A sharp lower bound on the mean curvature integral with critical power for integral varifolds

TL;DR

This paper establishes a precise lower bound on the mean curvature integral for integral varifolds at the critical power, advancing understanding of geometric measure theory and curvature estimates.

Contribution

It provides a sharp lower bound on the mean curvature integral with critical power for integral varifolds, a key result in geometric measure theory.

Findings

Sharp lower bound on mean curvature integral established

Results applicable to integral varifolds with critical integrability

Advances understanding of curvature behavior at critical power

Abstract

Part I Weakly differentiable functions Part II PDEs on varifolds 15 Sobolev spaces ... 75 16 Preliminaries ... 82 17 Maximum estimates ... 86 18 Second order flatness of Lebesgue spaces for integral varifolds with subcritical integrability of the mean curvature ... 94 19 Second order differentiability of the support of integral varifolds with critical integrability of the mean curvature ... 100 20 Harnack inequality ... 108 Part III The generalised Gauss map 21 Points of finite lower density ... 118 22 Points of infinite density ... 130 23 Area formula for the generalised Gauss map ... 137 24 A sharp lower bound on the mean curvature integral with critical power for integral varifolds ... 138 Appendix A Monotonicity identity ... 139 B An observation concerning the Lusin property ... 140