A note on Yamabe constants of products with hyperbolic spaces

TL;DR

This paper investigates the Yamabe constants of products of hyperbolic spaces with compact manifolds, providing explicit numerical estimates for specific dimensions by leveraging the uniqueness of solutions to a related PDE.

Contribution

It offers new numerical estimates for Yamabe constants on hyperbolic product spaces using the uniqueness of solutions to a PDE on hyperbolic space.

Findings

Numerical estimates for (n,m)=(2,2),(2,3),(3,2)

Explicit calculations based on PDE solution uniqueness

Insights into Yamabe constants for hyperbolic product manifolds

Abstract

We study the H^n-Yamabe constants of Riemannian products (H^n \times M^m, g_h^n +g), where (M,g) is a compact Riemannian manifold of constant scalar curvature and g_h^n is the hyperbolic metric on H^n. Numerical calculations can be carried out due to the uniqueness of (positive, finite energy) solutions of the equation \Delta u -\lambda u + u^q =0 on hyperbolic space H^n under appropriate bounds on the parameters \lambda, q, as shown by G. Mancini and K. Sandeep. We do explicit numerical estimates in the cases (n,m)=(2,2),(2,3) and (3,2).