Minimal hypersurfaces of least area

Mazet, Laurent; Rosenberg, Harold

doi:10.4310/jdg/1497405627

Cited by 28 publications

(43 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Because of this, we are also interested in describing the limit behavior of the least positive energy solutions as ε → 0, in terms of the minimal hypersurface that arises as its limit-interface. In some cases, we are able to conclude that this limit-interface is in fact a minimal hypersurface with least energy, in the sense of Rosenberg-Mazet [43].…”

Section: Low Energy Levels Of E εmentioning

confidence: 84%

“…. This section is motivated by the work of Mazet-Rosenberg on the minimal hypersurfaces of least area [43]. We show (see Theorem 2.1) Theorem 1. i ε is attained by solutions which are either stable or obtained by min-max and have index 1.…”

Section: Introductionmentioning

confidence: 94%

“…(4) If in addition, 3 ≤ n ≤ 7 and the metric on M is bumpy with Ric M > 0, then supp V is a smooth minimal hypersurface of least area among all minimal hypersurfaces, as studied by Mazet-Rosenberg in [43]. In particular, V has multiplicity one and it is realized by a connected orientable smooth hypersurface of index one.…”

Section: Low Energy Levels Of E εmentioning

confidence: 99%

See 2 more Smart Citations

The Allen–Cahn equation on closed manifolds

Gaspar

Guaraco

2018

Calc. Var.

View full text Add to dashboard Cite

We study global variational properties of the space of solutions to −ε 2 ∆u + W ′ (u) = 0 on any closed Riemannian manifold M . Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min-max and have index 1. We show that if ε is not small enough, in terms of the Cheeger constant of M , then there are no interesting solutions. However, we show that the number of minmax solutions to the equation above goes to infinity as ε → 0 and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed ε, as shown recently by G. Smith. We also show that the energy of the min-max solutions accumulate, as ε → 0, around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with RicM > 0, the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet-Rosenberg. Finally, we prove that the min-max energy values are bounded from below by the widths of the area functional as defined by Marques-Neves.Both authors were partly supported by CNPq-Brazil and NSF-DMS-1104592. 1 c ε (p) ≤ Cp 1 n for all p ∈ N.The proofs of these sublinear bounds are inspired by similar computations by Gromov [28], Guth [30] and Marques-Neves [40] for sweepouts of the area functional. In this sense, our work brings the analogy between phase transitions and minimal hypersurfaces to a high parameter global variational context.Combining Theorem 3 with the upper bound from Theorem 4, we obtainCorollary 5. On any closed Riemanian manifold the number of solutions to the elliptic Allen-Cahn equation (1) goes to infinity as ε → 0. Moreover, for every p, Theorem A implies that the min-max solutions at the level c ε (p) have a limit-interface that is a smooth embedded minimal hypersurface.This article ends in Section 6, where we compare the sequence of values lim inf ε→0 c ε (p) with the widths ω p (M ) of the area functional on M , as defined by Marques-Neves [40]. These widths form a sequence of real values which Gromov [28] proposed to consider as a nonlinear spectrum of M , by analogy

show abstract

Section: Low Energy Levels Of E εmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

The Allen–Cahn equation on closed manifolds

Gaspar

Guaraco

2018

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…If S denotes the collection of all smooth embedded stable minimal surfaces, we define A S (M ) = inf({|Σ|, Σ ∈ O∩S}∪{2|Σ|, Σ ∈ U ∩S}). Actually we proved in [10] that A 1 (M ) = min(W M , A S (M )). In order to simplify some notations, we will denote a(Σ) = |Σ| if Σ ∈ O and a(Σ) = 2|Σ| is Σ ∈ U.…”

Section: The Quantity a 1 (M )mentioning

confidence: 97%

“…In our paper [10], we introduce the quantity A 1 (M ), where M is a compact orientable 3-manifold. If O denotes the collection of all smooth orientable embedded closed minimal surfaces in M and U the collection of all smooth non-orientable ones, A 1 (M ) is defined by The main result in [10] says that A 1 (M ) is the area (or twice the area) of some minimal surface in M . Moreover it gives some characterization of this minimal surface in terms of its index and its genus.…”

Section: Introductionmentioning

confidence: 99%

Drilling short geodesics in hyperbolic 3-manifolds

Bromberg¹

2006

Spaces of Kleinian Groups

View full text Add to dashboard Cite

If M is a finite volume complete hyperbolic 3-manifold, the quantity A1(M ) is defined as the infimum of the areas of closed minimal surfaces in M . In this paper we study the continuity property of the functional A1 with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if A1(M ) is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in M is the main theme of this paper arXiv:1706.07742v2 [math.DG]

show abstract

Minimal Surfaces in Hyperbolic 3‐Manifolds

Coskunuzer

2020

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

Minimal hypersurfaces of least area

Cited by 28 publications

References 18 publications

The Allen–Cahn equation on closed manifolds

The Allen–Cahn equation on closed manifolds

Drilling short geodesics in hyperbolic 3-manifolds

Minimal Surfaces in Hyperbolic 3‐Manifolds

Contact Info

Product

Resources

About