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.
Bibliographical noteFunding 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.
© 2022 European Mathematical Society.
- Quantitative stability
- isoperimetric inequality
- Łojasiewicz inequality