The catenoid estimate and its geometric applications

By Daniel Ketover, F. C. Marques, A. Neves. Journal of differential geometry, 2016

60 citations recorded by the source corpus.

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the "doublings" of the Clifford torus by Kapouleas-Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved by an index 1 orientable minimal hypersurface.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/21aa3c07ceed0d2d67e5ab812ea3f17166cc2e97

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Minimal 2-spheres in 3-spheres (2017, methodology): “ery thin tube. As long as the tube in the surface Σs,t is very thin, it contributes negligible area and thus we obtain (1.4) sup s,t |Σs,t| ≈ 2|Σ0|. Usingthecatenoidestimate fromKetover-Marques-Neves [25], by opening up the neck near (s,t) ≈ (0,0) (where the area of Σs,t is close to 2|Σ0|) we can im”
The Allen–Cahn equation on closed manifolds (2016, background): “It follows from Propositions 10 and 13 in [43], combined with the recent catenoid estimate [35] to rule out the case unoriented surfaces, that ̃ is separating and orientable.”
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].”
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, result): “licity one conjecture for Almgren–Pitts (in dimensions 3 through 7) has now been proven by X. Zhou [Zho19]. 8 OTIS CHODOSH AND CHRISTOS MANTOULIDIS curvature, this was proven by Ketover–Marques–Neves [KMN16]. More precisely, our main results here are as follows (see Theorem 4.1 and Corollary 6.1 for”
Morse index of multiplicity one min-max minimal hypersurfaces (2015, background): “These results have been improved recently by Ketover and the authors in [26], where new index characterizations and multiplicity one theorems are proven as a consequence of the catenoid estimate.”