The Regularity of Critical Points to Scale-Invariant Curvature Energies in Dimension 4

TL;DR

This paper proves that critical points of certain scale-invariant curvature energies in 4D are analytic, using conservation laws and PDE techniques, advancing understanding of geometric variational problems.

Contribution

It establishes the regularity and analyticity of critical points for a class of scale-invariant curvature energies in four dimensions, employing Noether's theorem and PDE methods.

Findings

Critical points are analytic in local harmonic charts.

Conservation laws lead to elliptic PDE systems.

Methods from integrability by compensation are applied.

Abstract

We consider a class of scale-invariant curvature energies defined on immersed $4$-dimensional manifolds and prove that weak immersions that are critical points of such energies are analytic in any given local harmonic chart. Because of the criticality of this variational problem, the regularity result is obtained through the identification of conservation laws by applying Noether theorem. The resulting identities generate a lower order elliptic system of PDEs to which methods from integrability by compensation and interpolation theory are applied.