The Min-Max theory and the Willmore conjecture

2014/01/06 by Fernando Marques, Fernando C. Marques, André Neves · 23 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.4007/annals.2014.179.2.6

Abstract

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in R 3 is at least 2 2 . We prove this conjecture using the min-max theory of minimal surfaces. Contents 1. Introduction 2. Main ideas and organization Part I. Proof of the Willmore conjecture 3. Canonical family: First properties 4. Definitions from Geometric Measure Theory 5. Canonical family: Boundary blow-up 6. The min-max family 7. The Almgren-Pitts Min-Max Theory I 8. The Almgren-Pitts Min-Max Theory II 9. Lower bound on width 10. Proof of Theorem B 11. Proof of Theorem A Part II. Technical work 12. No area concentration 13. Interpolation results: Continuous to discrete 14. Interpolation results: Discrete to continuous 15. Pull-tight

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• Width Stability of Rotationally Symmetric Metrics. [europepmc]
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