Chen-Willmore submanifolds shaped on closed generalized elasticae

Abstract

In this survey we review some recent results on Willmore surfaces and ChenWillmores submanifolds. We first show how they are related to important models in the theory of elastic surfaces and membranes and show the strong connection between elastic curves in 2-dimensional space forms and the critical points of the elastic energy. Then, we will describe how the conformal invariance of the Chen-Willmore energy along with the Palais’ symmetric criticality principle can be used in order to produce effective methods for the construction of Chen-Willmore submanifolds.