Regularity of mean curvature flow of graphs on Lie groups free up to step 2

Luca Capogna,Giovanna Citti,Maria Manfredini

doi:10.1016/j.na.2015.05.008

Luca Capogna, Giovanna Citti + Show 1 more

Open Access

https://doi.org/10.1016/j.na.2015.05.008

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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 σϵ collapsing to a subRiemannian metric σ0 as ϵ→0. We establish Ck,α 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.

Highlights

The mean curvature flow is the motion of a surface where each point is moving in the direction of the normal with speed equal to the mean curvature
In this paper we study long time existence of graph solutions of the mean curvature flow in a special class of degenerate Riemannian ambient spaces: The setting of sub-Riemannian manifolds [22], [40]
In particular we will focus on a class of Lie groups endowed with a metric structure (G, σ0) that arises as limit of collapsing left-invariant Riemannian structures (G, σ )

Summary

Introduction

The mean curvature flow is the motion of a surface where each point is moving in the direction of the normal with speed equal to the mean curvature. In this paper we study long time existence of graph solutions of the mean curvature flow in a special class of degenerate Riemannian ambient spaces: The setting of sub-Riemannian manifolds [22], [40]. Xm be a bracket generating family of smooth vector fields, free up to step 2, denote by X1, ..., Xn its completion to a basis of of the tangent bundle T G and set for every u ∈ Rn,. Under the assumptions of the Theorem 1.2, as → 0 the solutions u converge uniformly (with all its derivatives) on compact subsets of Q to the unique, smooth solution u0 ∈ C∞(Ω × (0, ∞)) ∩ L∞((0, ∞), C1(Ω )) of the sub-Riemannian mean curvature flow (1.5) in Ω × (0, ∞) with initial data φ. Regularity of minimal surfaces in the special case of Heisenberg group has been investigated in [23, 36, 16, 15, 8, 9, 18, 33, 37, 39]

Structure stability in the Riemannian limit

Gradient estimates