Regularity of mean curvature flow of graphs on Lie groups free up to step 2. (October 2015)

• Record Type: Journal Article
• Authors: Capogna, Luca; Citti, Giovanna; Manfredini, Maria
• Is Part Of: Nonlinear analysis. Volume 126(2015)
• Page Start: 437
• Page End: 450
• Publication Date: 2015-10
• Subjects: Mean curvature flow -- Sub-Riemannian geometry -- Carnot groups; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.05.008
• Languages: English
• ISSNs: 0362-546X

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 ϵ collapsing to a subRiemannian metric σ 0 as ϵ → 0 . We establish C k, 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.