Abstract
In this paper, we establish the first variation formula of the lowest constant \(\lambda_{a}^{b}(g)\) along the Ricci flow and the normalized Ricci flow, such that to the following nonlinear equation there exist positive solutions:
with \(\int_{M}u^{2}\,dv=1\), where a is a real constant. In particular, the results proved in this paper generalize partial results in Cao (Proc. Am. Math. Soc. 136:4075-4078, 2008) and Li (Math. Ann. 338:927-946, 2007).
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1 Introduction
Let \((M, g)\) be an n-dimensional compact Riemannian manifold. In [3], Perelman introduced the functional
and proved that the \(\mathcal{F}\)-functional is nondecreasing under the Ricci flow coupled to a backward heat-type equation
where R is the scalar curvature depending on the metric g. More precisely, they proved that under the system (1.2),
If we define
where the infimum is taken over all smooth functions f which satisfy
then the nondecreasing of the \(\mathcal{F}\)-functional implies the nondecreasing of \(\lambda(g)\). In particular, \(\lambda(g)\) defined in (1.4) is the lowest eigenvalue of the operator
In [4], Cao considered the eigenvalues of the operator \(-\Delta+\frac{R}{2}\) on manifolds with nonnegative curvature operator and showed that the eigenvalues are nondecreasing along the Ricci flow. Using the same technique, Li [2] also obtained the same monotonicity of the first eigenvalue of the operator \(-\Delta+\frac{R}{2}\) by removing the assumption on a nonnegative curvature operator.
Later, Cao [1] proved the first eigenvalues of the operator \(-\Delta+bR\) with the constant \(b\geq1/4\) are nondecreasing along the Ricci flow. That is, they assume \(u=u(x, t)\) is the corresponding positive eigenfunction of \(\lambda(t)\):
with \(\int_{M}u^{2}\,dv=1\), then
by letting \(f=-2\log u\). Multiplying both sides of (1.7) with u and integrating on M, we see that the first eigenvalue given in (1.7) satisfies
where
In particular,
where
if we let \(f=-2\log u\). It is easy to see from (1.11) that the nondecreasing of the \(\tilde{\mathcal{F}}^{b}\)-functional is equivalent to the nondecreasing of \(\lambda(t)\).
In this paper, we consider the monotonicity along the Ricci flow of lowest constant \(\lambda_{a}^{b}(g)\) such that to the following nonlinear equation there exist positive solutions:
with
where a is a real constant. In particular, (1.7) can be seen a special case of (1.12) when \(a=0\). For the lowest constant \(\lambda_{a}^{b}(g)\) such that to the nonlinear equation (1.12) there exist positive solutions, we prove the following.
Theorem 1.1
Let \(g(t)\), \(t\in[0,T)\) be a solution to the Ricci flow
on a compact Riemannian manifold M. Then for \(b\geq\frac{1}{4}\), the lowest constant \(\lambda_{a}^{b}(g)\) such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies
where \(f=-2\log u\).
For the normalized Ricci flow, we can obtain the following.
Theorem 1.2
Let \(g(t)\), \(t\in[0,T)\) be a solution to the normalized Ricci flow
on a compact Riemannian manifold M, where \(r=(\int_{M}R \,dv)/(\int_{M} \,dv)\) is the average scalar curvature. Then the lowest constant \(\lambda_{a}^{b}(g)\) such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies
where \(f=-2\log u\) and \(\lambda^{b}\) is the lowest eigenvalue of (1.7).
In particular, when \(n=2\), we have \(R_{ij}=\frac{R}{2}g_{ij}\) and the normalized Ricci flow (1.16) becomes \(\frac{\partial}{\partial t}g_{ij}=-(R-r)g_{ij}\). Hence, \(\frac{d}{d t}r=0\), which implies that r is a constant (or see p.455 in [5] for an alternative proof). Then from the estimate (1.17), we obtain the following.
Theorem 1.3
Let \(g(t)\), \(t\in[0,T)\) be a solution to the normalized Ricci flow (1.16) on a compact surface \(M^{2}\). Then for \(b\geq\frac{1}{4}\), the lowest constant \(\lambda_{a}^{b}(g)\) such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies
where \(f=-2\log u\) and \(\lambda^{b}\) is the lowest eigenvalue of (1.7).
Remark 1.1
In particular, when \(a=0\), our estimate (1.15) reduces to Theorem 1.5 of Cao in [1] and the estimate (1.18) reduces to the Corollary 2.4 of Cao in [1], respectively.
On the other hand, under the transformation \(f=-2\log u=-\log v\) with \(u^{2}=v\), equation (1.2) becomes
In particular, the second equation in (1.19) is exactly the conjugate heat equation introduced by Perelman. In [6], Cao and Zhang obtained differential Harnack inequalities for positive solutions of the nonlinear parabolic equation of the type \(v_{t}=\Delta v-v\log v+Rv\). Extending the second equation in (1.19) to the following nonlinear version:
Guo and Ishida [7, 8] studied Harnack inequalities for positive solutions of equation (1.20) on a compact Riemannian manifold with a family of \(g(t)\) evolving by a geometric flow \(\frac{\partial}{\partial t}g_{ij}=-2S_{ij}\), where \(S_{ij}\) is a family of smooth symmetric two-tensor and \(S=g^{ij}S_{ij}\). Clearly, there is a one-to-one relation for the following two equations:
under \(f=-\log v\). Therefore, a natural problem is to consider the monotonicity of
under the Ricci flow coupled to a nonlinear backward heat-type equation
where c, d are two real constants.
