The volume of a slightly curved submanifold in a convex region

Author:
B. V. Dekster

Journal:
Proc. Amer. Math. Soc. **68** (1978), 203-208

MSC:
Primary 53C40

MathSciNet review:
0474147

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Abstract: Let *T* be a compact convex region in an *n*-dimensional Riemannian space, be the minimum sectional curvature in *T*, and be the minimum normal curvature of the boundary of *T*. Denote by a *v*-dimensional sphere, plane or hyperbolic plane of curvature . We assume that , *k* are such that on there exists a circumference of curvature *k*. Let be its radius. Now, let *Q* be a convex (in interior sense) *m*-dimensional surface in *T* whose normal curvatures with respect to any normal are not greater than *x* satisfying . Denote by the length of a circular arc of curvature *x* in with the distance between its ends. We prove that the volume of *Q* does not exceed the volume of a ball in of radius . These volumes are equal when *T* is a ball in and *Q* is its *m*-dimensional diameter.

**[1]**B. V. Dekster,*Estimates for the volume of a domain in a Riemannian space*, Mat. Sb. (N.S.)**88(130)**(1971), 61–87 (Russian). MR**0301671****[2]**B. V. Dekster,*An inequality of isoperimetric type for a domain in a Riemannian space*, Mat. Sb. (N.S.)**90(132)**(1973), 257–274, 326 (Russian). MR**0362159****[3]**B. V. Dekster,*Estimates of the length of a curve*, J. Differential Geometry**12**(1977), no. 1, 101–117. MR**0470906****[4]**D. Gromoll, W. Klingenberg, and W. Meyer,*Riemannsche Geometrie im Grossen*, Lecture Notes in Mathematics, No. 55, Springer-Verlag, Berlin-New York, 1968 (German). MR**0229177**

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DOI:
https://doi.org/10.1090/S0002-9939-1978-0474147-6

Article copyright:
© Copyright 1978
American Mathematical Society