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Comparing the relative volume with a revolution manifold as a model. (English) Zbl 0794.53036
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 P. Berard and S. Gallot [Semin. Gaulaouic-Meyer- Schwartz, Equations Dériv. Partielles 1983-1984, Exp. No. 15, 34 p. (1984; Zbl 0542.53025)] and 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.
Reviewer: I.Pop (Iaşi)
##### MSC:
 53C40 Global submanifolds 53C20 Global Riemannian geometry, including pinching
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##### References:
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