The hot spots conjecture on Gaussian spaces

We study the hot spots conjecture for domains in the Gaussian space $(\mathbb{R}^n, (2\pi)^{-n/2} e^{-|x|^2/2} dx)$ for $n \ge 2$. Given a bounded domain $\Omega$ with a piecewise smooth boundary, we consider the first nontrivial eigenfunction of the Ornstein--Uhlenbeck operator $L_\gamma = \Delta - \langle x, \nabla \rangle$ subject to Neumann or mixed Dirichlet--Neumann boundary conditions, and prove that its extrema are attained only on the boundary $\partial\Omega$. More precisely, we establish the conjecture for two classes of domains: (i) lip domains in Gaussian spaces with mixed boundary conditions, and (ii) $n$-symmetric domains whose intersection with some orthant is a lip domain. As a corollary, we show that any first nontrivial Neumann eigenfunction of a $2$-symmetric domain in the two-dimensional Gaussian space has no interior extrema, provided the second Neumann eigenvalue is simple. Our approach is based on a variational principle for the Hodge Laplacian on weighted manifolds and the Hodge decomposition of differential $1$-forms on Lipschitz domains, extending the variational method of Kennedy--Rohleder from the Euclidean setting to Gaussian spaces. Although de Dios Pont has shown that the hot spots conjecture can fail for certain convex domains endowed with suitable log-concave measures, our results identify broad classes of domains for which the conjecture remains valid in Gaussian spaces.