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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 estimate of the first eigenfunction plays a crucial role, which was established for convex domains in the Euclidean space and the round sphere. Joint with G. Khan, H. Nguyen, and G. Wei, we obtain log-concavity estimates of the first eigenfunction for convex domains in surfaces of positive curvature and consequently establish fundamental gap estimates. In a subsequent work, together with G. Khan and G. Wei, we improve the log-concavity estimates and obtain stronger gap estimates which recover known results on the round sphere.