Gradient estimates for the heat equation under the Ricci flow

Abstract

The paper considers a manifold $M$ evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on $M$. Among other results, we prove Li-Yau-type inequalities in this context. We consider both the case where $M$ is a complete manifold without boundary and the case where $M$ is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on $M$.

21 pages, 2 figures

References in corpus (2)

• Ricci Flow and the Poincare Conjecture
• Stability of the Ricci Yang-Mills flow at Einstein Yang-Mills metrics

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• Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures
• Elliptic gradient estimates for a nonlinear heat equation and applications
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• Hessian estimates for the conjugate heat equation coupled with the Ricci flow
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• New differential Harnack inequalities for nonlinear heat equations
• On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows
• Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry
• Exponential Convergence of Parabolic Optimal Transport on Bounded Domains
• Local derivative estimates for the heat equation coupled to the Ricci flow
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• The Dirichlet Problem for the Prescribed Ricci Curvature Equation on Cohomogeneity One Manifolds
• A Probabilistic Approach for Gradient Inequalities on Time-Inhomogeneous Manifolds
• Hamilton type entropy formula along the Ricci flow on surfaces with boundary
• Aronson-Bénilan estimates for the fast diffusion equation under the Ricci flow
• Some new gradient estimates for two nonlinear parabolic equations under Ricci flow