Existence of Integral $m$-Varifolds minimizing $\int |A|^p$ and $\int |H|^p$, $p>m$, in Riemannian Manifolds

TL;DR

This paper establishes the existence and partial regularity of m-dimensional varifolds that minimize curvature-based functionals in Riemannian manifolds, introducing new tools like monotonicity formulas and isoperimetric inequalities.

Contribution

It proves existence and partial regularity of curvature-minimizing varifolds in Riemannian manifolds, with new analytical tools for handling these variational problems.

Findings

Existence of minimizers for  and  functionals in Riemannian manifolds.

Development of monotonicity formulas involving  for varifolds.

Introduction of isoperimetric inequalities to control mass of varifolds.

Abstract

We prove existence and partial regularity of integral rectifiable $m$-dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$-dimensional manifold $(N,g)$, $2\leq m<n$ and $p>m$, under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in $\mathbb{R}^S$ involving $\int |H|^p$, to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.