Harmonic Analysis and Differential Equations Seminar: Yusen Xia (USCB)

Abstract: How small can the first nonzero Neumann eigenvalue of a triangle be when its diameter is fixed? I will discuss a sharp answer for geodesic triangles in two-dimensional space forms, extending a Euclidean result of Laugesen and Siudeja. In the spherical case, we assume the diameter is at most (\pi/2), and new equality phenomena occur at the endpoint.

The proof combines eigenvalue comparison with properties of nodal sets, giving a unified treatment of positive, zero, and negative curvature. I will also discuss a hot-spots theorem for non-acute spherical triangles: apart from a special equality family, the maximum and minimum of the first nonconstant Neumann eigenfunction occur at the endpoints of the longest side.

This is joint work with Shoo Seto and Guofang Wei.