For the functional \(\overline{\mathcal{F}}^{c}_{d}(g,f)\), we derive the following monotonicity formula.
Theorem 1.4
Let \(g(t)\), \(t\in[0,T)\) be a solution to the Ricci flow (1.14) on a compact Riemannian manifold M. Then all functionals \(\overline{\mathcal{F}}^{c}_{d}(g,f)\) defined by (1.22) under the system (1.23) satisfy
In particular, if \(R(t)\geq0\) for all t and \(a\geq0\), \(k\geq1\), then \(\frac{d}{d t}\overline{\mathcal{F}}_{\frac{nak}{8}}^{k}(g,f)\geq0\).
Remark 1.2
Choosing \(a=0\) in (1.24), we obtain Theorem 4.2 of Li in [2].
2 Proof of Theorems 1.1 and 1.2
Proof of Theorems 1.1
Let u be a positive solution to the following nonlinear elliptic equation:
Multiplying both sides of (2.1) with u and integrating on M, we have
If the metric \(g(t)\) evolves by (1.14), we have \(\frac{\partial}{\partial t}\,dv=-R \,dv\). It follows from (2.2) that
Applying
and
into (2.3) yields
where the last equality used
from (1.13). Noticing \(R_{t}=\Delta R+2|R_{ij}|^{2}\) for the Ricci flow, hence from (2.6) we have
Taking a transformation \(f=-2\log u\), which is equivalent to \(u^{2}=e^{-f}\), then
Thus, (2.8) can be written as
Using the second Bianchi identity \(R_{,i}=2R_{ij,}{}^{j}\) again, we have
Therefore, inserting (2.11) into (2.10) yields
Integrating by parts again, one has
and
where the last equality in (2.14) was used with
By virtue of (2.14), subtracting (2.13), we obtain
It follows from (2.13) and (2.14) that
and the desired estimate (1.15) is achieved. □
Proof of Theorem 1.2
If the metric \(g(t)\) evolves by (1.16), we have \(\frac{\partial}{\partial t}\,dv=-(R-r)\,dv\). It follows from (2.2) that
Applying (2.4) and
to (2.18) yields
Noticing \(R_{t}=\Delta R+2|R_{ij}|^{2}-\frac{2r}{n}R\) for the normalized Ricci flow, we obtain from (2.20)
Using (2.9), then (2.21) can be written as
By virtue of a similar computation, we can obtain
which gives
Then the desired estimate (1.17) is attained. □
3 Proof of Theorem 1.4
Under the following coupled system (1.23), by a direct computation, we have the following:
Thus, we have
and
By virtue of the Bochner formula with respect to the f-Laplacian, we have
and hence
Therefore, (3.6) becomes
Therefore, from (3.4) and (3.8), we obtain
Noticing (3.5) tells us that
Thus, (3.9) can be written as
Since (3.4) holds, we have
which gives
Therefore, we have
and the desired estimate (1.24) is obtained.
4 Conclusions
We establish the first variation formula of the lowest constant \(\lambda_{a}^{b}(g)\) along the Ricci flow and the normalized Ricci flow, such that to the following nonlinear equation there exist positive solutions:
with \(\int_{M}u^{2}\,dv=1\), where a is a real constant. Equation (4.1) can be seen as a nonlinear version of eigenvalue problem of the operator \(-\Delta u+bR\). In particular, when \(a=0\), our estimate (1.15) in Theorem 1.1 reduces to Theorem 1.5 of Cao in [1] and the estimate (1.18) in Theorem 1.3 reduces to the Corollary 2.4 of Cao in [1], respectively.
On the other hand, we obtained the first variation formula (1.24) of the functional
under the Ricci flow coupled to a nonlinear backward heat-type equation
where \(c,d\) are two real constants. In particular, when \(a=0\) in (1.24), we obtain Theorem 4.2 of Li in [2].
References
Cao, X-D: First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc. 136, 4075-4078 (2008)
Li, J-F: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338, 927-946 (2007)
Perelman, G: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
Cao, X-D: Eigenvalues of \((-\Delta+\frac{R}{2})\) on manifolds with nonnegative curvature operator. Math. Ann. 337, 435-441 (2007)
Cao, X-D, Hou, S, Ling, J: Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann. 354, 451-463 (2012)
Cao, X-D, Zhang, Z: Differential Harnack estimates for parabolic equations. In: Complex and Differential Geometry. Springer Proc. Math., vol. 8, pp. 87-98 (2011)
Guo, H, Ishida, M: Harnack estimates for nonlinear backward heat equations in geometric flows. J. Funct. Anal. 267, 2638-2662 (2014)
Guo, H, Ishida, M: Harnack estimates for nonlinear heat equations with potentials in geometric flows. Manuscr. Math. 148, 471-484 (2015)
Acknowledgements
The research of the first author is supported by NSFC (No. 11371018, 11171091), Henan Provincial Core Teacher (No. 2013GGJS-057) and IRTSTHN (14IRTSTHN023). The research of the second author is partially supported by NSFC (No. 11401179) and Henan Provincial Education department (No. 14B110017).
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Huang, G., Li, Z. Evolution of a geometric constant along the Ricci flow. J Inequal Appl 2016, 53 (2016). https://doi.org/10.1186/s13660-016-1003-6
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DOI: https://doi.org/10.1186/s13660-016-1003-6