On the second variation of the Graham-Witten energy

TL;DR

This paper derives an explicit formula for the second variation of the Graham-Witten energy, a conformal invariant generalizing Willmore energy, at minimal submanifolds in Einstein manifolds, and analyzes its critical points.

Contribution

It provides the first explicit second variation formula for Graham-Witten energy at minimal submanifolds in Einstein manifolds, extending understanding of its stability properties.

Findings

Even-dimensional totally geodesic spheres are critical points.

These spheres have non-negative second variation, indicating stability.

The formula aids in analyzing stability of submanifolds under conformal invariants.

Abstract

The area renormalization procedure gives an invariant of even-dimensional closed submanifolds in a conformal manifold, which we call the Graham-Witten energy, and it is a generalization of the classical Willmore energy. In this paper, we obtain an explicit formula for the second variation of this energy at minimal submanifolds in an Einstein manifold. As an application, we prove that the even-dimensional totally geodesic spheres in the unit sphere are critical points of the Graham-Witten energy with non-negative second variation.