\frac{n}{n+2}} vol\left( B\left( x, \frac{\rho }{2}\right) \right) ^{\frac{n}{n+2}}. \end{aligned}$$

(6.16)

Iterating (6.16), we obtain

$$\begin{aligned} \mathrm{{vol}}(B(x, \rho )) \ge \left( \frac{\rho ^2}{2A}\right) ^{\sum _{l=1}^m \left( \frac{n}{n+2}\right) ^l} 4^{-\sum _{l=1}^m l\left( \frac{n}{n+2}\right) ^l} \mathrm{{vol}}\left( B\left( x, \frac{\rho }{2^m}\right) \right) ^{\left( \frac{n}{n+2}\right) ^m} \end{aligned}$$

(6.17)

for all natural numbers $m\ge 1$. Letting $m\rightarrow \infty ,$ we finally arrive at

$$\begin{aligned} \mathrm{{vol}}(B(x, \rho ))\ge & {} \left( \frac{\rho ^2}{2A}\right) ^{\sum _{l=1}^{\infty } \left( \frac{n}{n+2}\right) ^l} 4^{-\sum _{l=1}^{\infty } l\left( \frac{n}{n+2}\right) ^l}\nonumber \\= & {} \left( \frac{\rho ^2}{2A}\right) ^{\frac{n}{2}} 4^{-\frac{n(n+2)}{4}}=\left( \frac{1}{2^{n+3}A}\right) ^{\frac{n}{2}} \rho ^n. \end{aligned}$$

(6.18)

$\square$

Appendices

Appendix 1: The Logarithmic Sobolev Inequalities on the Euclidean Space

In this appendix, we review several versions of the logarithmic Sobolev inequality on the euclidean space for the purpose of presenting the background of the logarithmic Sobolev inequalities. These versions are equivalent to each other.

1.1 The Gaussian version

This is the original version of L. Gross.

Theorem 7.1

Let $u\in W^{1,2}_{loc}(\mathbf{R}^n)$ satisfy $\int _{\mathbf{R}^n} u^2d\mu =1,$ where

$$\begin{aligned} d\mu =(2\pi )^{-\frac{n}{2}}e^{-\frac{|x|^2}{2}}\mathrm{{d}}x. \end{aligned}$$

(7.1)

Then

$$\begin{aligned} \int _{\mathbf{R}^n} u^2 \ln u^2 \mathrm{{d}}\mu \le 2 \int _{\mathbf{R}^n} |\nabla u|^2 \mathrm{{d}}\mu . \end{aligned}$$

(7.2)

1.2 The straight (Euclidean volume element) version

Theorem 7.2

There holds

$$\begin{aligned} \int u^2 \ln u^2 \mathrm{{d}}x \le 2\int |\nabla u|^2 \mathrm{{d}}x, \end{aligned}$$

(7.3)

provided that $u \in W^{1,2}(\mathbf{R}^n)$ and $\int u^2 \mathrm{{d}}x=(2 \pi )^{n/2} e^n.$ Equivalently, for $\beta >0,$

$$\begin{aligned} \int u^2 \ln u^2 \mathrm{{d}}x \le 2\int |\nabla u|^2\mathrm{{d}}x + \beta \ln \beta -\frac{n}{2} \beta \ln (2 \pi e^2), \end{aligned}$$

(7.4)

provided that $u \in W^{1,2}(\mathbf{R}^n)$ and $\int u^2=\beta$.

1.3 The log gradient version

It appears to be stronger than the other versions because of the logarithm in front of the Dirichlet integral of $u$.

Theorem 7.3

There holds

$$\begin{aligned} \int u^2 \ln u^2 \mathrm{{d}}x \le \frac{n}{2} \ln \left[ \frac{2}{\pi n e } \int |\nabla u|^2 \mathrm{{d}}x \right] , \end{aligned}$$

(7.5)

provided that $u \in W^{1,2}(\mathbf{R}^n)$ and $\int u^2 \mathrm{{d}}x =1$.

1.4 The entropy version (as formulated in [6])

This version is intimately related to Perelman’s entropy functional ${\mathcal {W}}$. Indeed, it can be viewed as the motivation for ${\mathcal {W}}$.

Theorem 7.4

There holds

$$\begin{aligned} \int \left( \frac{1}{2}|\nabla f|^2+f-n\right) e^{-f}\mathrm{{d}}x \ge 0, \end{aligned}$$

(7.6)

provided that $f \in W^{1,2}_{loc}(\mathbf{R}^n)$ and $\int e^{-f}\mathrm{{d}}x=(2\pi )^{n/2}$.

Appendix 2: The Estimate for $e^{tH}$

In this appendix, we present the proof of Theorem 5.3. The global case $\sigma ^*=\infty$ of this theorem follows from Corollary 2.2.8 in [2]. On the other hand, the proof of this corollary in [2] can easily be extended to cover the local case $\sigma ^*<\infty ,$ as is done below. The global case is customarily phrased in terms of “ultracontractivity,” i.e., the logarithmic Sobolev inequality implies the ultracontractivity of $e^{-tH},$ see e.g., [2]. Note that the global case suffices for the main purpose of this paper. The local case should be useful for further applications.

