Some sharp isoperimetric theorems for Riemannian manifolds (Q2710409)

This paper is divided into two parts. In the first one (sections~2-4) the authors investigate the behavior of isoperimetric regions of small volume in a compact Riemannian manifold. In the second one (section~5) they prove some isoperimetric inequalities in a Riemannian surface.NEWLINENEWLINENEWLINEIn the first part of the paper several results are proved. The authors show in theorem~2.2 that isoperimetric regions of small volume in a compact Riemannian manifold can be rescaled to get a round sphere in the limit. In theorem~3.5 they show that in a smooth, compact, connected, \((n+1)\)-dimensional Riemannian manifold \(M\) with Ricci curvature \(\text{Ric}\geq nK_{0}\), the isoperimetric profile of \(M\) is smaller than or equal to the isoperimetric profile of the model space with constant sectional curvature \(K_{0}\). Finally, a last comparison result is proved in theorem~4.4 in either of the following two cases: (i) when the sectional curvatures \(K_{s}\) of \(M\) satisfy \(K_{s}< K_{0}\), or (ii) when \(K_{s}\leq K_{0}\) and \(G\leq G_{0}\), where \(G\), \(G_{0}\) are the Gauss-Bonnet-Chern integrands in \(M\) and the model space, respectively. The conclusion of theorem 4.4 is that the isoperimetric profile \(M\) is bounded below, for small volumes, by the isoperimetric profile of the complete simply connected manifold with constant sectional curvature \(K_0\). When \(n\) is even then \(G=G_{0}=0\) and condition (ii) is reduced to \(K_{s}\leq K_{0}\).NEWLINENEWLINENEWLINEThe main tool in the proof of theorem 4.4 is the Gauss-Bonnet-Chern formula. Theorem 4.4 is related to the so called Cartan-Hadamard conjecture or Aubin conjecture: ``let \(M^n\) be a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant \(K_0\leq 0\). Then the isoperimetric profile of \(M\) is bounded below by the isoperimetric profile of the complete simply connected space of constant sectional curvature \(K_0\)''. This conjecture has only been solved in dimension \(2\) by \textit{A. Weil} [C. R. Acad. Sci., Paris 182, 1069-1071 (1926; JFM 52.0712.05)], in dimension \(3\) by \textit{B. Kleiner} [Invent. Math. 108, 37-47 (1992; Zbl 0770.53031)], and in dimension \(4\) when \(K_0=0\) by \textit{C. Croke} [Comment. Math. Helv. 59, 187-192 (1984; Zbl 0552.53017)]. \textit{O. Druet} [Proc. Am. Math. Soc. 130, 2351-2361 (2002; Zbl 1067.53026)] has reached the same conclusion as in Theorem 4.4 by replacing the assumptions on the sectional curvatures by the bound \(S< n(n+1) K_{0}\) on the scalar curvature \(S\) of \(M\).NEWLINENEWLINENEWLINEThe second part of the paper is devoted to the proof of some isoperimetric inequalities on a surface \(M\) under the upper bound \(K\leq K_{0}\) on the Gauss curvature \(K\) of \(M\). Let \(L_{0}\) be the infimum of the lengths of simple closed curves on \(M\). Then it is proved in proposition~5.2 that if \(M\) has convex boundary (may be empty) and it is convex at infinity (may be compact) then the perimeter \(P\) of any region of area \(A\leq \text{area}(M)/2\) satisfies \( P^{2}\geq\min\{L_{0}^{2}, 4\pi A-K_{0}A^{2}\}\). It is also proved in theorem~5.3 that if \(S\) is a sphere, a complete convex plane, or a compact convex disc, then the perimeter of any smooth region of area \(A\) on \(S\) satisfies NEWLINE\[NEWLINE P^{2}\geq\min\{(2L_{0})^{2}, 4\pi A-K_{0}A^{2}\}. NEWLINE\]NEWLINE Both results are proved by using \textit{M. A. Grayson}'s curve shortening flow [Ann. Math. (2) 129, 71-111 (1989; Zbl 0686.53036)], which was previously used by \textit{I.~Benjamini} and \textit{J.~Cao} [Duke Math. J. 85, 359-396 (1996; Zbl 0886.53031)] to obtain isoperimetric inequalities on surfaces. For results related to the second part of this work readers may look at the reviewer's paper [Commun. Anal. Geom. 9, 1093-1138 (2001; Zbl 1018.53003)] and the references therein.