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Published online: 2014-06-01
Mini-Workshop: The Willmore Functional and the Willmore ConjectureTobias Lamm, Jan Metzger and André Neves (1) Karlsruhe Institute of Technology (KIT), Germany
(2) Universität Potsdam, Germany
(3) Imperial College London, UK
The Willmore functional evaluated on a surface immersed into Euclidean space is given by the $L^2$-norm of its mean curvature. The interest for studying this functional comes from various directions. First, it arises in applications from biology and physics, where it is used to model surface tension in the Helfrich model for bilipid layers, or in General Relativity where it appears in Hawking’s quasi-local mass. Second, the mathematical properties justify consideration of the Willmore functional in its own right. The Willmore functional is one of the most natural extrinsic curvature functionals for immersions. Its critical points solve a fourth order Euler-Lagrange equation, which has all minimal surfaces as solutions.
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Lamm Tobias, Metzger Jan, Neves André: Mini-Workshop: The Willmore Functional and the Willmore Conjecture. Oberwolfach Rep. 10 (2013), 2119-2153. doi: 10.4171/OWR/2013/37