Study of a Four Dimensional Willmore Energy

TL;DR

This thesis explores a four-dimensional conformally invariant energy extending the Willmore energy, analyzing its critical points, conservation laws, and regularity, and contrasting it with other generalizations.

Contribution

It introduces a new four-dimensional Willmore-type energy, derives its variational properties, conservation laws, and establishes smoothness of critical points.

Findings

Critical points are smooth hypersurfaces.

The energy generalizes the classical Willmore energy to four dimensions.

Minimal hypersurfaces are critical points of the energy.

Abstract

In this thesis, a four dimensional conformally invariant energy is studied. This energy generalises the well known two-dimensional Willmore energy. Although not positive definite, it includes minimal hypersurfaces as critical points. We compute its first variation and by applying the Noether theorem to the invariances, we derive some conservation laws which are satisfied by its critical points and with good analytical dispositions. In particular, we show that critical points are smooth. We also investigate other possible four dimensional generalisations of the Willmore energy, and give strong evidence that critical points of such energies do not include minimal hypersurfaces.