Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Sabourau, Stéphane

doi:10.1515/crelle-2014-0147

Cited by 9 publications

(9 citation statements)

References 32 publications

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“…The precise statement of their result is actually rather explicit about the geometric dependence of the constant they obtain, see Theorems 5.1 therein; in particular, their estimate implies that all unit volume positive Ricci curvature metrics have their Almgren-Pitts widths uniformly bounded from above as well. The latter bound was also proven by Sabourau [38]. We remark that the Almgren-Pitts width of a three-sphere with positive Ricci curvature is precisely the least possible area of an embedded minimal surface (see [43], Theorem 15), while the Simon-Smith width of such three-sphere is equal to the least possible area of an embedded minimal two-sphere (see [24], proof of Theorem 3.4).…”

Section: Introductionmentioning

confidence: 60%

“…Considering examples of Riemannian metrics constructed by Burago and Ivanov [11], Guth showed that there are unit volume metrics on S 3 with arbitrarily large Almgren-Pitts width (see [20], Appendix A, and also [38], Section 7, for further results in this direction). Thus, no upper bound for the normalised width can be valid for all Riemannian metrics on S 3 .…”

Section: Introductionmentioning

confidence: 99%

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On the min-max width of unit volume three-spheres

Ambrozio,

Montezuma

2018

Preprint

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

On the min-max width of unit volume three-spheres

Ambrozio,

Montezuma

2018

Preprint

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“…In this paper, we consider two different notions of sweepout and waist similar but somewhat different to those introduced by Gromov in [11, §6]; see Definitions 1.1 and 1.10. With these notions, universal upper bounds on the waist of one-parameter families of one-cycles sweeping out essential surfaces of closed Riemannian manifolds were obtained in [24]. Our first theorem extends this result to the waist of multi-parameters families of one-cycles sweeping out any closed Riemannian manifold.…”

Section: Introductionmentioning

confidence: 75%

“…However, such a construction is not always possible in general. A different path was taken in [24], where a retraction from a different filling Q was constructed by considering all the simplices lying in the 2-skeleton of Q at the same time (and not only one at a time) and by proceeding by induction on the dimension of the higher-dimensional skeleta of Q from there, using a topological assumption on the manifold. In the proof of Theorem 1.3, where the manifold M is arbitrary, we take yet a different approach.…”

Section: Introductionmentioning

confidence: 99%

Sweepouts of closed Riemannian manifolds

Nabutovsky¹,

Rotman²,

Sabourau³

2020

Preprint

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We show that for every closed Riemannian manifold there exists a continuous family of 1-cycles (defined as finite collections of disjoint closed curves) parametrized by a sphere and sweeping out the whole manifold so that the lengths of all connected closed curves are bounded in terms of the volume (or the diameter) and the dimension n of the manifold, when n ≥ 3. An alternative form of this result involves a modification of Gromov's definition of waist of sweepouts, where the space of parameters can be any finite polyhedron (and not necessarily a pseudomanifold). We demonstrate that the so-defined polyhedral 1-dimensional waist of a closed Riemannian manifold is equal to its filling radius up to at most a constant factor. We also establish upper bounds for the polyhedral 1-waist of some homology classes in terms of the volume or the diameter of the ambient manifold. In addition, we provide generalizations of these results for sweepouts by polyhedra of higher dimension using the homological filling functions. Finally, we demonstrate that the filling radius and the hypersphericity of a closed Riemannian manifold can be arbitrarily far apart.

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“…Remark 1.9. In [33] Stephane Sabourau independently obtained upper bounds on the width and volume of the smallest minimal hypersurface on Riemannian manifolds with Ricci ≥ 0.…”

Section: Previous Workmentioning

confidence: 99%

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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Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Cited by 9 publications

References 32 publications

On the min-max width of unit volume three-spheres

On the min-max width of unit volume three-spheres

Sweepouts of closed Riemannian manifolds

Width, Ricci curvature, and minimal hypersurfaces

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