Compactness of minimal hypersurfaces with bounded index
By B. Sharp. 2015
115 citations recorded by the source corpus.
We prove a compactness result for minimal hypersurfaces with bounded index and volume, which can be thought of as an extension of the compactness theorem of Choi-Schoen (Invent. Math. 1985) to higher dimensions.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/d1d0812937e6a54c36a780c233f14c86b3d0a72b
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Minimal hypersurfaces with bounded index (2015, methodology): “ndex in a three-manifold with a metric of positive Ricci curvature. This was preceded by the higher dimensional (i.e., allowing the ambient manifold to be n-dimensional for 3 ≤ n ≤ 7) result of Sharp [Sha15], showing that for a metric of positive Ricci curvature, the space of embedded minimal surfac”
- Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “Thus by Sharp’s compactness result [34], for a well chosen sequence ǫk → 0, the restriction of the support S∞ of V∞ to C(U)\∂U is a smooth minimal hypersurface, since the metrics hǫ converge smoothly in this region by Lemma 5.”
- Free boundary minimal surfaces in the unit ball : recent advances and open questions (2019, result): “By studying the relationship between the eigenvalues of the Jacobi operator J and the eigenvalues of the Laplacian on 1-forms, Sargent [79] and Ambrozio-Carlotto-Sharp [2] independently obtained new Morse index bounds in terms of the dimension n and the topology of the minimal hypersurface Σ.”
- Morse index of multiplicity one min-max minimal hypersurfaces (2015, background): “Sharp’s Compactness Theorem [48] implies WL,j is a finite set if the metric (M,g) is bumpy.”
- Density of minimal hypersurfaces for generic metrics. (2017, methodology): “The Compactness Theorem of Sharp (Theorem 2.3 of [16]) implies that there exists a finite disjoint collection { Σ 1 , . . . , Σ N } of closed, smooth, embedded minimal hypersurfaces in M , satisfying and integers { m 1 , . . . , m N } ⊂ N such that, after passing to a subsequence, =1 m j vol g (Σ j…”
- Compactness analysis for free boundary minimal hypersurfaces (2017, background): “A version of the theorem below appeared in [32], assuming a uniform upper bound on the Morse index, and later in [3] assuming a lower bound on the p-the eigenvalue of the Jacobi operator, for what concerns the case of closed minimal hypersurfaces (see also [1], Proposition 4.”
- On the Multiplicity One Conjecture in min-max theory (2019, unknown): “Now we present an adaption of Sharp’s compactness theorem [35] (for minimal hypersurfaces) to the PMC setting and necessary modifications of the proof.”