In this talk, we prove the hot spots conjecture for rotationally symmetric domains in $\mathbb R^n$by the continuity method. More precisely, we show that the odd Neumann eigenfunction in $x_n$ associated with lowest nonzero eigenvalue is a Morse function on the boundary, which has exactly two critical points and is monotone in the direction from its minimum point to its maximum point. As a consequence, we prove that the Jerison and Nadirashvili's conjecture B.3 holds true for rotationally symmetric domains and are also able to obtain a sharp lower bound for the Neumann eigenvalue. We will also discuss some recent results on n-axes symmetry or hyperbolic drum type domains.
