Mean curvature flow in a Riemannian manifold endowed with a Killing vector field
type: Item
instance of: scholarly article
title: Mean curvature flow in a Riemannian manifold endowed with a Killing vector field
zbMATH Open document ID: 1459.53085
author: Liangjun Weng
published in: Pacific Journal of Mathematics
publication date: 2021-01-08T00:00:00Z
review text: Let \((N^{n+1},g)\) be an \((n+1)\)-dimensional Riemannian manifold endowed with a Killing vector field \(\nu\). Suppose that the distribution orthogonal to \(\nu\) is of constant rank and integrable. Let \(\mathbb{L}^n\) be a complete integral leaf of that distribution and let \(M\subset \mathbb{L}^n\) be a bounded domain with boundary \(\partial M \in C^3\). The Killing graph of a differentiable function \(u:\bar{M}\to \mathbb{R}\) is the hypersurface \(\Sigma_u \subset N\) parametrized by the map \(X(x)=\zeta(x,u(x))\), \(x\in \bar{M}\), where \[ \zeta :\bar{M}\times \mathbb{R} \to N \] is the flow generated by the Killing vector field \(\nu\) with inital values in \(N\). In this paper, the author studies the Killing graphs over a bounded regular domain \(M\) in an integral distribution orthogonal to a Killing vector field with prescribed variable contact angle. Under some condition between the geometry of the domain and the contact angle, the author shows that the solutions of the mean curvature flow of Killing graphs with capillarity type boundary condition converge to a translating solution.
reviewed by: Abdoul Salam Diallo
Mathematics Subject Classification ID: 53E10
Mathematics Subject Classification ID: 35J66
zbMATH DE Number: 7291160
zbMATH Keywords: Killing vector field
zbMATH Keywords: warped product manifold
zbMATH Keywords: mean curvature flow
zbMATH Keywords: uniform gradient estimate
zbMATH Keywords: variable contact angle
Wikidata QID: Q115230555
MaRDI profile type: Publication
full work available at URL: pjm.2020.308.435
OpenAlex ID: W3111898836
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DOI: 10.2140/PJM.2020.308.435
description: scientific article; zbMATH DE number 7291160