Proof of Theorem 5.3

Part 1 We first assume $\Psi \ge 0,$ i.e., $\min \Psi ^-=0$. It follows from (5.9)

$$\begin{aligned} \int _M u^2 \ln u^2 \le \sigma Q(u)+ \beta (\sigma )\Vert u\Vert _2^2+\Vert u\Vert _2^2 \ln \Vert u\Vert _2^2 \end{aligned}$$

(8.1)

for all $u \in W^{1,2}(M)$. Here the notation of the volume form is omitted. Replacing $u$ by $|u|^{p/2}$ for $p>2$ and $u \in W^{1,2}(M) \cap L^{\infty }(M),$ we deduce

$$\begin{aligned} p \int _M |u|^p \ln |u| \le \sigma Q(|u|^{\frac{p}{2}})+ \beta (\sigma ) \Vert u\Vert _p^p +p \Vert u\Vert _p^p \ln \Vert u\Vert ^2_p. \end{aligned}$$

(8.2)

Since

$$\begin{aligned} Q(|u|^{\frac{p}{2}})=\frac{p^2}{4(p-1)} Q(|u|, |u|^{p-1}), \end{aligned}$$

(8.3)

we arrive at

$$\begin{aligned} \int _M |u|^p \ln |u| \le \frac{\sigma p}{4(p-1)} Q(|u|, |u|^{p-1})+ \frac{\beta (\sigma )}{p} \Vert u\Vert _p^p +\Vert u\Vert _p^p \ln \Vert u\Vert _p. \end{aligned}$$

(8.4)

By the nonincreasing property of $\beta ,$ we then infer, replacing $\sigma$ by $\frac{4(p-1)}{p} \sigma$

$$\begin{aligned} \int _M |u|^p \ln |u| \le \sigma Q(|u|, |u|^{p-1})+ \frac{\beta (\sigma )}{p} \Vert u\Vert _p^p +\Vert u\Vert _p^p \ln \Vert u\Vert _p \end{aligned}$$

(8.5)

for $\sigma \in (0, \frac{p}{4(p-1)}\sigma ^*]$.

Part 2 We continue with the assumption $\Psi \ge 0$. Consider $0<t \le \frac{1}{4} \sigma ^*$. Let $\sigma (p)$ be a nonnegative continuous function for $p\ge 2$ such that $\sigma (p) \in (0, \frac{p}{4(p-1)}\sigma ^*]$ for $p>2,$ which will be chosen later. Then we have

$$\begin{aligned} \int _M |u|^p \ln |u| \le \sigma (p) Q(|u|, |u|^{p-1})+ \Gamma (p) \Vert u\Vert _p^p +\Vert u\Vert _p^p \ln \Vert u\Vert _p \end{aligned}$$

(8.6)

for each $p>2$ and all $u \in W^{1,2}(M) \cap L^{\infty }(M),$ where $\Gamma (p)=\frac{\beta (\sigma (p))}{p}.$ Define the function $p(s)$ for $0\le s <t$ by

$$\begin{aligned} \frac{\mathrm{{d}}p}{\mathrm{{d}}s}=\frac{p}{\sigma (p)}, p(0)=2. \end{aligned}$$

(8.7)

Assume that

$$\begin{aligned} p(s) \rightarrow \infty \end{aligned}$$

(8.8)

as $s \rightarrow t$. We also define the function $N(s)$ for $0\le s<t$ by

$$\begin{aligned} \frac{\mathrm{{d}}N}{\mathrm{{d}}s}=\frac{\Gamma (p(s))}{\sigma (s)}, N(0)=0 \end{aligned}$$

(8.9)

and set

$$\begin{aligned} N^*=\lim \limits _{s\rightarrow t} \equiv \int _2^{\infty } \frac{\Gamma (p)}{p}\mathrm{{d}}p. \end{aligned}$$

(8.10)

For $u\in W^{1,2}(M)\cap L^{\infty }(M)$ with $u\ge 0,$ we set $u_s=e^{-sH}u$ for $0<s<t$. By the contraction properties of $e^{-sH},$ we have $u_s \in W^{1,2}(M) \cap L^{\infty }(M)$ for all $s$. If $\Psi \in C^{\infty }(M),$ we have for a fixed $q>2$

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}s}\Vert u_s\Vert _q^q =q \int _M \frac{\partial u_s}{\partial s} \cdot u_s^{q-1} = -q \int _M Hu_s \cdot u_s^{q-1}. \end{aligned}$$

(8.11)

Hence,

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}s}\Vert u_s\Vert _q^q =-qQ(u_s, u_s^{q-1}). \end{aligned}$$

(8.12)

