The isoperimetric profile of a noncompact Riemannian manifold for small volumes

Nardulli, Stefano

doi:10.1007/s00526-012-0577-1

Cited by 15 publications

(21 citation statements)

References 10 publications

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“…As a last consequence of Theorem 5 we get Corollary 3 which gives an asymptotic expansion of the isoperimetric profile in Puiseux's series up to the second non trivial order generalizing previous results of [Nar14b]. Before to state the corollary we recall here the definition of the isoperimetric profile.…”

Section: Resultssupporting

confidence: 66%

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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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Section: Resultssupporting

confidence: 66%

“…Remark 2.10. Via the same smoothing results of [Shi89], [Kap05] one can prove the preceding asymptotic expansion just using the theory of pseudo-bubbles developped by the first author in [Nar14b]. However the theory of pseudo bubbles does not give the sharp isoperimetric comparison of Theorem 5.…”

Section: Resultsmentioning

confidence: 81%

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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“…where v * depends only on m, α, n, Q, r is obtained in Theorem 1 of [Nar10]. Remark: In (II) if we assume (M, g) ∈ M m,α (n, Q, r) for some n, Q, r, instead of M satisfying the assumption of Definition 1.8, one has to replace β ≤ α, with β < α.…”

Section: Resultsmentioning

confidence: 99%

Generalized existence of isoperimetric regions in non-compact Riemannian manifolds and applications to the isoperimetric profile

Nardulli

2014

Asian Journal of Mathematics

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“…Its proof is modeled on a previous one of existence of clusters minimizing perimeter under given volume constraints in Euclidean space [23]. See also the paper by Nardulli [26].…”

Section: Introductionmentioning

confidence: 99%