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The fundamental gap is the difference of the first two eigenvalues of the Laplace operator, which is important both in mathematics and physics and has been extensively studied. For the Dirichlet boundary condition the log-concavity of the first eigenfunction plays a crucial role in proving lower bounds, which was established for convex domains in the Euclidean space and the round sphere. In recent works, there has been progress in proving gap estimates on perturbations of the round sphere in dimension two and conformal deformations in higher dimensions. For negatively curved spaces, it turns out that there is no uniform lower bound of the fundamental gap. Hence it is natural to ask whether one can prove a fundamental gap estimate, assuming a stronger notion of convexity. In recent work there has been progress on answering this question. This is based on joint work with G. Khan, H. Nguyen, S. Saha and G. Wei in various subsets

Fundamental Gap Estimates in Various Geometries