## Fundamental Gap of Convex Domains in Space Forms and Surface

### Abstract

The fundamental gap is the difference of the first two eigenvalues of the Laplacian, which is important both in mathematics and physics. We will review many recent fantastic results for convex domains in $\mathbb R^n, \mathbb S^n, \mathbb H^n$ with Dirichlet boundary conditions. Then we will present a very recent estimate for the convex domain in surfaces with positive curvature. The last result is joint with G. Khan, H. Nguyen, M. Tuerkoen.