Hamilton's gradient estimate for the heat kernel on complete manifolds

By Brett Kotschwar. 2007

66 citations recorded by the source corpus.

In this paper we extend a gradient estimate of R. Hamilton for positive solutions to the heat equation on closed manifolds to bounded positive solutions on complete, non-compact manifolds with Re ≥ -Kg. We accomplish this extension via a maximum principle of L. Karp and P. Li and a Berstein-type estimate on the gradient of the solution. An application of our result, together with the bounds of P. Li and S.T. Yau, yields an estimate on the gradient of the heat kernel for complete manifolds with non-negative Ricci curvature that is sharp in the order of t for the heat kernel on R".

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/545cf9e459b90c091286f7beb8c0315435f0b0fb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A formula for the time derivative of the entropic cost and applications (2019, unknown): “Then let us recall that under (H1) Hamilton’s gradient estimate is satisfied (see [21]): for any positive function u ∈ Lp ∩ L∞(m) for some p ∈ [1,∞) it holds t|∇ log Ptu|(2) ≤ (1 + 2κ−t) log ‖u‖L∞(m) Ptu ) , ∀t > 0 (2.”
Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below (2018, background): “erm is nonnegative, since vi is subharmonic. For the last term, we can do integration by parts to transform derivatives to the heat kernel and φ. By the estimate for the derivative of the heat kernel [27], we ﬁnd that on B(pi,6), the last term is bounded by Ψ(t). Therefore, we ﬁnd that on B(pi,6), (”
Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature (2014, methodology): “emannian manifold with Ricn,m V ≥ −K. Thirdly, we derive Hamilton’s Harnack inequality for ∆V operator. Setting V ≡ 0 in Theorem 1.11, we obtain the classical result of Hamilton [16]. Later Kotschwar [21] extended Hamilton’s gradient estimate to complete noncompact Riemannian manifold. Li [25] prove”
L2 curvature bounds on manifolds with bounded Ricci curvature (2016, background): “ery minor modiﬁcations of classical estimates which are better suite d to our purposes. Let us summarize the basic estimates on heat kernels, which follow from the results in [LiYau86], [SY], [Ha93], [Kot07]: Theorem 2.19 (Heat Kernel Estimates). Let (Mn,g,x) be a pointed Riemannian manifold with Vo”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “This type of estimate is well-known on Riemannian manifolds with non-negative Ricci curvature (see for instance [69]), but is new in our general framework.”