Paper · Oct 23, 2023
Parabolic frequency monotonicity for two nonlinear equations under Ricci flow
As on arXiv
Abstract
In this paper, we consider the parabolic frequency for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds. We obtain the monotonicity of parabolic frequency for the solution of two nonlinear parabolic equations with bounded Ricci curvature, then we apply the parabolic frequency monotonicity to get some integral type Harnack inequalities and we use -K1 instead of the lower bound 0 of Ricci curvature from Theorem 4.3 in 16, where K1 is any positive constant.
Comments: 14 pages. accepted by Acta Mathematica Scientia
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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.
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.