# The spherical image of singular varieties of bounded mean curvature

## TL;DR

This paper extends geometric measure theory to singular varieties with bounded mean curvature, introducing new second-order properties, a generalized second fundamental form, and extending classical inequalities to these complex structures.

## Contribution

It introduces a new second fundamental form for varifolds with bounded mean curvature and extends classical geometric inequalities to singular varieties in the viscosity sense.

## Key findings

- Generalized normal bundle satisfies a Lusin (N) condition
- Extension of the Coarea formula for the Gauss map
- Characterization of equality cases in Almgren's inequality

## Abstract

In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part 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, which allows to extend the classical Coarea formula for the Gauss map of smooth varieties, and to introduce for all integral varifolds of bounded mean curvature a natural definition of second fundamental form, whose trace equals the generalized varifold mean curvature. In the second part, we use this machinery to extend a sharp geometric inequality of Almgren to all compact varieties of bounded mean curvature in the viscosity sense and we characterize the equality case. As a consequence we formulate sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.