In the general case $\Psi \in L^{\infty }(M),$ this formula follows from the spectral formula for $e^{-sH}$. Using this formula, we compute

$$\begin{aligned}&\frac{\mathrm{{d}}}{\mathrm{{d}}s}\ln (e^{-N(s)}\Vert u_s\Vert _{p(s)})=\frac{\mathrm{{d}}}{\mathrm{{d}}s}\left( -N(s)+\frac{1}{p(s)} \ln \Vert u_s\Vert _{p(s)}^{p(s)}\right) \nonumber \\&\quad =\frac{\Gamma }{\sigma }-\frac{1}{p^2}\frac{p}{\sigma } \ln \Vert u_s\Vert _p^p +\frac{1}{p} \Vert u_s\Vert _p^{-p} \left( -pQ(u_s, u_s^{p-1})+\frac{p}{\sigma } \int _M u_s^p \ln u_s\right) \nonumber \\&\quad =\frac{1}{\sigma }\Vert u_s\Vert ^{-p}_p \left( \int _M u_s^p \ln u_s - \sigma Q(u_s, u_s^{p-1})-\Gamma \Vert u_s\Vert _p^p-\Vert u_s\Vert _p^p \ln \Vert u_s\Vert _p \right) .\nonumber \\ \end{aligned}$$

(8.13)

By (8.6), this is nonpositive. Hence, $e^{-N(s)}\Vert u_s\Vert _{p(s)}$ is nonincreasing, which leads to

$$\begin{aligned} \Vert e^{-sH}u\Vert _{p(s)} \le e^{N(s)} \Vert f\Vert _2 \end{aligned}$$

(8.14)

for all $0\le s<t$. By the contraction properties, we have $\Vert e^{-tH}u\Vert _{p(s)} \le \Vert e^{-sH}u\Vert _{p(s)},$ whence

$$\begin{aligned} \Vert e^{-tH}u\Vert _{p(s)} \le e^{N(s)} \Vert f\Vert _2 \end{aligned}$$

(8.15)

for all $0\le s<t$. It follows that

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le e^{N^*} \Vert u\Vert _2. \end{aligned}$$

(8.16)

This estimate extends to $u\in L^2(M)$ with $u\ge 0$ by an approximation. For a general $u \in L^2(M),$ we use the pointwise inequality $|e^{-tH}u| \le e^{-tH}|f|$ (a consequence of the positivity preserving property) to deduce

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty }\le \Vert e^{-tH}|u|\Vert _{\infty } \le e^{N^*} \Vert u\Vert _2. \end{aligned}$$

(8.17)

Now we choose

$$\begin{aligned} \sigma (p)=\frac{2t}{p} \end{aligned}$$

(8.18)

for $p \ge 2$. Then $p(s)=\frac{2t}{t-s}$. One readily sees that $\sigma (p)\in (0, \frac{p}{4(p-1)} \sigma ^*]$ for $p>2$ and $p(s)\rightarrow \infty$ as $s \rightarrow t$. We have for this choice

$$\begin{aligned} N^*=\frac{1}{2t} \int _0^t \beta (\sigma )\mathrm{{d}}\sigma . \end{aligned}$$

(8.19)

Hence, we arrive at

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le e^{\tau (t)} \Vert u\Vert _2 \end{aligned}$$

(8.20)

for all $u \in L^2(M)$ and $0<t\le \frac{1}{4}\sigma ^*$.

Part 3 For a general $\Psi ,$ we consider ${\bar{\Psi }}=\Psi -\min \Psi ^-$ and denote the corresponding operator and quadratic form by ${\bar{H}}$ and ${\bar{Q}},$ respectively. We have by (5.9)

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{{dvol}} \le \sigma {\bar{Q}}(u)+ {\bar{\beta }}(\sigma ) \end{aligned}$$

(8.21)

for all $u \in L^2(M)$ with $\Vert u\Vert _2=1,$ where ${\bar{\beta }}(\sigma )=\beta (\sigma )+\sigma \min \Psi ^-$. We apply (8.20) to deduce for $0<t\le \frac{1}{4}\sigma ^*$ and $u \in L^2(M)$

$$\begin{aligned} \Vert e^{-t{\bar{H}}} u\Vert _{\infty } \le e^{\frac{1}{2t} \int _0^t {\bar{\beta }}(\sigma ) \mathrm{{d}}\sigma } \Vert u\Vert _2 = e^{\tau (t)+\frac{t}{4} \min \Psi ^-} \Vert u\Vert _2. \end{aligned}$$

(8.22)

The desired estimate (5.11) follows.

