Fundamental gap of convex domains in the sphere
Abstract：For a bounded convex domain on a Riemannian manifold, the fundamental gap is the difference of the first two non-trivial Dirichlet eigenvalues. In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture for convex domains in the Euclidean space, showing that the gap is at least as large as the one for a one-dimensional model. They also conjectured that similar results hold for spaces with constant sectional curvature. Very recently, on the unit sphere, Seto-Wang-Wei proved that the fundamental gap is greater than the gap of the one dimensional sphere model, in particular, ≥ 3 π^2/D^2 (n ≥ 3), provided the diameter of the domain D ≤ π/2. In a joint work with Guofang Wei at UCSB, we extend Seto-Wang-Wei’s lower bound estimate to all convex domains in the hemisphere.