A priori bounds for geodesic diameter. Part II. Fine connectedness properties of varifolds

TL;DR

This paper introduces a new connectedness property called indecomposability for varifolds with integrable first variation, which aids in deriving geometric consequences in variational problems involving sets, chains, and immersions.

Contribution

It defines indecomposability for varifolds, explores its properties, and demonstrates its inheritance and geometric implications in variational contexts.

Findings

Indecomposability is weaker than indecomposability but still preserves connectedness properties.

The paper develops concepts of generalized bounded variation and partitions for varifolds.

Substantial geometric consequences are derived from the indecomposability property.

Abstract

For varifolds whose first variation is representable by integration, we introduce the notion of indecomposability with respect to locally Lipschitzian real valued functions. Unlike indecomposability, this weaker connectedness property is inherited by varifolds associated with solutions to geometric variational problems phrased in terms of sets, $G$ chains, and immersions; yet it is strong enough for the subsequent deduction of substantial geometric consequences therefrom. Our present study is based on several further concepts for varifolds put forward in this paper: real valued functions of generalised bounded variation thereon, partitions thereof in general, partition thereof along a real valued generalised weakly differentiable function in particular, and local finiteness of decompositions.