Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension
By Mu-Tao Wang. 2001
151 citations recorded by the source corpus.
Abstract.Let f:Σ1↦Σ2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in Σ1×Σ2 by the mean curvature flow. Under suitable conditions on the curvature of Σ1 and Σ2 and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map ft and ft converges to a constant map as t approaches infinity. This also provides a regularity estimate for Lipschitz initial data.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/31837c284200e23890eda7f27c10d689ed44dff5
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, background): “In addition, the result can fail, if the ambient space is a Riemannian manifold since in some cases one gets long-time existence and convergence (for example in [38, 75, 76, 78, 84, 87]).”
- Geometric flows with rough initial data (2009, background): “Moreover, for M = Σ1×Σ1, where Σ1 and Σ2 are compact manifolds of constant curvature, and initial maps f0 : Σ1 → Σ2 which are Lipschitz with small Lipschitz norm, the mean curvature flow has been studied by Wang [24] (see also [23]).”
- On graphic Bernstein type results in higher codimension (2002, background): “Thewell-known Bernstein theorem assertsanycom pletem inim alsurfacethat can bewritten asthegraph ofa function on R 2 m ustbea plane.”
- Mean curvature flows and isotopy of maps between spheres (2003, methodology): “A crucial observation in [15] is that ∗ is a monotone quantity under the mean curvature flow when ∗ > 1/ √ 2.”