The estimate (5.12) follows from (5.11) in terms of duality, namely we have for $u, v \in L^2(M)$

$$\begin{aligned} \int _M v e^{-tH}u = \int _M u e^{-tH}v \le \Vert e^{-tH}v\Vert _{\infty } \Vert u\Vert _1 \le e^{\tau (t)-\frac{3t}{4} \min \Psi ^-}\Vert v\Vert _2 \Vert u\Vert _1.\qquad \end{aligned}$$

(8.23)

It follows that

$$\begin{aligned} \Vert e^{-tH}u\Vert _2 \le e^{\tau (t)-\frac{3t}{4} \min \Psi ^-}\Vert u\Vert _1 \end{aligned}$$

(8.24)

and then

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le e^{\tau (\frac{t}{2})-\frac{3t}{8} \min \Psi ^-}\Vert e^{-\frac{t}{2}H}u\Vert _2 \le e^{2\tau (\frac{t}{2})-\frac{3t}{4} \min \Psi ^-} \Vert u\Vert _1. \end{aligned}$$

(8.25)

By Lemma 5.1, we arrive at (5.12) for all $u \in L^1(M)$. The estimate (5.12) also follows from the arguments in Part 2 by choosing $\sigma (p)=\frac{t}{p}$ and $p(s)=\frac{t}{t-s}$.$\square$

Appendix 3: From the Estimate for $e^{-tH}$ to the Sobolev Inequality

In this appendix, we present the proof of Theorem 5.4. We also present a more general result Theorem 9.5, and its implication for the Ricci flow. Consider a compact Riemannian manifold $(M, g)$ of dimension $n \ge 1$ and $\Psi \in L^{\infty }(M)$ as in the set-up for Theorem 5.4. If $Q \ge 0,$ then we define the spectral square root $H^{\frac{1}{2}}$ of the operator $H=-\Delta +\Psi$ as follows. For $u=\sum _{i \ge 1} a_i \phi _i \in L^2(M),$ we set

$$\begin{aligned} H^{\frac{1}{2}}u=\sum _{i\ge 1} \lambda _i^{\frac{1}{2}} a_i \phi _i, \end{aligned}$$

(9.1)

whenever the series converges in $L^2(M)$.

Lemma 9.1

Assume $Q \ge 0$. Then $H^{\frac{1}{2}}$ is a bounded operator from $W^{1,2}(M)$ to $L^2(M)$. Indeed, there holds for all $u \in W^{1,2}(M)$

$$\begin{aligned} \Vert H^{\frac{1}{2}}u\Vert _2^2=Q(u). \end{aligned}$$

(9.2)

Proof

For $,$ there holds $Q(u)=<Hu, u>_2 = \sum _{i\ge 1} \lambda _i a_i^2.$ By approximation, we derive $Q(u)=\sum _{i \ge 1} \lambda _i a_i^2$ for all $u \in W^{1,2}(M)$. Now we have for $N \ge 1$

$$\begin{aligned} \left\| \sum _{1\le i \le N} \lambda _i^{\frac{1}{2}} a_i \phi _i \right\| ^2_2 =\sum _{1\le i \le N} \lambda _i a_i^2. \end{aligned}$$

(9.3)

Taking the limit as $N \rightarrow \infty ,$ we infer $\Vert H^{\frac{1}{2}}u\Vert ^2_2 = Q(u)$.$\square$

If $Q>0,$ i.e., the first eigenvalue of $H$ is positive, then the inverse $H^{-\frac{1}{2}}: L^2(M) \rightarrow W^{1,2}(M)$ of $H^{\frac{1}{2}}$ exists. We have $H^{-\frac{1}{2}}u=\sum _{i\ge 1} \lambda _i^{-\frac{1}{2}} a_i \phi _i$ for $u=\sum _{i\ge 1} a_i \phi _i \in L^2(M)$. More generally, we define $H^{-\frac{1}{2}}$ in the case $Q \ge 0$ by $H^{-\frac{1}{2}}u=\sum _{\lambda _i>0} \lambda _i^{-\frac{1}{2}} a_i \phi _i$ for $u=\sum _{i\ge 1} a_i \phi _i \in L^2(M)$.

Lemma 9.2

Assume $Q \ge 0$. We set $\phi _1^*=\phi _1$ if $\lambda _1=0$ and $\phi _1^*=0$ if $\lambda _1>0$. There holds

$$\begin{aligned} H^{-\frac{1}{2}} u =\Gamma (\frac{1}{2})^{-1} \int _0^{\infty } t^{-\frac{1}{2}} e^{-tH} u \mathrm{{d}}t \end{aligned}$$

(9.4)

for all $u \in L^2(M)$ with $u \perp \phi _1^*$. Moreover, if $u \in L^2(M)$ with $u \perp \phi _1^*$ satisfies $\Vert e^{-tH}u\Vert _{\infty } \le \phi (t)$ on an open interval $(a, b) \subset (0, \infty )$ for a nonnegative continuous function $\phi ,$ then there holds

$$\begin{aligned} \left\| \int _a^b t^{-\frac{1}{2}} e^{-tH} u dt\right\| _{\infty } \le \int _a^b t^{-\frac{1}{2}} \phi (t) \mathrm{{d}}t. \end{aligned}$$

(9.5)

