Eigenfunctions in Finsler Gaussian solitons

Gaussian solitons are important examples in the theory of Riemannian measure space. In the first part, we explicitly characterize the first eigenfunctions of the drift Laplacian in a Gaussian shrinking soliton, which shows that apart from each coordinate function, other first eigenfunctions must involve exponential functions and the so-called error functions. Moreover, the second eigenfunctions are also described. In the second part, we discuss the corresponding issues in Finsler Gaussian shrinking solitons, which is a natural generalization of Gaussian shrinking solitons. For the first eigenfunction, we complement an example to show that if a coordinate function is a first eigenfunction, then the Finsler Gaussian shrinking soliton must be a Euclidean measure space. For the second eigenfunction, we give some necessary and sufficient conditions for these spaces to be Euclidean measure spaces.