# Higher-dimensional Willmore energies via minimal submanifold asymptotics

## TL;DR

This paper introduces a conformally invariant higher-dimensional Willmore energy derived from minimal submanifold asymptotics, analyzing its properties, explicit form in four dimensions, and variational behavior in Euclidean and spherical backgrounds.

## Contribution

It generalizes the Willmore energy to higher even dimensions using minimal submanifold asymptotics and studies its variational properties and explicit form in four dimensions.

## Key findings

- Explicit identification of the energy for four-dimensional submanifolds.
- Analysis of critical embeddings and boundedness properties.
- Second variation calculations in Euclidean space and spheres.

## Abstract

A conformally invariant generalization of the Willmore energy for compact immersed submanifolds of even dimension in a Riemannian manifold is derived and studied. The energy arises as the coefficient of the log term in the renormalized area expansion of a minimal submanifold in a Poincare-Einstein space with prescribed boundary at infinity. Its first variation is identified as the obstruction to smoothness of the minimal submanifold. The energy is explicitly identified for the case of submanifolds of dimension four. Variational properties of this four-dimensional energy are studied in detail when the background is a Euclidean space or a sphere, including identifications of critical embeddings, questions of boundedness above and below for various topologies, and second variation.