New developments on the p-Willmore energy of surfaces (Q1980255)

The Willmore energy is an important conformal invariant of immersed surfaces \(\Sigma\) in the Euclidean 3-space. The article surveys some recent results [the second author et al., Ann. Global Anal. Geom. 56, No. 1, 147--165 (2019; Zbl 1417.58009)] on the \(p\)-Willmore energy \(W^p(\Sigma) = \int_\Sigma H^p dS\) (i.e., different powers of the mean curvature \(H\) of \(\Sigma\) in the integrand) for nonnegative integer \(p\) of surfaces (with boundary) in 3-dimensional space forms. Section 2 gives the first and second variations of \(W^p\), and presented a flux formula, which reveals (in Section 3) a connection between its critical points and the minimal surfaces. Section 4 reformulates the \(p\)-Willmore flow problem and presents (and visualizes through computer implementation) a model for the \(p\)-Willmore flow of graphs. For the entire collection see [Zbl 1445.53003].