Differential geometry seminar

A geometric energy landscape is usually investigated locally through its critical points and their Morse indices. Understanding its global structure is challenging and requires both analytic and topological approaches. We present a special case where such global analysis is feasible: the Willmore energy of immersed 2-spheres in $\mathbb{R}^3$ . We show that the subset of immersions with energy at most $12\pi$ consists of four path components. As a consequence, we obtain insight into the singular behavior of the Willmore flow and a relation between self-intersections and the Willmore energy beyond the Li-Yau inequality. To prove these results, we glue together different instances of the Willmore flow and devise an invariant for triple-point-free immersed spheres.

arXiv: https://arxiv.org/abs/2506.23359 and https://arxiv.org/abs/2506.21130