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

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 $(N,g)$, $2\le m<n$ 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.

33 pages; this second submission corresponds to the published version of the paper, minor typos are fixed

References in corpus (3)

• Second order rectifiability of integral varifolds of locally bounded first variation
• The Conformal Willmore Functional: a Perturbative Approach
• Some Applications of the Isoperimetric Inequality for Integral Varifolds

Cited by in corpus (7)

• Willmore Spheres in Compact Riemannian Manifolds
• Weakly differentiable functions on varifolds
• Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds
• Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities
• Immersions with bounded second fundamental form
• Quantitative Estimates on the Singular Set of Minimal Hypersurfaces with Bounded Index
• Integral decompositions of varifolds