New Developments on the p-Willmore Energy of Surfaces

Abstract

The p-Willmore energy $\mathcal{W}^p$, which extends the venerable Willmore energy by accommodating different powers of the mean curvature in its integrand, is a relevant geometric functional that bears both similarities and differences to its namesake. To elucidate this, some recent results in this area are surveyed. In particular, the first and second variations of $\mathcal{W}^p$ are given, and a flux formula is presented which reveals a connection between its critical points and the minimal surfaces. Finally, a model for the p-Willmore flow of graphs is presented, and this connection is visualized through computer implementation.