Mean Curvature Flow of Spacelike Graphs

We prove the mean curvature flow of a spacelike graph in \((\Sigma_1\times \Sigma_2, g_1-g_2)\) of a map \(f:\Sigma_1\to \Sigma_2\) from a closed Riemannian manifold \((\Sigma_1,g_1)\) with \(Ricci_1> 0\) to a complete Riemannian manifold \((\Sigma_2,g_2)\) with bounded curvature tensor and deriv...

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Bibliographic Details
Published in:arXiv.org 2010-08
Main Authors: Li, Guanghan, Salavessa, Isabel M C
Format: Article
Language:English
Subjects:
Brownian motion
Curvature
Mathematical analysis
Riemann manifold
Tensors
Online Access:Get full text
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Summary:We prove the mean curvature flow of a spacelike graph in \((\Sigma_1\times \Sigma_2, g_1-g_2)\) of a map \(f:\Sigma_1\to \Sigma_2\) from a closed Riemannian manifold \((\Sigma_1,g_1)\) with \(Ricci_1> 0\) to a complete Riemannian manifold \((\Sigma_2,g_2)\) with bounded curvature tensor and derivatives, and with sectional curvatures satisfying \(K_2\leq K_1\), remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption \(K_2\leq K_1\), that if \(K_1>0\), or if \(Ricci_1>0\) and \(K_2\leq -c\), \(c>0\) constant, any map \(f:\Sigma_1\to \Sigma_2\) is trivially homotopic provided \(f^*g_20\), and \(\rho=+\infty\) in case \(K_2\leq 0\). This largely extends some known results for \(K_i\) constant and \(\Sigma_2\) compact, obtained using the Riemannian structure of \(\Sigma_1\times \Sigma_2\), and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.
ISSN:2331-8422
DOI:10.48550/arxiv.0804.0783