Calculus of Variations and Geometric Measure Theory

D. Inauen - A. Marchese

Quantitative minimality of strictly stable minimal submanifolds in a small flat neighbourhood

created by marchese on 28 Aug 2017
modified on 08 Nov 2018


Published Paper

Inserted: 28 aug 2017
Last Updated: 8 nov 2018

Journal: J. Funct. Anal.
Year: 2017


In this paper we extend the results of "A strong minimax property of nondegenerate minimal submanifolds" by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of "A selection principle for the sharp quantitative isoperimetric inequality" by Cicalese and Leonardi, which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands.

Keywords: Geometric measure theory, minimal surfaces, Integral currents