Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the\n Willmore Conjecture

2020/07/10 by Tristan Rivière, Rivière, Tristan
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2007.05467

Abstract

We introduce a general scheme that permits to generate successive min-max\nproblems for producing critical points of higher and higher indices to\nPalais-Smale Functionals in normal Banach manifolds equipped with complete\nFinsler structures. We call the resulting tree of minmax problems a minmax\nhierarchy. We give several examples and in particular we explain how to\nimplement this scheme in the framework of the viscosity method introduced by\nthe author some years ago in order to give a new proof of the Willmore\nconjecture after the famous result by Marques and Neves.\n

Citations

• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces
• Harmonic Maps from S³ into S² with low Morse Index

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