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Daily Overview |
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Gvp1: Geometric variational problems
Session Topics: Geometric variational problems
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| Presentations | ||
Rotational symmetric Willmore surfaces with umbilic lines 1: Ulm University, Germany; 2: Eberhard-Karls-Universität Tübingen The most famous Willmore surface is a round sphere, a totally umbilic surface. In this talk we construct rotationally symmetric Willmore surfaces with countably many umbilic lines. Further we discuss how such surfaces, after an appropriate Möbius transformation, are minimal in hyperbolic space outside its umbilic lines. Weak immersions with second fundamental form in a critical Sobolev space ETH Zürich, Switzerland
Over the past decade, generalizations of the Willmore energy to even dimensional submanifolds of the Euclidean space have been subject to numerous works, due to their relation to renormalized volume of minimal submanifolds in Poincaré-Einstein manifolds and their applications to AdS/CFT correspondence.
I will present an analytical framework, recently developed in a collaboration with Tristan Rivière, in order to address variational problems concerning these generalized Willmore energies.
Bubble classification of immersions at the boundary of the moduli space with 8π Willmore energy University of Bonn, Germany In this talk, we study the asymptotic bubbling behavior of sequences of weak genus-p immersions with diverging conformal classes and limiting Willmore energy of 8π. After applying suitable Möbius transformations, we show that the sequences resemble two round spheres at the largest scales and p+1 catenoids at the smallest scales. In order to make this precise, we will introduce the notion of a bubble graph. Moreover, we apply this classification result to several constrained minimization problems when the minimizers degenerate. This talk is based on joint work with Christian Scharrer and Manuel Schlierf. The Willmore Energy Landscape of Spheres Technische Universität Darmstadt, Germany 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. | ||
