Minimal Graphs and Graphical Mean Curvature Flow in M × \mathbb R

2013/11/14 by Matthew McGonagle, Ling Xiao, McGonagle, Matthew +1
Computer Science · Mathematics · #53C21 #58J05 #58J32 #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1311.3699

openalex publication_date 2013/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the problem of finding minimal graphs in Mⁿ×\mathbb R with general boundary conditions using a variational approach. We look at so called generalized solutions of the Dirichlet Problem that minimize a functional adapted from the area functional. We construct barriers to show that for certain conditions on our boundary data, ϕ(x), the solutions obtain the boundary data ϕ(x). Following Oliker-Ural'tseva we also consider solutions u^ε of a perturbed mean curvature flow for ε> 0. We show that there are subsequences ε_i where u^ε_i converges to a function u satisfying the mean curvature flow, and subsequences u(⋅, t_i) converge to a generalized solution u of the Dirichlet problem. Furthermore, u depends only on the choice of sequence ε_i.

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