The celebrated hot spots conjecture says that the second Neumann eigenfunctions attain their (global) maximum (hottest point) only on the boundary of the domain.
Each vertex of a convex polygon in the plane is always a critical point of a Neumann eigenfunction. In addition, Judge and Mondal [Ann. Math., 2022] showed that there are no critical points in the interior of a triangle.
However, several open problems regarding the second Neumann eigenfunction in triangles remained open.
In this talk, I answer some of these unresolved problems, including
the uniqueness of non-vertex critical points,
the sufficient and necessary conditions for the existence of a critical point,
the exact location of the global maximum,
the location of the nodal line,
a new proof of the simplicity of the second Neumann eigenvalue,
and other. Our approach relies on the continuity method via domain deformation.
This is joint work with Prof. Hongbin Chen and Prof. Changfeng Gui.