Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions

Mondino, Andrea; Nardulli, Stefano

doi:10.4310/cag.2016.v24.n1.a5

Cited by 35 publications

(53 citation statements)

References 37 publications

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“…Let us briefly comment on the relevance of Corollary 1.3 despite the triviality of its proof. Recall that, from the note of Christodoulou and Yau [5], if (M, g) has non negative scalar curvature then isoperimetric spheres (and more generally stable CMC spheres) have positive Hawking mass; on the other hand it is known (see for instance [6] or [28]) that, if M is compact, then small isoperimetric regions converge to geodesic spheres centered at a maximum point of the scalar curvature as the enclosed volume converges to 0 (see also [22] for the non-compact case). Therefore a link between regions with positive Hawking mass and critical points of the scalar curvature was already present in literature, but Corollary 1.3 expresses this link precisely.…”

Section: Introductionmentioning

confidence: 99%

Concentration of small Willmore spheres in Riemannian 3-manifolds

Laurain

Mondino

2014

Anal. PDE

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Given a 3-dimensional Riemannian manifold (M, g), we prove that if (Φ k ) is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by 8π, and Hausdorff converging to a pointp ∈ M , then Scal(p) = 0 and ∇ Scal(p) = 0 (resp. ∇ Scal(p) = 0). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the euclidean 3-dimensional space. This generalizes previous results of Lamm and Metzger contained in [14]- [15]. An application to the Hawking mass is also established.

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Section: Introductionmentioning

confidence: 99%

Concentration of small Willmore spheres in Riemannian 3-manifolds

Laurain

Mondino

2014

Anal. PDE

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“…It is known that mild bounded geometry implies weak bounded geometry, but the converse is not true. For more details about this point the reader is referred to Remark 2.5 of [MN16] and to the references therein. Definition 2.3.…”

Section: Resultsmentioning

confidence: 99%

Sharp Isoperimetric Inequalities for Small Volumes in Complete Noncompact Riemannian Manifolds of Bounded Geometry Involving the Scalar Curvature

Nardulli

Acevedo

2018

International Mathematics Research Notices

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We provide an isoperimetric comparison theorem for small volumes in an n-dimensional Riemannian manifold (M n , g) with strong bounded geometry, as in Definition 2.3, involving the scalar curvature function. Namely in strong bounded geometry, if the supremum of scalar curvature function S g < n(n − 1)k 0 for some k 0 ∈ R, then for small volumes the isoperimetric profile of (M n , g) is less then or equal to the isoperimetric profile of M n k 0 the complete simply connected space form of constant sectional curvature k 0 . This work generalizes Theorem 2 of [Dru02b] in which the same result was proved in the case where (M n , g) is assumed to be just compact. As a consequence of our result we give an asymptotic expansion in Puiseux's series up to the second nontrivial term of the isoperimetric profile function for small volumes. Finally, as a corollary of our isoperimetric comparison result, it is shown, in the special case of manifolds with strong bounded geometry, and S g < n(n − 1)k 0 that for small volumes the Aubin-Cartan-Hadamard's Conjecture in any dimension n is true.

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“…Therefore Theorem 1.1 in particular implies that P(B r (x 0 )) ≤ P R n (B R n (V )), but is a strictly stronger statement which at best of our knowledge is original. The aforementioned counterpart of Theorem 1.1 for the isoperimetric problem was proved instead by Morgan-Johnson [43,Theorem 3.5] for compact manifolds and extended to non-compact manifolds in [42,Proposition 3.2].…”

Section: 3)mentioning

confidence: 99%

On an isoperimetric-isodiametric inequality

Mondino

Spadaro

2017

Anal. PDE

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Abstract. The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed isoperimetric-isodiametric inequalities in Riemannian manifolds. We first prove that the same inequality, with the sharp Euclidean constants, holds on Cartan-Hadamard spaces as well as on minimal submanifolds of R n . The equality cases are also studied and completely characterized; in particular, the latter gives a new link with free boundary minimal submanifolds in a Euclidean ball. We also consider the case of manifolds with non-negative Ricci curvature and prove a new comparison result stating that metric balls in the manifold have product of boundary area and radius bounded by the Euclidean counterpart and equality holds if and only if the ball is actually Euclidean. We then pass to consider the problem of the existence of optimal shapes (i.e. regions minimizing the product of boundary area and radius under the constraint of having fixed enclosed volume), called here isoperimetric-isodiametric regions. While it is not difficult to show existence if the ambient manifold is compact, the situation changes dramatically if the manifold is not compact: indeed we give examples of spaces where there exists no isoperimetric-isodiametric region (e.g. minimal surfaces with planar ends and more generally C 0 -locally-asymptotic Euclidean CartanHadamard manifolds), and we prove that on the other hand on C 0 -locally-asymptotic Euclidean manifolds with non-negative Ricci curvature there exists an isoperimetric-isodiametric region for every positive volume (this class of spaces includes a large family of metrics playing a key role in general relativity and Ricci flow: the so called Hawking gravitational instantons and the Bryanttype Ricci solitons). Finally we pass to prove the optimal regularity of the boundary of isoperimetric-isodiametric regions: in the part which does not touch a minimal enclosing ball the boundary is a smooth hypersurface outside of a closed subset of Hausdorff co-dimension 8, and in a neighborhood of the contact region the boundary is a C 1,1 -hypersurface with explicit estimates on the L ∞ -norm of the mean curvature.

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Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions

Cited by 35 publications

References 37 publications

Concentration of small Willmore spheres in Riemannian 3-manifolds

Concentration of small Willmore spheres in Riemannian 3-manifolds

Sharp Isoperimetric Inequalities for Small Volumes in Complete Noncompact Riemannian Manifolds of Bounded Geometry Involving the Scalar Curvature

On an isoperimetric-isodiametric inequality

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