Gauss Maps of the Mean Curvature Flow

2002/09/16 by Mu-Tao Wang, Mu‐Tao Wang, Wang, Mu-Tao
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.math/0209202

final version, to appear in Mathematical Research Letter

openalex publication_date 2002/09/16 · arxiv created 2003/03/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F:Σⁿ × [0,T)→ \R^n+m be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps γ:(Σⁿ, g_t)→ G(n, m) form a harmonic heat flow with respect to the time-dependent induced metric g_t. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on G(n,m) produces a subsolution of the nonlinear heat equation on (Σ, g_t). We also show the condition that the image of the Gauss map lies in a totally geodesic submanifold of G(n, m) is preserved by the mean curvature flow. Since the space of Lagrangian subspaces is totally geodesic in G(n,n), this gives an alternative proof that any Lagrangian submanifold remains Lagrangian along the mean curvature flow.

Citations

• A canonical way to deform a Lagrangian submanifold

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