Abstract

In a Riemannian manifold a regular convex domain is said to be λ\lambda-convex if its normal curvature at each point is greater than or equal to λ>0\lambda>0. In a Hadamard manifold, the asymptotic behaviour of the quotient vol(Ωt)/vol(∂Ωt)\vol(\Omega_{t})/\vol(\partial\Omega_{t}) for a family of λ\lambda-convex domains Ωt\Omega_{t} expanding over the whole space has been studied and general bounds for this quotient are known. In this paper we improve this general result in the complex hyperbolic space CHn(−4k2)\mathbb{C}H^n(-4k^2), a Hadamard manifold with constant holomorphic curvature equal to −4k2-4k^2. Furthermore, we give some specific properties of convex domains in CHn(−4k2)\mathbb{C}H^n(-4k^2) and we prove that λ\lambda-convex domains of arbitrary diameter exists if λ≤k\lambda\leq k.

Author information

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Judit Abardia

Dep. de Matemàtiques, Facultat de Ciències, Universitat Autònoma, 08193–Bellaterra / Barcelona, Spain

juditab@mat.uab.cat

Eduardo Gallego

Dep. de Matemàtiques, Facultat de Ciències, Universitat Autònoma, 08193–Bellaterra / Barcelona, Spain

egallego@mat.uab.cat

Keywords

  • Complex hyperbolic space
  • convex domain
  • volume
  • area

Mathematics Subject Classification

52A20; 52A55

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Suggested citation

J. Abardia, E. Gallego. “Convexity on Complex Hyperbolic Space.” Journal of Convex Analysis 20 (2013), No. 2, 329–338. https://doi.org/10.68381/jca20021

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.