On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

Songzi Li; Xiang-Dong Li

Paper · Jun 16, 2017

On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

Songzi Li, Xiang-Dong Li

math.DGarXiv:1706.05304

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Abstract

In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation $\partial_{t} u = Lu$ associated with the time dependent Witten Laplacian on complete Riemannian manifolds equipped with a variant of the $(K, m)$ -super Perelman Ricci flows and the $K$ -super Perelman Ricci flows.

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Submitted: Jun 16, 2017
Subjects: math.DG

Comments: To appear in the Special Issue of the Asian Journal of Mathematics in honor of Prof. N. Mok's 60th birthday

Ricci flow Harnack inequalities Weak & singular flows

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

6D.5
Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation
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.2
Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities
Comparing curvature across space and time.
12A.6
𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing
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.

[1] 6D.5
Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation
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.

[2] 11B.2
Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities
Comparing curvature across space and time.

[3] 12A.6
𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing
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.

On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

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