For isosceles trapezoids, parallelograms, and kites, the symmetry and non-vertex critical points of the second Neumann eigenfunction are now characterized by height thresholds.
Rohleder, A new approach to the hot spots conjecture
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Critical points of the second Neumann eigenfunctions on the quadrangles with symmetry
For isosceles trapezoids, parallelograms, and kites, the symmetry and non-vertex critical points of the second Neumann eigenfunction are now characterized by height thresholds.