Proof

For $u\in L^2(M)$ with $u \perp \phi _1^*,$ we write $u =\sum _{\lambda _i>0} a_i \phi _i,$ where the series converges in $L^2(M)$. We have $e^{-tH}u=\sum _{\lambda _i>0} e^{-\lambda _i t} a_i \phi _i$. We have

$$\begin{aligned} \sum _{\lambda _i>0} \int _0^{\infty } t^{-\frac{1}{2}} e^{-\lambda _i t} a_i \phi _i \mathrm{{d}}t =\Gamma \left( \frac{1}{2}\right) \sum _{\lambda _i>0} \lambda _i^{-\frac{1}{2}} a_i \phi _i=\Gamma \left( \frac{1}{2}\right) H^{-\frac{1}{2}}u. \end{aligned}$$

(9.6)

Hence, the formula (9.4) follows. Next we note that convergence in $L^2$ implies almost everywhere convergence. Moreover, if $u_k$ converges to $u$ almost everywhere, then $\Vert u\Vert _{\infty } \le \liminf \Vert u_k\Vert _{\infty }$. These two facts lead to (9.5).$\square$

Next we recall, for the sake of clarity and precise estimates, the Marcinkiewicz interpolation theorem [8] and the Riesz–Thorin interpolation theorem [8], which we formulate in the special case of the measure space $(M, \mu ),$ where $\mu$ denotes the Lebesgue measure associated with the volume element $\mathrm{{dvol}}$ of $g$.

Theorem 9.3

(Marcinkiewicz interpolation theorem) Let $L$ be an additive operator from $L^{\infty }(M)$ to the space of measurable functions on $M$. Let $1 \le p_0 \le q_0 \le \infty$ and $1 \le p_1 \le q_1 \le \infty$ with $q_0 \not =q_1$. Assume that $L$ is of weak type $(p_0, q_0)$ with constant $K_0$ and of weak type $(p_1, q_1)$ with constant $K_1,$ i.e.,

$$\begin{aligned} \mu (\{|L(u)| >\alpha \}) \le \left( K_0 \frac{\Vert u\Vert _{p_0}}{\alpha }\right) ^{q_0} \end{aligned}$$

(9.7)

and

$$\begin{aligned} \mu (\{|L(u)| >\alpha \}) \le \left( K_1 \frac{\Vert u\Vert _{p_1}}{\alpha }\right) ^{q_1} \end{aligned}$$

(9.8)

for all $u \in L^{\infty }(M)$. Then $L$ is of type $(p_t, q_t)$ on $L^{\infty }(M)$ with constant $K_t$ for each $0<t<1,$ i.e.,

$$\begin{aligned} \Vert L(u)\Vert _{q_t} \le K_t \Vert u\Vert _{p_t} \end{aligned}$$

(9.9)

for all $u \in L^{\infty }(M)$ and $\alpha >0,$ where

$$\begin{aligned}&\frac{1}{p_t}=\frac{1-t}{p_0}+\frac{t}{p_1}, \frac{1}{q_t}=\frac{1-t}{q_0}+ \frac{t}{q_1}, \end{aligned}$$

(9.10)

$$\begin{aligned}&K_t\le KK_0^{1-t}K_1^t, \end{aligned}$$

(9.11)

and $K=K(p_0, q_0, p_1, q_1, t)$ is bounded for $0<\epsilon \le t \le 1-\epsilon$ with each given $\epsilon >0,$ but tends to infinity as $t\rightarrow 0$ or $t \rightarrow 1$.

It follows that for each $0<t<1$, $L$ extends uniquely to an additive operator $L: L^{p_t}(M) \rightarrow L^{q_t}(M)$ with the bound (9.10).

This follows from [8, Theorem 5.2]. The space of simple functions is used in [8, Theorem 5.2] instead of $L^{\infty }(M)$. Moreover, $L$ is only assumed to be sublinear. Note that Theorem 9.3 holds both in the set-up of real-valued functions and the set-up of complex-valued functions.

Theorem 9.4

(Riesz–Thorin interpolation theorem) Let $L$ be a linear operator from $L^{\infty }_{{C}}(M),$ i.e., the complex-valued $L^{\infty }(M),$ to the space of complex-valued measurable functions on $M$. Let $1\le p_0, p_1, q_0, q_1 \le \infty$. Assume that $L$ is of type $(p_0, q_0)$ on $L^{\infty }_C(M)$ with constant $K_0,$ and of type $(p_1, q_1)$ on $L^{\infty }_C(M)$ with constant $K_1$. Then $L$ is of type $(p_t, q_t)$ on $L^{\infty }_C(M)$ with constant $K_t$ for each $0\le t \le 1,$ where $p_t$ and $q_t$ are given by (9.10) and

$$\begin{aligned} K_t \le K_0^{1-t} K_1^t. \end{aligned}$$

(9.12)

Consequently, for each $0\le t \le 1$, $L$ extends uniquely to a linear operator $L: L^{p_t}_C(M) \rightarrow L^{q_t}_C(M)$ with the bound

$$\begin{aligned} \Vert L(u)\Vert _{q_t} \le K_t \Vert u\Vert _{p_t} \end{aligned}$$

(9.13)

for all $u \in L^{p_t}_C(M),$ where $L^p_C(M)$ denotes the complex-valued $L^p(M)$.

