The Conformal Willmore Functional: a Perturbative Approach

TL;DR

This paper investigates the conformal Willmore functional using a perturbative Lyapunov-Schmidt reduction, establishing existence of critical points in nearly Euclidean 3-manifolds and non-existence in general Riemannian 3-manifolds.

Contribution

It introduces a perturbative approach to analyze the conformal Willmore functional, proving existence and non-existence results in specific Riemannian settings.

Findings

Existence of critical points in $( ^3, g_ )$ with metrics close to Euclidean.

Non-existence of critical points in general 3D Riemannian manifolds.

Application of Lyapunov-Schmidt reduction to conformal Willmore functional.

Abstract

The conformal Willmore functional (which is conformal invariant in general Riemannian manifold $(M,g)$) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds $(\mathbb{R}^3, g_\epsilon)$ -where $g_\epsilon$ is a metric close and asymptotic to the euclidean one. With the same technique a non existence result is proved in general Riemannian manifolds $(M,g)$ of dimension three.