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

2022/09/13 by Ulrich Menne, Menne, Ulrich, Christian Scharrer +1 · 1 citation
Engineering · Mathematics · #3D Shape Modeling and Analysis #46E35 #49Q05 #49Q15 (Primary) #49Q20 #53A07 #53A10 #53C42 #54D05 (Secondary) #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2209.05955

openalex publication_date 2022/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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