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

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

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 of the volume of a region in a Riemannian space*, Math. USSR-Sb.**17**(1972), 61-87. MR**0301671 (46:827)****[2]**-,*An inequality of the isoperimetric type for a domain in a Riemannian space*, Math. USSR-Sb.**19**(1973), 257-274. MR**0362159 (50:14601)****[3]**-,*Estimates of the length of a curve*, J. Differential Geometry**12**(1977), 99-115. MR**0470906 (57:10650)****[4]**D. Gromoll. W. Klingenberg and W. Meyer,*Riemannische Geometrie im Grossen*, Lecture Notes in Math., vol. 55 Springer-Verlag, Berlin and New York, 1968. MR**0229177 (37:4751)**

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

Article copyright:
© Copyright 1978
American Mathematical Society