If we replace the complex-valued spaces by real-valued spaces, then the same holds except that (9.12) is replaced by

$$\begin{aligned} K_t \le 2 K_0^{1-t} K_1^t. \end{aligned}$$

(9.14)

The bound (9.12) still holds in the set-up of real-valued functions, provided that $p_0 \le q_0, p_1 \le q_1$, or $T$ is a positive operator.

Now we are ready to prove Theorem 5.4. We present a more general result which implies Theorem 5.4.

Theorem 9.5

• (1) Let $\mu >1$. Assume that $\Psi \ge 0$ and for some $c>0$ the inequality

  $$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le c t^{-\frac{\mu }{4}} \Vert u\Vert _2 \end{aligned}$$

  (9.15)

  holds true for each $t>0$ and all $u\in L^2(M)$. Let $1<p<\mu$. Then there holds

  $$\begin{aligned} \Vert H^{-\frac{1}{2}}u\Vert _{\frac{\mu p}{\mu -p}} \le C(c, \mu , p) \Vert u\Vert _p \end{aligned}$$

  (9.16)

  for all $u \in L^p(M)$, where the positive constant $C(\mu , c, p)$ can be bounded from above in terms of upper bounds for $c,$ $\mu$, $\frac{1}{\mu -p}$ and $\frac{1}{p-1}$. Consequently, there holds

  $$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le C(c, \mu , p) \Vert H^{\frac{1}{2}}u\Vert _p \end{aligned}$$

  (9.17)

  for all $u \in W^{1,p}(M)$.

• (2) Let $\mu >1$. Assume that for some $c>0$ the inequality

  $$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le c t^{-\frac{\mu }{4}}\Vert u\Vert _2 \end{aligned}$$

  (9.18)

  holds true for each $0<t<1$ and all $u\in L^2(M)$. Set $H_0=H-\inf \Psi ^-+1$. Let $1<p<\mu$. Then there holds

  $$\begin{aligned} \Vert H_0^{-\frac{1}{2}}u\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu ,c, p) \Vert u\Vert _p \end{aligned}$$

  (9.19)

  for all $u \in L^p(M)$, where the positive constant $C(\mu ,c, p)$ has the same property as the $C(\mu , c, p)$ above. Consequently, there holds

  $$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu , c, p) \Vert H_0^{\frac{1}{2}}u\Vert _p \end{aligned}$$

  (9.20)

  for all $u \in W^{1,p}(M)$.

Proof

(1) For simplicity, we work in the set-up of real-valued functions. The case $p=2$ follows from [2, Theorem 2.4.2]. The proof of that theorem in [2] extends in a standard way to the general case of (9.16), so we follow it here. By the proof of Theorem 5.3 in Appendix 2, we have with $\tau (t)=\ln c_1-\frac{\mu }{4} \ln t$

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le e^{2 \tau (\frac{t}{2})}\Vert u\Vert _1 = \frac{2^{\frac{\mu }{2}}c^2}{t^{\frac{\mu }{2}}} \Vert u\Vert _1 \end{aligned}$$

(9.21)

for all $u \in L^1(M)$ and all $t>0$. On the other hand, we have by Lemma 5.2 $\Vert e^{-tH}u\Vert _{\infty } \le \Vert u\Vert _{\infty }$ for all $u \in L^{\infty }(M)$. By Theorem 9.4, we then have

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le \left( \frac{2^{\frac{\mu }{2}}c^2}{t^{\frac{\mu }{2}}}\right) ^{\frac{1}{p}} \Vert u\Vert _p \end{aligned}$$

(9.22)

for each $1 \le p \le \infty$ and all $u \in L^p(M)$.

Next we consider $1 \le p<\mu$ and set

$$\begin{aligned} \frac{1}{q}=\frac{1}{p}-\frac{1}{\mu }, \,\,\, \text{ i.e., } \, \, \, q=\frac{\mu p}{\mu -p}. \end{aligned}$$

(9.23)

Observe that (9.15) implies that the first eigenvalue $\lambda _1$ of $H$ is positive. Otherwise, since $Q \ge 0,$ $\lambda _1$ would be zero. Then $e^{-tH}\phi _1=\phi _1$ for all $t>0$. This contradicts (9.15). Thus (9.4) is valid for all $u \in L^2(M)$. We show that $H^{-\frac{1}{2}}:L^{\infty }(M) \rightarrow L^2(M)$ is of weak type $(p, q)$. For a given $T \in (0, \infty ),$ we write $H^{-\frac{1}{2}}(u)=G_{0, T}(u)+G_{T,\infty }(u),$ where

$$\begin{aligned} G_{a, b}(u)=\Gamma \left( \frac{1}{2}\right) ^{-1} \int _a^b t^{-\frac{1}{2}} e^{-tH}u\mathrm{{d}}t. \end{aligned}$$

