Harnack Estimates for Ricci Flow on a Warped Product
نویسندگان
چکیده
In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric (N × F, gN + fgF ) with (F, gF ) Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman’s differential Harnack inequality in [18].
منابع مشابه
Warped product and quasi-Einstein metrics
Warped products provide a rich class of physically significant geometric objects. Warped product construction is an important method to produce a new metric with a base manifold and a fibre. We construct compact base manifolds with a positive scalar curvature which do not admit any non-trivial quasi-Einstein warped product, and non compact complete base manifolds which do not admit any non-triv...
متن کاملDifferential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flow
In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman’s Harnack inequality for the fundamental solution of the conjugate heat equatio...
متن کاملSharp Differential Estimates of Li-Yau-Hamilton Type for Positive .p; p/-Forms on Kähler Manifolds
In this paper we study the heat equation (of Hodge Laplacian) deformation of .p; p/-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a .p; p/-form solution is preserved under such an invariant condition, we prove the sharp differential Harnack (in the sense of LiYau-Hamilton) estimates for the positive solutions of the Hodge Laplacian heat equa...
متن کاملOn an alternate proof of Hamilton’s matrix Harnack inequality for the Ricci flow
In [LY] a differential Harnack inequality was proved for solutions to the heat equation on a Riemannian manifold. Inspired by this result, Hamilton first proved trace and matrix Harnack inequalities for the Ricci flow on compact surfaces [H0] and then vastly generalized his own result to all higher dimensions for complete solutions of the Ricci flow with nonnegative curvature operator [ H2]. So...
متن کاملDifferential Harnack Inequalities on Riemannian Manifolds I : Linear Heat Equation
Abstract. In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with Ricci(M) ≥ −k, k ∈ R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type L...
متن کامل