Evolution and monotonicity of a geometric constant under the Ricci flow

Let $(M,g(t))$ be a compact Riemannian manifold and the metric $g(t)$ evolve by the Ricci flow. In the paper we derive the evolution equation for a geometric constant $\lambda$ under the Ricci flow and the normalized Ricci flow, such that there exist positive solutions to the nonlinear equation\[-\Delta_{\phi} f + af \: \ln \, f + bRf = \lambda f \: \textrm{,}\]where $\Delta \phi$ is the Witten–Laplacian operator, $\phi \in C^\infty (M)$, $a$ and $b$ are both real constants, and $R$ is the scalar curvature with respect to the metric $g(t)$. As an application, we obtain the monotonicity of the geometric constant along the Ricci flow coupled to a heat equation for manifold $M$ with some Ricci curvature condition when $b \gt \frac{1}{4}$.

Keywords

eigenvalue, Perelman’s $\mu$-entropy, Witten–Laplacian operator, Ricci flow

2010 Mathematics Subject Classification

Primary 53C21, 53C44. Secondary 58C40.

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This work was supported by PRC grant NSFC 11771377, NSFC 11371310, NSFC 11801229, the Natural Science Foundation of Jiangsu Province BK20191435, the Foundation of Yangzhou University 2019CXJ002, and the Qing Lan Project.

Received 20 February 2020

Accepted 29 September 2020

Published 11 April 2021