Four-Dimensional Willmore-type Energy and the Bach Equation

Abstract

In this talk, I will introduce new conformally invariant functionals for a four-dimensional hypersurface. I will show how they extend the Willmore 2-dimensional problem and more importantly how they relate to a well-known conformally-invariant *intrinsic* energy whose critical points are known as Bach-flat manifolds. Using techniques inspired and adapted from 2-d, I will prove that Bach-flat immersions that lie in the critical space $W^{3,2}\cap W^{1,\mathrm{\infty }}$ are in fact smooth.