(9.24)

We have by Lemma 9.2 and (9.22)

$$\begin{aligned} \Vert G_{T, \infty }(u)\Vert _{\infty }\le & {} \Gamma \left( \frac{1}{2}\right) ^{-1}\int _T^{\infty } t^{-\frac{1}{2}} \left( \frac{2^{\frac{\mu }{2}}c_1^2}{t^{\frac{\mu }{2}}}\right) ^{\frac{1}{p}} \Vert u\Vert _p \mathrm{{d}}t \nonumber \\= & {} c_1(\mu , p) T^{\frac{1}{2}-\frac{\mu }{2p}}\Vert u\Vert _p \end{aligned}$$

(9.25)

for all $u \in L^{p}(M),$ where

$$\begin{aligned} c_1(\mu , p)=\frac{2p}{\mu -p}\Gamma \left( \frac{1}{2}\right) ^{-1} \left( 2^{\frac{\mu }{2}}c^2\right) ^{\frac{1}{p}}. \end{aligned}$$

(9.26)

On the other hand, we have by Lemma 5.2

$$\begin{aligned} \Vert G_{0, T}(u)\Vert _{p} \le \Gamma \left( \frac{1}{2}\right) ^{-1} \int _0^T t^{-\frac{1}{2}} \Vert u\Vert _p\mathrm{{d}}t =2 \Gamma \left( \frac{1}{2}\right) ^{-1} T^{\frac{1}{2}} \Vert u\Vert _p \end{aligned}$$

(9.27)

for all $u \in L^p(M)$. Given $u \in L^{\infty }(M)$ and $\alpha >0,$ we define $T$ by

$$\begin{aligned} \frac{\alpha }{2}=c_1(\mu , p) \Vert u\Vert _p T^{\frac{1}{2}-\frac{\mu }{p}}. \end{aligned}$$

(9.28)

Then $\Vert G_{T, \infty }(u)\Vert _{\infty } \le \frac{\alpha }{2},$ and hence

$$\begin{aligned} \mu (\{|H^{-\frac{1}{2}}(u)|>\alpha \})\le & {} \mu \left( \left\{ |G_{0,T}(u)|>\frac{\alpha }{2}\right\} \right) \le \left( \frac{\alpha }{2}\right) ^{-p} \Vert G_{0, T}(u)\Vert _p^p \nonumber \\\le & {} \left( \frac{\alpha }{2}\right) ^{-p}\left( 2 \Gamma \left( \frac{1}{2}\right) ^{-1} T^{\frac{1}{2}}\right) ^p \Vert u\Vert _p^p \nonumber \\= & {} c_2(\mu , p)\left( \frac{\Vert u\Vert _p}{\alpha }\right) ^q, \end{aligned}$$

(9.29)

where

$$\begin{aligned} c_2(\mu , p)=2^{\frac{p(2\mu -p)}{\mu -p}}c_1(\mu , p)^{\frac{p^2}{\mu -p}} \Gamma \left( \frac{1}{2}\right) ^{-p}. \end{aligned}$$

(9.30)

It follows that $H^{-\frac{1}{2}}$ is of weak type $(p, q)$ with constant $c_2(\mu , p)^{1/q}$.

Given $1<p<\mu ,$ we set $\gamma =\max \{\frac{p}{p-1}, 2, \frac{2\mu -p}{\mu -p}\}+1,$ $p_0=\frac{\gamma -1}{\gamma }p$ and $p_1=\frac{\gamma -1}{\gamma -2}p$. Then $1<p_0<p_1<\mu$ and

$$\begin{aligned} \frac{1}{p_0}+\frac{1}{p_1}=\frac{2}{p},\, \, \, \frac{1}{q_0}+\frac{1}{q_1}=\frac{2}{q}, \end{aligned}$$

(9.31)

where ${1}/{q_0}={1}/{p_0}-{1}/{\mu }$ and ${1}/{q_1}={1}/{p_1}-{1}/{\mu },$ and $q$ is the same as before, i.e., $1/q=1/p-1/\mu$. Applying (9.29) and Theorem 9.3 with $t=\frac{1}{2},$ we then arrive at (9.16) with

$$\begin{aligned} C(\mu , c, p)=K\left( p_0, q_0, p_1, q_1, \frac{1}{2}\right) c_2(\mu , p_0)^{\frac{1}{2q_0}} c_2(\mu , p_1)^{\frac{1}{2q_1}}. \end{aligned}$$

(9.32)

The property of $C(\mu , c, p)$ is easy to see from this formula.

