This paper investigates the second Neumann eigenfunction ${u}$ of a planar triangle ${T}$ . In a recent paper by Judge and Mondal (Ann. Math. (2) 195(1):337–362, 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. (2) 191(1):167–211, 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. Am. Math. Soc. 144(6):2479–2493, 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.
