Higher-dimensional Willmore energies via minimal submanifold asymptotics

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 Poincaré–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.

Keywords

Willmore energies, minimal submanifolds, renormalized area, Poincaré–Einstein metrics

2010 Mathematics Subject Classification

Primary 53A07. Secondary 53A30, 53B25, 53C42, 58E30.

Full Text (PDF format)

Received 5 September 2018

Accepted 25 October 2019

Published 18 February 2021