Energy reduction for Fourth order Willmore energy

Abstract

We introduce a fourth-order Willmore-type problem for closed four-dimensional submanifolds immersed in ${\mathbb{R}}^n$ and establish a connected sum energy reduction for the general fourth-order Willmore energy, analogous to the seminal result of Bauer and Kuwert \cite{Bauer-Kuwert03}.

47 pages. Comments are welcome. There is a mistake in Lemma 5.7 of the previous version, which implies that the energy reduction result holds only for manifolds that are both non-umbilic. Consequently, we can establish an energy reduction criterion that may help prevent topology loss, rather than a non-existence result