A Curvature Varifold Whose Weak Second Fundamental Form is Not Preserved Under Decompositions

TL;DR

This paper constructs a specific curvature varifold that demonstrates the failure of preservation of the weak second fundamental form under decomposition, highlighting limitations in the decomposition theory of curvature varifolds.

Contribution

It introduces a curvature varifold with a weak second fundamental form that is not preserved under any decomposition into curvature varifolds, revealing new limitations in the theory.

Findings

Existence of a curvature varifold with non-preserved weak second fundamental form

Demonstration that certain curvature varifolds cannot be decomposed into simpler curvature components

Insights into the structural limitations of curvature varifold decompositions

Abstract

We construct a curvature varifold that does not admit a decomposition whose components are curvature varifolds.