Every immersed 2-sphere with Willmore energy at most 12π admits an energy-nonincreasing regular homotopy to a round sphere or to a surface in the explicit family J, giving exactly four regular homotopy classes below 12π.
Conformally Invariant Variational Problems
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abstract
Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl tensors, $Q$-curvature, Yang-Mills fields, etc... We shall be concerned with the study of conformal invariance in analysis. More precisely, we will focus on the study of nonlinear PDEs arising from conformally invariant two dimensional variational problems (e.g. harmonic maps, prescribed mean curvature surfaces, Willmore and Constrained conformal surfaces, isothermic surfaces). The present manuscript are lecture notes of courses given by the author at several places including UBC Vancouver, SNS Pisa, IHP Paris, ICTP Trieste.
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The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow
Every immersed 2-sphere with Willmore energy at most 12π admits an energy-nonincreasing regular homotopy to a round sphere or to a surface in the explicit family J, giving exactly four regular homotopy classes below 12π.