Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
By F. C. Marques, A. Neves. Inventiones Mathematicae, 2013
202 citations recorded by the source corpus.
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/92dd661d1b6572cffc9b72477daf84ba8866b775
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- 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]).”
- Existence of hypersurfaces with prescribed mean curvature I – generic min-max (2018, methodology): “Recently, Marques-Neves [43, 1, 45] used the Almgren-Pitts min-max theory to resolve a number of longstanding open problems in geometry, including their celebrated proof of the Willmore conjecture.”
- On the Multiplicity One Conjecture in min-max theory (2019, unknown): “By adapting the celebrated min-max theory developed by Almgren [3], Pitts [31] (for 3 ≤ (n+1) ≤ 6), and Schoen-Simon [33] (for n + 1 = 7), Marques-Neves [28, 27] proved that each ωk(M,g) is associated with an integral varifold Vk whose support is a disjoint collection of smooth, connected, closed, e”
- Weyl law for the volume spectrum (2016, methodology): “It has been used by Marques and Neves to prove existence of infinitely many minimal hypersurfaces in manifolds with positive Ricci curvature [16].”
- Compactness of minimal hypersurfaces with bounded index (2015, background): “Due to the recent work of Marques-Neves [13] minimal hypersurfaces exist in abundance at least in the case of positive Ricci curvature for the ambient metric.”
- 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.”
- The Allen–Cahn equation on closed manifolds (2016, methodology): “Both arise from the nature of the p-sweepouts in [40].”
- Morse index of multiplicity one min-max minimal hypersurfaces (2015, methodology): “φi : X(ki)0 → Zn(M ;Z2), with ki < ki+1, and a sequence of positive numbers {δi} converging to zero such that (i) S = {φi} is an (X,M)-homotopy sequence of mappings into Zn(M ;M;Z2) with f(φi) < δi (see Section 2 of [34] for a definition); (ii) sup{F(φi(x)− Φ(x)) : x ∈ X(ki)0} ≤ δi; (iii)”
- Density of minimal hypersurfaces for generic metrics. (2017, methodology): “[8]), the existence of infinitely many closed, smooth, embedded minimal hypersurfaces was only known for manifolds of positive Ricci curvature Mn+1, 3 ≤ (n + 1) ≤ 7, as proven by the last two authors in [12].”