Sharp isoperimetric inequalities for small volumes in complete noncompact Riemannian manifolds of bounded geometry involving the scalar curvature
Abstract
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 $k_0\in\mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of $\mathbb{M}^n_{k_0}$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2016
- DOI:
- 10.48550/arXiv.1611.01638
- arXiv:
- arXiv:1611.01638
- Bibcode:
- 2016arXiv161101638N
- Keywords:
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- Mathematics - Differential Geometry;
- 49Q20;
- 58E99;
- 53A10;
- 53C23;
- 53C21;
- 49Q05;
- 49Q10;
- 49Q15;
- 53C42;
- 83C99
- E-Print:
- 72 pages, 1 figure. Some typos and cut paste errors are corrected a new entry in the bibliography is added