Relation between area and volume for λ-convex sets in Hadamard manifolds
✍ Scribed by A.A. Borisenko; E. Gallego; A. Reventós
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 117 KB
- Volume
- 14
- Category
- Article
- ISSN
- 0926-2245
No coin nor oath required. For personal study only.
✦ Synopsis
It is known that for a sequence { t } of convex sets expanding over the whole hyperbolic space H n+1 the limit of the quotient vol( t )/vol(∂ t ) is less or equal than 1/n, and exactly 1/n when the sets considered are convex with respect to horocycles. When convexity is with respect to equidistant lines, i.e., curves with constant geodesic curvature λ less than one, the above limit has λ/n as lower bound. Looking how the boundary bends, in this paper we give bounds of the above quotient for a compact λ-convex domain in a complete simply-connected manifold of negative and bounded sectional curvature, a Hadamard manifold. Then we see that the limit of vol( t )/vol(∂ t ) for sequences of λ-convex domains expanding over the whole space lies between the values λ/nk 2 2 and 1/nk 1 .