As on arXiv
Abstract
In this paper, we consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying a mean curvature flow in a Ricci flow background. One of them is a weighted extended version of the Gibbons-Hawking-York action on Riemannian metrics in compact manifolds with boundary. We compute its variational properties from which naturally arise boundary conditions to the analysis of its time-derivative under Perelman's modified extended Ricci flow. For instance, the boundary integrand term provides an extension of Hamilton's differential Harnack expression for mean curvature flows in Euclidean space. We also derive the evolution equations for both the second fundamental form and the mean curvature under mean curvature flow in an extended Ricci flow background. In the special case of gradient solitons to the extended Ricci flow, we discuss mean curvature solitons and establish a Huisken's monotonicity-type formula. We show how to construct a family of mean curvature solitons and establish a characterization of such a family. Also, we show how for constructing examples of mean curvature solitons in an extended Ricci flow background.
Comments: 24 pages. Suggestions and comments are welcome
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 2021, plus the canon.
-
A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background
Topics Boundaries & non-compact, Ricci solitons · Same authors José N. V. Gomes, Matheus Hudson · Shared terms “extended Ricci”, “monotonicity type”, “Huisken monotonicity”
-
Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow
Topics Harnack inequalities, Ricci solitons · Same authors José N. V. Gomes, Matheus Hudson · Shared terms “Harnack expression”, “Hamilton differential”, “monotonicity type”
-
Monotonicity of Perelman -Entropy of Mean Curvature Flow
Topics Harnack inequalities · Shared terms “Perelman”, “Hamilton”
-
Harnack Inequality for -Mean Curvature Flow
Topics Harnack inequalities · Shared terms “Euclidean space”, “Hamilton”
-
Log gradient estimates for heat type equations on manifolds
Topics Harnack inequalities · Shared terms “Hamilton”, “Perelman”
-
Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds
Topics Harnack inequalities · Shared terms “Perelman”
-
Evolution of Functionals Under Extended Ricci Flow
Topics Ricci flow · Shared terms “extended Ricci”, “time derivative”, “expression”
-
Parabolic frequency monotonicity for two nonlinear equations under Ricci flow
Topics Harnack inequalities
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.