Critical points of the second Neumann eigenfunctions on the quadrangles with symmetry

In this paper, we focus primarily on the symmetry properties of the second Neumann eigenfunction $u$ with respect to the symmetry axis or symmetry center of the relevant domain $Q$, such as isosceles trapezoids, parallelograms, kite domains, and we provide some affirmative answers to the Hot Spots Conjecture for these domains. Our proofs combine symmetry decomposition, comparison of eigenvalues, and the continuity method. Precisely, we have the following three aspects of results. (1) when $Q$ is an isosceles trapezoid, if the base angle $\alpha\le \frac{\pi}{3}$, $u$ is antisymmetric about the symmetric axis; if the base angle $\alpha> \frac{\pi}{3}$, there exists a critical height $\hat{h}$, when height $h<\hat{h}$, $u$ is antisymmetric about the symmetric axis; when height $h>\hat{h}$, $u$ is symmetric about the symmetric axis; when height $h=\hat{h}$, the multiplicity of second Neumann eigenvalue is 2. Meanwhile, we fully characterize the location of non-vertex critical points of $u$ on $\overline{Q}$. (2) When $Q$ is a parallelogram, $u$ is centrally antisymmetric about the center of $Q$ and does not have any non-vertex critical points. In particular, when $Q$ is a rhombus, $u$ is symmetric with respect to the longer diagonal and is antisymmetric with respect to the short diagonal. (3) When $Q$ is a kite $P_1P_2P_3P_4$, where $P_1$ is the origin, $P_2=(a,-h)$ lies in the four quadrant, $P_3=(1,0)$ lies on the positive $x$-axis, and $P_4=(a,h)$ is symmetric with $P_2$ about $x$-axis which lies in the first quadrant. If $a\ge 2$, $u$ is antisymmetric about $x$-axis; if $0<a<2$, there exist two constants $h_0$ and $h_1$ ($h_0\le h_1$), when $h<h_0$, $u$ is symmetric about $x$-axis; when $h>h_1$, $u$ is antisymmetric about $x$-axis. Meanwhile, we fully characterize the location of non-vertex critical points of $u$ on $\overline{Q}$.