Paper · Jul 6, 2017
Hamilton differential Harnack inequality and -entropy for Witten Laplacian on Riemannian manifolds
As on arXiv
Abstract
In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the -condition, where and are two constants. Moreover, we introduce the -entropy and prove the -entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the -condition and on compact manifolds equipped with -super Ricci flows.
- Submitted
- Jul 6, 2017
- Last revised
- Oct 10, 2017 (v2)
- Subjects
- math.DG
Comments: To appear in Journal of Functional Analysis. This paper is an improved version of a part of our previous preprint [14] (arxiv:1412.7034, version1 (22 December 2014) and version 2 (7 February 2016))
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.
-
On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows
Topics Harnack inequalities, Weak & singular flows · Same authors Songzi Li, Xiang-Dong Li · Shared terms “Witten Laplacian”, “super Ricci”, “Riemannian manifold”
-
-entropy, super Perelman Ricci flows and -Ricci solitons
Topics Harnack inequalities, Weak & singular flows · Same authors Songzi Li, Xiang-Dong Li · Shared terms “Witten Laplacian”, “complete Riemannian”, “heat equation”
-
-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds
Topics Weak & singular flows · Same authors Songzi Li, Xiang-Dong Li · Shared terms “Witten Laplacian”, “entropy formula”, “super Ricci”
-
On the Shannon entropy power on Riemannian manifolds and Ricci flow
Topics Weak & singular flows · Same authors Songzi Li, Xiang-Dong Li · Shared terms “Witten Laplacian”, “complete Riemannian”, “Riemannian manifold”
-
On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spaces
Topics Harnack inequalities, Weak & singular flows · Same author Xiang-Dong Li · Shared terms “super Ricci”, “entropy formula”, “Hamilton”
-
Harnack Inequality for -Mean Curvature Flow
Topics Harnack inequalities · Same author Xiang-Dong Li · Shared terms “Hamilton”
-
Differential Harnack inequalities via Concavity of the arrival time
Topics Harnack inequalities, Weak & singular flows · Shared terms “differential Harnack”, “Hamilton differential”
-
New differential Harnack inequalities for nonlinear heat equations
Topics Harnack inequalities · Shared terms “differential Harnack”, “heat equation”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6D.5Diffusions on Manifolds and Curvature
Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.
- 11B.2Ancient Solutions and the Harnack Inequality
Comparing curvature across space and time.
- 12A.6𝓛-Geodesics and the Variational Formulas
Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.