Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

Mario Santilli

doi:10.1142/s1664360720500083

Abstract

In this paper we deal with a class of varieties of bounded mean curvature in the viscosity sense that has the remarkable property to contain the blow up sets of all sequences of varifolds whose mean curvatures are uniformly bounded and whose boundaries are uniformly bounded on compact sets. We investigate the second-order properties of these varieties, obtaining results that are new also in the varifold’s setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, a property that allows to prove a Coarea-type formula for their generalized Gauss map. Then we use this formula to extend a sharp geometric inequality of Almgren and the associated soap bubble theorem. As a consequence of the geometric inequality we obtain sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.

Highlights

The (m, h) sets can be roughly described as ”varieties with mean curvature bounded by h in the viscosity sense”
They were introduced by Brian White in [Whi16] to study the area-blow-up of sequences of submanifolds and they contain all m dimensional varifolds V such that δV ≤ h V, see [Whi16, 2.8]2
Similar notions have been considered in the theory of viscosity solutions of PDE’s; see [CC93], [Sav17] and [Sav18]

Summary

Introduction

It follows from a recent result of Schneider [Sch15] that a typical (in the sense of Baire category) compact convex hypersurface in Rn is C1 but does not possess the n−1 dimensional Lusin (N) condition The validity of this condition (which is new in the varifold case) is a consequence of the weak maximum principle, which is the defining property of (m, h) sets. This crucial quantitative estimate is obtained working directly on the projection of the contact set C of Γ, combining the Coarea formula 1.3 and the Barrier principle of White [Whi16, 7.1], and with no structural or smoothness assumptions at the touching points This argument originates from the approach to Almgren’s theorem developed in [Men12] and it is somewhat more direct than Almgren’s method, which instead uses the convex hull of Γ. The sharp geometric inequality for (m, h) sets readily implies sufficient conditions (see 4.4 and 4.5) to conclude that the area-blow-up set of certain sequences of varifolds is empty

Preliminaries

Area formula for the spherical image

Almgren’s sharp geometric inequality