By [9], the operator $H^{\frac{1}{2}}$ is a pseudo-differential operator of order $1$. Since $M$ is compact, it follows that $H^{\frac{1}{2}}$ is a bounded operator from $W^{1,p}(M)$ into $L^p(M)$ for all $1<p<\infty$. (The special case $p=2$ is contained in Lemma 9.1). For $2 \le p <\mu$ (assuming $\mu >2$), we have $W^{1,p}(M) \subset W^{1,2}(M),$ and hence $H^{-\frac{1}{2}}H^{\frac{1}{2}}u=u$ by Lemma 9.1. Replacing $u$ in (9.16) by $H^{\frac{1}{2}}u$ for $u \in W^{1, p}(M),$ we then arrive at (9.17). For $1<p<\min \{2, \mu \},$ we can argue this way to arrive at (9.16) for $u \in C^{\infty }(M)$. By the boundedness of $H^{\frac{1}{2}}: W^{1,p}(M) \rightarrow L^p(M),$ we then arrive at (9.16) for all $u \in W^{1,p}(M)$ via approximation.

By [9], the operator $H^{-\frac{1}{2}}$ is a pseudo-differential operator of order $-1$. It follows that $H^{-\frac{1}{2}}$ is a bounded map from $L^p(M)$ into $W^{1,p}(M)$ for all $1<p<\infty$. It also follows that $H^{-\frac{1}{2}}: L^p(M) \rightarrow W^{1,p}(M)$ is the inverse of $H^{\frac{1}{2}}: W^{1,p}(M) \rightarrow L^p(M)$. Moreover, by approximation, the inequality (9.17) also implies the inequality (9.16). (2) For $0<t<1,$ we have for $u\in L^2(M)$

$$\begin{aligned} \Vert e^{-tH_0}u\Vert _{\infty } =e^{-t(1-\inf \Psi ^-) }\Vert e^{-tH}u\Vert _{\infty } \le c t^{-\frac{\mu }{4}} \Vert u\Vert _2. \end{aligned}$$

(9.33)

For $t\ge 1,$ we write $t=\frac{m}{2}+t_0$ for a natural number $m$ such that $1/2 \le t_0 <1$. Then we have for $u \in L^2(M)$

$$\begin{aligned} \Vert e^{-tH_0}u\Vert _{\infty }= & {} e^{-t(1-\inf \Psi ^-)} \Vert e^{-\frac{m}{2}tH} e^{-t_0H}u\Vert _{\infty } \le e^{-t}\Vert e^{-t_0H}u\Vert _{\infty } \nonumber \\\le & {} c e^{-t} t_0^{-\frac{\mu }{4}} \Vert u\Vert _2 \le c2^{\frac{\mu }{4}} e^{-t}\Vert u\Vert _2\nonumber \\\le & {} c2^{\frac{\mu }{4}}e^{-\frac{\mu }{4}} \left( \frac{\mu }{4}\right) ^{\frac{\mu }{4}} t^{-\frac{\mu }{4}} \Vert u\Vert _2. \end{aligned}$$

(9.34)

Hence, we can apply the result in (1) to arrive at the desired inequalities (9.19) and (9.20). (Note that by the above arguments they are equivalent to each other.)$\square$

Proof of Theorem 5.4

1) Let $u \in W^{1,2}(M)$. Applying (9.17) with $p=2,$ we arrive at

$$\begin{aligned} \Vert u\Vert _{\frac{2\mu }{\mu -2}} \le C(\mu , c, 2)\Vert H^{\frac{1}{2}}u\Vert _2. \end{aligned}$$

(9.35)

Combining this with (9.2), we then obtain the desired inequality. 2) This is similar to 1). Note that the quadratic form of $H-\inf \Psi ^-+1$ is $Q(u)+(1-\inf \Psi ^-) \Vert u\Vert _2^2$. $\square$

Combining Theorems 1.1, 1.3, 5.3 and 9.5, we obtain the following two results for the Ricci flow, which extend Theorems 1.5 and 1.6. Let $g=g(t)$ be a smooth solution of the Ricci flow on $M \times [0, T)$ as before.

Theorem 9.6

Assume that $R_{g_0} \ge 0$ and $\lambda _0(g_0)>0$. Let $1<p<n$. There is a positive constant $C$ depending only on the dimension $n$, a positive lower bound for $\lambda _0(g_0),$ a positive lower bound for $\mathrm{{vol}}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{n-p}$, such that for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \Vert \left( -\Delta +\frac{R}{4}\right) ^{\frac{1}{2}}u\Vert _p. \end{aligned}$$

(9.36)

Theorem 9.7

Assume $T<\infty$ and $1<p<n$. There is a positive constant $C$ depending only on the dimension $n$, a nonpositive lower bound for $R_{g_0}$, a positive lower bound for $\mathrm{{vol}}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $T$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{n-p}$, such that for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \Vert H_0^{\frac{1}{2}}u\Vert _p, \end{aligned}$$

(9.37)

where

$$\begin{aligned} H_0=-\Delta +\frac{R}{4}-\frac{\min R_{g_0}^-}{4}+1. \end{aligned}$$

(9.38)