Symmetry of Solutions to Fractional Semilinear Equations on Hyperbolic Spaces

TL;DR

This paper investigates symmetry properties of solutions to fractional semilinear equations on hyperbolic spaces, deriving explicit Green's functions and extending maximum principles, using Helgason-Fourier analysis and the method of moving planes.

Contribution

It provides the first explicit Green's function for the fractional Laplacian on hyperbolic space and applies a novel symmetry proof technique in this setting.

Findings

Nonnegative solutions are symmetric on hyperbolic space.

Explicit Green's function for fractional Laplacian on hyperbolic space derived.

Extended maximum principles applicable to hyperbolic space.

Abstract

We study a semilinear equation involving the fractional Laplacian on the hyperbolic space $\mathbb{H}^n$. Unlike in conformally compact Einstein manifolds, the fractional Laplacian on $\mathbb{H}^n$ does not enjoy conformal covariance. By employing Helgason-Fourier analysis, we explicitly derive the Green's function of the fractional Laplacian on $\mathbb{H}^n$ as well as its asymptotic behaviors. We then apply a direct method of moving planes to the integral form of the equation, and show that nonnegative weak solutions are symmetric. In addition, we extend several maximum principles to hyperbolic space.