The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds
We prove a sharp lower bound for the Tanaka–Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are “semi-isometrically” immersed in a Kähler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka–Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb C ^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac 12$-positivity in the sense of Cao, Chang, and Chen, for compact “convex” real hypersurfaces in a Kähler manifold of nonnegative holomorphic sectional curvature. Our approach is rather simple and uses a version of the Gauss equation for semi-isometric CR immersions of pseudohermitian manifolds into Kähler manifolds.