Rigidity of min-max minimal spheres in three-manifolds

By F. C. Marques, A. Neves. 2011

120 citations recorded by the source corpus.

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than $4\pi$ and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a4049248cfb7689969c338d743d19344c3229568

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On static three-manifolds with positive scalar curvature (2015, background): “In view of that, Theorem 2 gives an interesting rigidity result for the standard round hemisphere under a nice extra geometric assumption (for related results, see for example [32], [22] and [42]).”
Morse index of multiplicity one min-max minimal hypersurfaces (2015, background): “Previously, the one-parameter case in manifolds with positive Ricci curvature had been studied before by Zhou [62, 63], who did important work extending to high dimensions previous work of the authors [31].”
Min-max for phase transitions and the existence of embedded minimal hypersurfaces (2015, methodology): “We refer the reader to [7, 17, 20, 23, 43, 44] for applications to compact manifolds; [6, 26] for applications to non-compact manifolds; [18] for applications to a free-boundary situation; and to Marques-Neves [22, 21] for the solution of the long-standing Willmore conjecture and the existence of in”
On the Multiplicity One Conjecture in min-max theory (2019, unknown): “This conjecture was proved earlier for 1-parameter min-max constructions under positive Ricci curvature assumption by Marques-Neves [25], the author [48, 49], and Ketover-Marques-Neves [22].”
Min-max hypersurface in manifold of positive Ricci curvature (2015, unknown): “ore classical methods in nonlinear analysis (c.f. [St00]) do not extend to this situation. The current progress on this problem mainly focused on the case of one-parameter families. Marques and Neves [MN11] have conﬁrmed this conjecture in three dimension when the Ricci curvature of the ambient mani”
Minimal 2-spheres in 3-spheres (2017, background): “1 in [27]) that Di contains in its interior a minimal two-sphere, necessarily disjoint from Σ.”