Comparing the relative volume with a revolution manifold as a model (Q1310422)

Given a pair \((P,M)\), where \(M\) is an \(n\)-dimensional connected compact Riemannian manifold and \(P\) is a connected compact hypersurface of \(M\), the relative volume of \((P,M)\) is the quotient \(\text{volume}(P)/ \text{volume} (M)\). In this paper the authors give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature of \(M\) and the mean curvature of \(P\), with respect to that of a model pair \(({\mathcal P},{\mathcal M})\) where \({\mathcal M}\) is a revolution manifold and \({\mathcal P}\) a ``parallel'' of \({\mathcal M}\). From \textit{P. Berard} and \textit{S. Gallot} [Semin. Gaulaouic-Meyer- Schwartz, Equations Dériv. Partielles 1983-1984, Exp. No. 15, 34 p. (1984; Zbl 0542.53025)] and \textit{S. Gallot} [On the geometry of differentiable manifolds, Workshop, Rome/Italy 1986, Astérisque 163- 164, 31-91 (1988; Zbl 0674.53001)], a compact revolution manifold is a \(C^ \infty\) (compact) Riemannian manifold \((M^ \varphi; \langle\;,\;\rangle)\) of dimension \(n\) for which there are two points \({\mathcal N}\) and \({\mathcal S}\) of \(M^ \varphi\), a real number \(L>0\), a function \(\varphi: [0,2L]\to [0,+\infty)\) and a diffeomorphism \(f: (0,2L)\times S^{n-1}\to M-\{{\mathcal N},{\mathcal S}\}\) such that at each \((s,x)\in (0,2L)\times S^{n- 1}\) we have \(f^*\langle\;,\;\rangle= ds^ 2+ \varphi^ 2(s)\langle\;,\;\rangle_{S^{n-1}}\). A compact revolution manifold \(M^ \varphi\) is symmetric if the curvature \(\varphi(t)\) is symmetric with respect to \(L\) (i.e. \(\rho(2L-t)= \rho(t)\), or, equivalently, \(\rho(L-t)= \rho(L+t)\)). When \(M^ \varphi\) is symmetric it is said that \(M^ \varphi\) is lengthened if \(\rho\) is decreasing on the interval \([0,L]\), and \(M^ \varphi\) is called flattened if \(\rho\) is increasing on \([0,L]\). The main result: Theorem 1.6. Let \(M^ \varphi\) be a compact lengthened convex revolution manifold with sectional curvature \(\rho(t)\). Let \(R\in [0,L]\), and let \(M\) and \(P\) be as before. If \(\text{Ric} (\gamma_ p'(t), \gamma_ p'(t))\geq (n-1) \rho(R-t)\) for every \(t\) such that \(-c(-N(p))\leq t\leq c(N(p))\), \(| H|\leq \varphi'(R)/ \varphi(R)\), then \(v(P,M)\geq v(S_ R^ \varphi, M^ \varphi)\). Moreover, the equality implies that there is an isometry between \(M\) and \(M\) sending \(P\) onto \(S_ R\). A similar theorem when \(M\) is a compact flattened convex revolution manifold, if this exists, is not proved so far.