A Liouville-type theorem for Finslerian exponentially harmonic functions
Bin Shen
TL;DR
The paper generalizes Liouville-type results for exponentially harmonic functions to forward complete Finsler metric measure spaces by formulating the exponential energy and its Euler–Lagrange equation, and by establishing a Bochner-type identity under bounds on mixed weighted Ricci curvature and non-Riemannian curvatures. A key contribution is a Laplacian-comparison framework adapted to the Finsler setting, which, together with a maximum-principle argument, yields that every bounded exponentially harmonic function must be constant when $^mRic^{\infty}_{\nabla r}\ge 0$, the S-curvature and non-Riemannian tensors are bounded, and the misalignment is finite. This Liouville-type theorem extends rigidity phenomena from Riemannian to anisotropic Finsler spaces and provides tools for analyzing nonlinear, anisotropic PDEs on such spaces. The results enhance understanding of harmonic-type PDEs in metric-measure spaces with rich Finslerian geometry and may inform future studies of nonlinear variational problems on these spaces.
Abstract
In this manuscript, we investigate the exponentially harmonic equation on noncompact forward complete Finsler metric measure spaces. We demonstrate that this Finslerian equation represents a critical point of an exponential energy functional. Furthermore, we establish that any bounded solution to this equation is constant, provided that the mixed weighted Ricci curvature is nonnegative and certain additional non-Riemannian tensors are bounded.