Lectures on mean curvature flows in higher codimensions

Abstract

Mean curvature flows of hypersurfaces have been extensively studied and there are various different approaches and many beautiful results. However, relatively little is known about mean curvature flows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then introduces mean curvature flows in general dimensions and co-dimensions. The related techniques in the so called "blow-up" analysis are also discussed. At the end, we present some global existence and convergence results for mean curvature flows of two-dimensional surfaces in four-dimensional ambient spaces.

19 pages

References in corpus (2)

• A canonical way to deform a Lagrangian submanifold
• A convergence result of the Lagrangian mean curvature flow

Cited by in corpus (6)

• Mean curvature flow of arbitrary codimension in spheres and sharp differentiable sphere theorem
• Mean Curvature Flow of Higher Codimension in Hyperbolic Spaces
• Surfaces pinched by normal curvature for mean curvature flow in space forms
• Anisotropic flow of convex hypersurfaces by the square root of the scalar curvature
• On the mean curvature flow of submanifolds in the standard Gaussian space ${}^{†}$
• A penalized Allen-Cahn equation for the mean curvature flow of thin structures