Albert Borbély

The nonsolvability of the Dirichlet problem on negatively curved manifolds

Abstract The nonsolvability of the Dirichlet problem at infinity for negatively curved manifolds was proved recently by A. Ancona, using Brownian motion and probability theory. The aim of this paper is to give a different, nonprobabilistic proof of this result. The existence of bounded nontrivial harmonic functions which cannot be constructed via Chois method using convex sets, is also shown.

Some results on the convex hull of finitely many convex sets

A better than quadratic estimate is given for the volume of the convex hull of n points on Hadamard manifolds with pinched curvature. It was known previously that the volume is bounded by some polynomial in n. The estimate comes from the study of the convex hull of finitely many convex sets on Hadamard manifolds.

Sharp gradient estimates for eigenfunctions on Riemannian manifolds

Sharp gradient estimates are derived for positive eigenfunctions on complete Riemannian manifolds with Ricci curvature bounded below.

On the smoothness of the convex hull in negatively curved manifolds

In Hadamard manifolds the existence of suitable large convex sets is important for solving the Dirichlet problem at infinity. In this note we proveC1 boundary regularity of the convex hull of any compact setK away from points which lie on geodesics connecting points inK.

Geodesics avoiding subsets in Hadamard manifolds

Let M n , n ≥ 3, be an s-hyperbolic (in the sense of Gromov) Hadamard manifold. Let us assume that we are given a family of disjoint convex subsets and a point o outside these sets. It is shown that if one shrinks these sets by the constant s, then it is possible to find a complete geodesic through o that avoids the shrunk sets.

Volume Estimate Via Total Curvature in Hyperbolic Spaces

Let

On the total curvature of convex hypersurfaces in hyperbolic spaces

D\subset H^n(-k^2)

Elastic Splines I: Existence

be a convex compact subset of the hyperbolic space

IMMERSION OF MANIFOLDS WITH UNBOUNDED IMAGE AND A MODIFIED MAXIMUM PRINCIPLE OF YAU

H^n(-k^2)

ON MINIMAL SURFACES SATISFYING THE OMORI–YAU PRINCIPLE

with non-empty interior and smooth boundary. It is shown that the volume of D can be estimated by the total curvature of