The Riemannian quantitative isoperimetric inequality

Otis Chodosh; Max Engelstein; Luca Spolaor

doi:10.4171/JEMS/1223

The Riemannian quantitative isoperimetric inequality

Otis Chodosh, Max Engelstein, Luca Spolaor

School of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We study the Riemannian quantitive isoperimetric inequality. We show that a direct analogue of the Euclidean quantitative isoperimetric inequality is-in general-false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Łojasiewicz-Simon inequality. The main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.

Original language	English (US)
Pages (from-to)	1711-1741
Number of pages	31
Journal	Journal of the European Mathematical Society
Volume	25
Issue number	5
DOIs	https://doi.org/10.4171/JEMS/1223
State	Published - 2023

Bibliographical note

Funding Information:
Funding. O.C. was partially supported by an NSF grant DMS-1811059. M.E. was partially supported by an NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison’s grant DMS 1500771. L.S. was partially supported by an NSF grant DMS-1810645.

Publisher Copyright:
© 2022 European Mathematical Society.

Keywords

Quantitative stability
isoperimetric inequality
Łojasiewicz inequality

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10.4171/JEMS/1223

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title = "The Riemannian quantitative isoperimetric inequality",

abstract = "We study the Riemannian quantitive isoperimetric inequality. We show that a direct analogue of the Euclidean quantitative isoperimetric inequality is-in general-false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the {\L}ojasiewicz-Simon inequality. The main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.",

keywords = "Quantitative stability, isoperimetric inequality, {\L}ojasiewicz inequality",

author = "Otis Chodosh and Max Engelstein and Luca Spolaor",

note = "Funding Information: Funding. O.C. was partially supported by an NSF grant DMS-1811059. M.E. was partially supported by an NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison{\textquoteright}s grant DMS 1500771. L.S. was partially supported by an NSF grant DMS-1810645. Publisher Copyright: {\textcopyright} 2022 European Mathematical Society.",

year = "2023",

doi = "10.4171/JEMS/1223",

language = "English (US)",

volume = "25",

pages = "1711--1741",

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T1 - The Riemannian quantitative isoperimetric inequality

AU - Chodosh, Otis

AU - Engelstein, Max

AU - Spolaor, Luca

N1 - Funding Information: Funding. O.C. was partially supported by an NSF grant DMS-1811059. M.E. was partially supported by an NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison’s grant DMS 1500771. L.S. was partially supported by an NSF grant DMS-1810645. Publisher Copyright: © 2022 European Mathematical Society.

PY - 2023

Y1 - 2023

N2 - We study the Riemannian quantitive isoperimetric inequality. We show that a direct analogue of the Euclidean quantitative isoperimetric inequality is-in general-false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Łojasiewicz-Simon inequality. The main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.

AB - We study the Riemannian quantitive isoperimetric inequality. We show that a direct analogue of the Euclidean quantitative isoperimetric inequality is-in general-false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Łojasiewicz-Simon inequality. The main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.

KW - Quantitative stability

KW - isoperimetric inequality

KW - Łojasiewicz inequality

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JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 5

ER -

The Riemannian quantitative isoperimetric inequality

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