Min-Max theory and the Willmore conjecture

By F. C. Marques, A. Neves. 2012

528 citations recorded by the source corpus.

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in Euclidean three-space is at least 2\pi^2. We prove this conjecture using the min-max theory of minimal surfaces.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dee58e240aa1c5c9d4fa11da0d1aafc482b6c899

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Min-max hypersurface in manifold of positive Ricci curvature (2015, methodology): “ is a very common problem in the study of min-max theory (c.f. [MN12, Z12, MN13, Mo14]). Pitts already developed some tools in his book [P81]. Marques-Neves, in their proof of the Willmore conjecture [MN12], ﬁrst gave a complete theory to connect families continuous under ﬂat topology to families sa”
Embedded minimal tori in S3 and the Lawson conjecture (2012, methodology): “The method used in [18] is completely different from ours; it relies on the min-max theory for minimal surfaces and the rigidity theorem of Urbano.”
A sharp lower bound for the entropy of closed hypersurfaces up to dimension six (2014, background): “For instance, in proving the Willmore conjecture, Mar ques-Neves [19] showed that after the equatorial sphere, the (oriented) minimal surfac e inS(3) of smallest area is the Clifford torus.”
Minimal surfaces in $$S^3$$: a survey of recent results (2013, methodology): “ure holds for tori with antipodal symmetry. We note that Topping [51] has obtained an alternative proof of Theorem 4.3, which is based on techniques from integral geometry. In 2012, Marques and Neves [41] veriﬁed the Willmore conjecture in full generality. Their proof relies on the min-max theory fo”
Min-max for phase transitions and the existence of embedded minimal hypersurfaces (2015, methodology): “Finally, we present the min-max theorem from Almgren-Pitts (see [30, 22, 21, 26]).”
The catenoid estimate and its geometric applications (2016, background): “ity 1. Indeed, if the multiplicity were k>1, then by (3.17) we have k|Σm| <2|C|, where |C| = 2π2, the area of the Cliﬀord torus. Thus |Σ m| <|C|. By the resolution of the Willmore conjecture [MN2], in this case Σm can only be a round equator, which is not Gm-equivariant. Thus the multiplici”
Free boundary minimal surfaces in the unit ball : recent advances and open questions (2019, methodology): “The arguments in [69] also used crucially a result of Urbano [91] saying that the only non-totally geodesic closed minimal surface Σ in S(3) with index(Σ) ≤ 5 is (up to isometries of S(3)) the Clifford torus.”
On the Multiplicity One Conjecture in min-max theory (2019, unknown): “D G ] 5 F eb 2 01 the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one.”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “By the work of Marques and Neves [26], the manifold (M, g) also necessarily contains at least two disjoint minimal hypersurfaces, otherwise it would contain infinitely many minimal hypersurfaces.”
Morse index of multiplicity one min-max minimal hypersurfaces (2015, methodology): “The following proposition was proved in [32] (Proposition 13.”
Energy quantization for Willmore surfaces and applications (2011, background): “Thanks to the recent proof of The Willmore conjecture by F. C. Marques and A. Neves, [MN14], one knows that ω 3 g > 8 π ; thus for m = 3, the assumption in Theorem I.5 is then reduced to The paper is organized as follows.”