Invariant Hulls and Geometric Variational Principles

TL;DR

This paper explores the concept of invariant hulls for functionals on manifolds, examining their properties, relationships, and computational challenges, especially in the context of geometric variational problems in multiple dimensions.

Contribution

It introduces the notion of invariant hulls for functionals, analyzes their properties, and provides initial computational insights, particularly for the two-dimensional case.

Findings

Explicit computations are feasible in one dimension.

Higher-dimensional cases are significantly more complex.

Some conjectures and problems are proposed for future research.

Abstract

We investigate functionals defined on manifolds through parameterizations. If they are to be meaningful, from a geometrical viewpoint, they ought to be invariant under reparameterizations. Standard, local, integral functionals with this invariance property are well-known. We would like to focus though on the passage from a given arbitrary functional to its invariant realization or invariant hull through the use of inner-variations, much in the same way as with the convex or quasiconvex hulls of integrands in the vector Calculus of Variations. These two processes are, however, very different in nature. After examining some basic, interesting, general properties about the mutual relationship between a functional and its invariant realization, we deal with the one dimensional case to gain some initial familiarity with such a transformation and calculations, before proceeding to the higher dimensional situation. As one would anticipate, explicit computations in the latter are much harder to perform, if not impossible, as one is to work with vector variational problems. In particular, we are able to reach some modest conclusion about the volume functional of a piece of a manifold in the general $N$-dimensional situation, especially in the two-dimensional case $N=2$. Various problems and conjectures are stated along the way.