In this manuscript, we investigate Finslerian exponentially harmonic functions. Analogous to the Riemannian case, a function $u$ defined on a Finsler metric measure space $(M,F,\backslash mu)$ is said to be exponentially harmonic if it satisfies

$$\backslash tilde\{\backslash Delta\}\_\backslash mu\; u:=\backslash mathrm\{div\}\_\backslash mu(V(u)Du)=0,$$ (1.1)

where $V(u)=\backslash exp(\backslash frac\{1\}\{2\}F^\{*2\}(Du))$ represents the energy density of $u$ . Such a function arises as a critical point of the exponential energy functional (see Theorem 3.1). Finsler metric measure spaces encompass a richer set of geometric tensors compared to Riemannian manifolds (cf. [11]). It is important to emphasize that (1.1) is not merely a quasilinear elliptic equation, as the underlying space is an anisotropic and asymmetric manifold. Within this framework, we establish the following theorem.

Theorem 1.1. Let $(M,F,\backslash mu)$ be a forward complete $n$ -dimensional Finsler metric measure space with finite misalignment $\backslash alpha$ . Assume that the mixed weighted Ricci curvature $^m\backslash mathrm\{Ric\}^\backslash infty$ of $M$ is nonnegative, and that the $S$ -curvature as well as the non-Riemannian curvatures $U$ , $T$ and $\backslash mathrm\{div\}C=FC^i\_\{jk|i\}dx^j\backslash otimes\; dx^k$ satisfy the norm bound $F^\{-1\}|S|+F(U)+F(T)+\backslash |\backslash mathrm\{div\}C\backslash |\_\{HS(V)\}\backslash le\; K\_0$ , for any $(x,V)\backslash in\; SM$ . Then, any bounded exponentially harmonic function $u$ on $M$ is constant.

2 Related concepts and notations of the Finsler metric measure spaces

A Finsler metric measure space is a triple $(M,F,\backslash mu)$ , where $M$ is a differentiable manifold equipped with a Finsler metric $F$ and a Borel measure $\backslash mu$ . The Cartan tensor is defined by

$$C(X,Y,Z):=C\_\{ijk\}X^iY^jZ^k=\backslash frac\{1\}\{4\}\backslash frac\{\backslash partial^3F^2(x,y)\}\{\backslash partial\; y^i\backslash partial\; y^j\backslash partial\; y^k\}X^iY^jZ^k,$$

for any local vector fields $X,Y,Z$ . We always adopt the Chern connection, which is uniquely determined by

$$\backslash nabla\_X\; Y\; -\; \backslash nabla\_Y\; X\; =\; [X,\; Y];\; \backslash \backslash \; Z(g\_y(X,\; Y))\; -\; g\_y(\backslash nabla\_Z\; X,\; Y)\; -\; g\_y(X,\; \backslash nabla\_Z\; Y)\; =\; 2C\_y(\backslash nabla\_Z\; y,\; X,\; Y),$$

for any $X,Y,Z\backslash in\; TM\backslash setminus\backslash \{0\backslash \}$ , where $C\_y$ is the Cartan tensor. The coefficients of the Chern connection are locally expressed as $\backslash Gamma^i\_\{jk\}(x,y)$ in natural coordinates. These coefficients induce the spray coefficients as $G^i=\backslash frac\{1\}\{2\}\backslash Gamma^i\_\{jk\}y^jy^k$ . The spray is given by

$$G=y^i\backslash frac\{\backslash delta\}\{\backslash delta\; x^i\}=y^i\backslash frac\{\backslash partial\}\{\backslash partial\; x^i\}-2G^i\backslash frac\{\backslash partial\}\{\backslash partial\; y^i\},$$ (2.1)

where $\backslash frac\{\backslash delta\}\{\backslash delta\; x^i\}=\backslash frac\{\backslash partial\}\{\backslash partial\; x^i\}-N^j\_i\backslash frac\{\backslash partial\}\{\backslash partial\; y^j\}$ , and the nonlinear connection coefficients $N^i\_j$ are locally derived from the spray coefficients as $N^i\_j=\backslash frac\{\backslash partial\; G^i\}\{\backslash partial\; y^j\}$ . By convention, the horizontal Chern derivative is denoted by “ $|$ ” and the vertical Chern derivative by “ $;$ ”. Let $\backslash hat\{D\}$ denote the Levi-Civita connection of the induced Riemannian metric $\backslash hat\{g\}=g\_Y$ , and let $\backslash \{e\_i\backslash \}$

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