Power convexity of the torsion function on horo-convex domains in hyperbolic space
Wei Zhang, Qi Zhou
Abstract
In this paper, we study the torsion problem on bounded, smooth, strictly horo-convex domains in hyperbolic space. We prove that if the diameter of the domain is sufficiently small, then -u is strictly convex, where u denotes the torsion function. The main ingredients of our proof are the constant rank theorem, a boundary convexity estimate, and a deformation argument for the domain. We also show that the exponent 1/2 is optimal, even among strictly horo-convex domains of arbitrarily small diameter.
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