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

Abstract

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 ǫ collapsing to a subRiemannian metric σ 0 as ǫ → 0. We establish C k,α estimates for this flow, that are uniform as ǫ → 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 [10] to the total variation flow. IntroductionThe mean curvature flow is the motion of a surface where each points is moving in the direction of the normal with speed equal to the mean curvature In the case where the evolution of graphs S t = {(x, u(x, t))} ⊂ R n × R is considered, then, provided enough regularity is assumed, the function u satisfies the equationGiven appropriate boundary/initial conditions, global in time solutions asymptotically converge to minimal graphs.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 so-called sub-Riemannian Hörmander type setting [22], [40]. In particular we will focus on a class of Lie groups endowed with a metric structure (G, σ 0 ) that arises as limit of collapsing left-invariant tame Riemannian structures (G, σ ǫ ).Our approach to the existence of global (in time) smooth solutions is based on a Riemannian approximation scheme. We study graph solutions of the mean curvature flow in the Riemannian spaces (G, σ ǫ ) where G is a group and σ ǫ is a family of Riemannian metrics that 'collapse' as ǫ → 0 to a sub-Riemannian metric σ 0 in G.

Regularity for subelliptic PDE through uniform estimates in multi-scale geometries

“…This is accomplished by using the uniform Gaussian bounds established in in the previous section. This result extends to Hörmander type operators the analogous assertion proved by Manfredini and the authors in [14] in the setting of Carnot Groups.…”

“…The main results of this section, which generalizes to the Hörmander vector fields setting our previous result with Manfredini in [14] is …”

“…We outline how the structure stability results, the stability of the Schauder estimates and of the Harnack inequalities can be used to prove regularity and long time existence theorems for solutions of the subRiemannian mean curvature flow and the total curvature flow of graphs over bounded sets in step 2 Carnot groups and even in some non-nilpotent Lie groups. This is part of the work developed by the authors jointly with Maria Manfredini in [14,17]. The notion of horizontal, or p-mean curvature has arisen in the last 10 years thanks to the work of many researchers.…”

“…The mean curvature flow and the total curvature flow arise in connection to gradient descent for the perimeter functional and as such can be used for both applications. Very little is known about both flows in the subRiemannian setting and as far as we know the results in [14,17] are the first to establish existence of long time smooth flows. For other contributions to this topics, from different points of view, we recall the recent work in [35,36].…”

“…In the following we will remind the reader of the definition and basic properties of p-admissible structures and sketch some of the main regularity results from the recent papers [1] and [18]. We will conclude the section with a sample application of these techniques to the global (in time) existence of solutions for a class of evolution equations that include the subRiemannian total variation flow [14].…”