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This paper deals with the second Neumann eigenfunction ${u}$ of any planartriangle ${T}$. In a recent work by C. Judge and S. Mondal [Ann. Math., 2022],it was established that ${u}$ does not have any critical point within theinterior of ${T}$. In this paper, we show the uniqueness of non-vertex criticalpoint and the monotonicity property of the second eigenfunction. To be moreprecise, when ${T}$ is not an equilateral triangle, the non-vertex criticalpoint exists if and only if ${T}$ is an acute triangle that is not asuper-equilateral triangle, and the global extrema of ${u}$ are achieved at andonly at the endpoints of the longest side. This establishes the origin theoremand conjecture 13.6 initially posed by C. Judge and S. Mondal [Ann. Math.,2020]. Our proof relies heavily on continuity methods, eigenvalue inequalities,and the maximum principle to establish these results.

Uniqueness of critical points of the second Neumann eigenfunctions on triangles