Symmetry of Solutions to Fractional Semilinear Equations on Hyperbolic Spaces

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.