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 {\it 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. 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 $P_1=(0,0)$, $P_2=(a,-h)$, $P_3=(1,0)$, and $P_4=(a,h)$. If $0<a\le 1$, there exists a critical height $\tilde{h}$, when $h<\tilde{h}$, $u$ is symmetric about $x$-axis; when $h>\tilde{h}$, $u$ is antisymmetric about $x$-axis. If $1<a<2$, 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. If $a\ge 2$, $u$ is antisymmetric about $x$-axis. Meanwhile, we fully characterize the location of non-vertex critical points of $u$ on $\overline{Q}$.