Uniqueness of critical points of the second Neumann eigenfunctions on triangles

This paper investigates the second Neumann eigenfunction $u$ of a planar triangle $T$. In a recent paper by Judge and Mondal [Ann. Math., 2022], it was shown that $u$ has no critical points in the interior of $T$. In this paper, we show that $u$ has at most one non-vertex critical point and that $u$ is monotone in a certain direction in $T$. More precisely, when $T$ is not equilateral, we show that $u$ vanishes at some vertex if and only if $T$ is superequilateral, and that $u$ has a non-vertex critical point if and only if $T$ is acute and not superequilateral. These results confirm both the original theorem and Conjecture 13.6 of Judge and Mondal [Ann. Math., 2020]. We also resolve the objective of Polymath 7 (research thread 1), namely, that the extrema of $u$ are attained only at the endpoints of the longest side. In addition, we settle a conjecture of Siudeja [Proc. Amer. Math. Soc., 2016] on the ordering of mixed Dirichlet--Neumann Laplacian eigenvalues for triangles. Our proofs combine the continuity method, eigenvalue inequalities, the maximum principle, and the moving plane method.