JournalsjemsVol. 25, No. 5pp. 1711–1741

The Riemannian quantitative isoperimetric inequality

  • Otis Chodosh

    Stanford University, USA

  • Max Engelstein

    University of Minnesota, Minneapolis, USA

  • Luca Spolaor

    University of California San Diego, La Jolla, USA
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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.

Cite this article

Otis Chodosh, Max Engelstein, Luca Spolaor, The Riemannian quantitative isoperimetric inequality. J. Eur. Math. Soc. 25 (2023), no. 5, pp. 1711–1741

DOI 10.4171/JEMS/1223