Regularity of Mean Curvature Flow of Graphs on Lie Groups Free Up to Step 2

Authors

Luca Capogna, Worcester Polytechnic InstituteFollow
Giovanna Citti, Alma Mater Studiorum Università di Bologna
Maria Manfredini, Alma Mater Studiorum Università di Bologna

Document Type

Article

Publication Date

10-26-2015

Publication Title

Nonlinear Analysis, Theory, Methods and Applications

Abstract

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics σ ε collapsing to a subRiemannian metric σ0 as ε → 0. We establish C^kα estimates for this flow, that are uniform as ε → 0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in Capogna et al. (2013) to the total variation flow.

Keywords

Carnot groups, Mean curvature flow, Sub-Riemannian geometry

Volume

126

First Page

437

Last Page

450

DOI

10.1016/j.na.2015.05.008

ISSN

0362546X

Comments

Peer reviewed accepted manuscript.

Recommended Citation

Capogna, Luca; Citti, Giovanna; and Manfredini, Maria, "Regularity of Mean Curvature Flow of Graphs on Lie Groups Free Up to Step 2" (2015). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/130