Existence of infinitely many minimal hypersurfaces in positive Ricci curvature

Marques, Fernando C.; Neves, André

doi:10.1007/s00222-017-0716-6

Cited by 125 publications

(258 citation statements)

References 40 publications

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“…Using Almgren-Pitts min-max theory it is possible to obtain minimal hypersurfaces from sweep-outs of M . In [26] Marques and Neves proved the following results.…”

Section: Definition 22mentioning

confidence: 84%

“…Guth [19] derived similar bounds for min-max quantities corresponding to the Steenrod algebra generated by the fundamental class λ. Marques and Neves [26], building on the work of Gromov and Guth, proved existence of infinitely many minimal hypersurfaces on a manifold M of dimension n, for 3 ≤ n ≤ 7, under the assumption that M has positive Ricci curvature.…”

Section: W(m ) ≤ C (G + 1) Area(m )mentioning

confidence: 99%

“…Let K be a k-dimensional polyhderal complex and σ : K → Z 0 n−1 (M ) be continuous and assume that σ(x) has finite mass for all x. Following [26] we define k-parametric width W k as follows.…”

Section: Definition 22mentioning

confidence: 99%

“…Marques and Neves [26] showed that every manifold M of dimension n, for 3 ≤ n ≤ 7, with positive Ricci curvature possesses infinitely many embedded minimal hypersurfaces. Theorem 1.6 is an effective version of their result.…”

Section: Volumes Of Hypersurfacesmentioning

confidence: 99%

“…Theorem 1.1 follows from a bound on the width of M . One can find background and many results about widths of manifolds in [12,App.1F], [18], [3], [26]. Informally, the width W(M ) of a manifold M is the smallest number such that every sweep-out of M by hypersurfaces contains a hypersurface of volume at least W(M ).…”

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confidence: 99%

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Width, Ricci curvature, and minimal hypersurfaces

Glynn-Adey¹,

Liokumovich²

2017

J. Differential Geom.

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Abstract. Let (M, g0) be a closed Riemannian manifold of dimension n, for 3 ≤ n ≤ 7, and non-negative Ricci curvature. Let g = φ 2 g0 be a metric in the conformal class of g0. We show that there exists a smooth closed embedded minimal hypersurface in (M, g) of volume bounded by C(n)V n−1 n , where V is the total volume of (M, g). When Ric(M, g0) ≥ −(n − 1) we obtain a similar bound with constant C depending only on n and the volume of (M, g0).Our second result concerns manifolds (M, g) of positive Ricci curvature and dimension at most seven. We obtain an effective version of a theorem of F. C. Marques and A. Neves on the existence of infinitely many minimal hypersurfaces on (M, g) . We show that for any such manifold there exists k minimal hypersurfaces of volume at most CnV sys n−1 (M ), where V denotes the volume of (M, g0) and sys n−1 (M ) is the smallest volume of a non-trivial minimal hypersurface.

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“…Using Almgren-Pitts min-max theory it is possible to obtain minimal hypersurfaces from sweep-outs of M . In [26] Marques and Neves proved the following results.…”

Section: Definition 22mentioning

confidence: 84%

Section: W(m ) ≤ C (G + 1) Area(m )mentioning

confidence: 99%

Section: Definition 22mentioning

confidence: 99%

Section: Volumes Of Hypersurfacesmentioning

confidence: 99%

mentioning

confidence: 99%

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Width, Ricci curvature, and minimal hypersurfaces

Glynn-Adey¹,

Liokumovich²

2017

J. Differential Geom.

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Existence of constant mean curvature 2‐Spheres in Riemannian 3‐spheres

Cheng

Zhou

2023

Comm Pure Appl Math

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We prove the existence of branched immersed constant mean curvature (CMC) 2‐spheres in an arbitrary Riemannian 3‐sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3‐sphere is positively curved. To achieve this, we develop a min‐max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi‐harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min‐max values to gain energy upper bounds for min‐max sequences for almost every choice of mean curvature, and a Morse index estimate to obtain another uniform energy bound required to reach the remaining constant mean curvatures in the presence of positive curvature.

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Multiplicity‐1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature

Bellettini

2023

Comm Pure Appl Math

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We address the one‐parameter minmax construction for the Allen–Cahn energy that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold with (Guaraco's work, relying on works by Hutchinson, Tonegawa, and Wickramasekera when sending the Allen–Cahn parameter to 0). We obtain the following result: if the Ricci curvature of N is positive then the minmax Allen–Cahn solutions concentrate around a multiplicity‐1 minimal hypersurface (possibly having a singular set of dimension ). This multiplicity result is new for (for it is also implied by the recent work by Chodosh–Mantoulidis). We exploit directly the minmax characterization of the solutions and the analytic simplicity of semilinear (elliptic and parabolic) theory in . While geometric in flavour, our argument takes advantage of the flexibility afforded by the analytic Allen–Cahn framework, where hypersurfaces are replaced by diffused interfaces; more precisely, they are replaced by sufficiently regular functions (from N to ), whose weighted level sets give rise to diffused interfaces. We capitalise on the fact that (unlike a hypersurface) a function can be deformed both in the domain N (deforming the level sets) and in the target (varying the values). We induce different geometric effects on the diffused interface by using these two types of deformations; this enables us to implement in a continuous way certain operations, whose analogues on a hypersurface would be discontinuous. An immediate corollary of the multiplicity‐1 conclusion is that every compact Riemannian manifold with and positive Ricci curvature admits a two‐sided closed minimal hypersurface, possibly with a singular set of dimension at most . (This geometric corollary also follows from results obtained by different ideas in an Almgren–Pitts minmax framework.)

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Existence of infinitely many minimal hypersurfaces in positive Ricci curvature

Cited by 125 publications

References 40 publications

Width, Ricci curvature, and minimal hypersurfaces

Width, Ricci curvature, and minimal hypersurfaces

Existence of constant mean curvature 2‐Spheres in Riemannian 3‐spheres

Multiplicity‐1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature

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