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

Ambrozio, Lucas; Montezuma, Rafael

doi:10.48550/arxiv.1809.03638

Cited by 3 publications

(8 citation statements)

References 41 publications

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“…The main difference between our theorem and Proposition 1.4.1 of [2] is that in free boundary case, we can not rule out the case when {Σ i } is not properly embedded (Σ ∩ ∂M = ∅), due to the lack of convexity of ∂M . Readers can see [6] for a possible example of non-properly embedded free boundary minimal hypersurface in an Euclidean domain.…”

Section: Introductionmentioning

confidence: 83%

“…Here we include an abstract theorem used in the proof of Theorem 1.1 and 1.2., for the proof see Theorem B.1 of [2]. Theorem 4.2.…”

Section: Compactness Theorem and Equidistribution Theoremmentioning

confidence: 98%

“…Proof. (cf [2]) First, due to the density of bumpy metric on M and Rademacher's theorem, we can perturb the smooth family g : [0, 1] → Γ q to h : [0, 1] → Γ q which is arbitrarily close to g in smooth topology, and a set J ⊂ [0, 1] of full measure such that h(τ ) is a bumpy metric and W (M, h(t)) is differentiable at τ , for all τ ∈ J. For all τ ∈ J, fix a sequence t i → τ , we have…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…In a recent work of Ambrosio and Montezuna [2], the equidistribution phenomenon of minimal S 2 in S 3 is studied. With the assumption that the metric g 0 on S 3 is a local maximizer(in its conformal class) of the Simon-Smith width functional W (S 3 , g), the authors proved the existence of equi-distributed minimal 2-spheres in measure theoretic sense.…”

Section: Introductionmentioning

confidence: 99%

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Locally Maximizing Metric of Width on Manifolds with Boundary

Tu¹

2020

Preprint

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Section: Introductionmentioning

confidence: 83%

“…Here we include an abstract theorem used in the proof of Theorem 1.1 and 1.2., for the proof see Theorem B.1 of [2]. Theorem 4.2.…”

Section: Compactness Theorem and Equidistribution Theoremmentioning

confidence: 98%

Section: Proof Of Theorem 11mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Locally Maximizing Metric of Width on Manifolds with Boundary

Tu¹

2020

Preprint

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“…Let RP 3 be the three-dimensional real projective space, and F denote the non-empty set consisting of all embedded surfaces in RP 3 that are diffeomorphic to the two-dimensional projective plane RP 2 . Given a Riemannian metric g on RP 3 , we define…”

Section: Introductionmentioning

confidence: 99%

On the two-systole of real projective spaces

Ambrozio

Montezuma

2020

Rev. Mat. Iberoam.

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We establish an integral-geometric formula for minimal twospheres inside homogeneous three-spheres, and use it to provide a characterisation of each homogeneous metric on the three-dimensional real projective space as the unique metric with the largest possible twosystole among metrics with the same volume in its conformal class. IntroductionLet RP 3 be the three-dimensional real projective space, and F denote the non-empty set consisting of all embedded surfaces in RP 3 that are diffeomorphic to the two-dimensional projective plane RP 2 . Given a Riemannian metric g on RP 3 , we defineIn this paper, the geometric invariant above will be called the two-systole of (RP 3 , g). The term has been used to name slightly different invariants in the literature, depending on the choice of the set F (cf. [10], Sections 1 and 4.A.7, and [5], Section 5). The first systematic study of such invariants was done by Berger in [5], where he sought generalisations of Pu's inequality [15] for the (one)-systole of real projective planes, i.e. the smallest length of a non-trivial loop in (RP 2 , g). Berger computed that the two-systole of the standard round metric g 1 on RP 3 , with constant sectional curvature one, is equal to 2π (see [5], Théorème 7.1). This number is precisely the area of the totally geodesic projective planes in (RP 3 , g 1 ).In [6], Bray, Brendle, Eichmair and Neves studied how the two-systole behaves under the Ricci flow, proving along the way a sharp upper bound for A(RP 3 , g) in terms of the minimum value of the scalar curvature of (RP 3 , g) (see [6], Theorems 1.1 and 1.2). An important part of their analysis was to show that the infimum defining A(RP 3 , g) is actually attained by an embedded area-minimising projective plane in (RP 3 , g) (see [6], Proposition 2.3).In this paper, we investigate how large can be the normalised two-systole, A(RP 3 , g) vol(RP 3 , g) 2 3, MSC classification codes: 53A10, 53C30.

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Area, Scalar Curvature, and Hyperbolic 3-Manifolds

Lowe¹

2021

Preprint

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Let M be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if g is a Riemannian metric on M with scalar curvature greater than or equal to −6, we find lower bounds for the areas of stable immersed minimal surfaces Σ in M . Our bounds improve the closer Σ is to being homotopic to a totally geodesic surface in the hyperbolic metric.We also consider a functional introduced by Calegari-Marques-Neves that is defined by an asymptotic count of minimal surfaces in (M, g). We show this functional to be uniquely maximized, over all metrics of scalar curvature greater than or equal to −6, by the hyperbolic metric. Our proofs use the Ricci flow with surgery.

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

Cited by 3 publications

References 41 publications

Locally Maximizing Metric of Width on Manifolds with Boundary

Locally Maximizing Metric of Width on Manifolds with Boundary

On the two-systole of real projective spaces

Area, Scalar Curvature, and Hyperbolic 3-Manifolds

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