A New Conformal Invariant for four-Dimensional Hypersurfaces

TL;DR

This paper introduces a novel conformally invariant energy for four-dimensional hypersurfaces, enabling analysis of curvature energies and proving regularity of their critical points, including analogues of classical energies and Bach-flat hypersurfaces.

Contribution

It develops a new conformal invariant energy for four-dimensional hypersurfaces and demonstrates the smoothness of its critical points, extending results for classical curvature energies.

Findings

Critical points of the new energy are smooth.

Regularity results for four-dimensional Willmore and Q-curvature energies.

Bach-flat hypersurfaces are shown to be smooth.

Abstract

A new conformally invariant energy for four-dimensional hypersurfaces is devised. It renders possible the study of a large class of curvature energies, and we show that their critical points are smooth. As corollaries, we obtain the regularity of the critical points of the four-dimensional analogues of the Willmore energy, of the $Q$-curvature energy, but also that Bach-flat hypersurfaces are smooth, along with relevant estimates.