In this paper, we study the hot spots conjecture on Riemannian manifolds with isothermal coordinates and analytic metrics, such as hyperbolic spaces $\mathbb{ D}^n$ and spheres $S^n$ for $n\geq 2$. We prove that for some (possibly non-convex) Lipschitz domains in such a Riemannian manifold, which are generalizations of lip domains and symmetric domains with two axes of symmetry in $\mathbb{ R}^2$, the hot spot conjecture holds.
Categories: math.SP math.DG