The holomorphic sectional curvature and "convex" real hypersurfaces in K\"ahler manifolds

TL;DR

This paper establishes a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of certain convex real hypersurfaces in K"ahler manifolds, advancing understanding of their curvature properties and positivity conditions.

Contribution

It provides a new lower bound for the Tanaka-Webster curvature of semi-isometrically immersed hypersurfaces in K"ahler manifolds with nonnegative holomorphic sectional curvature, under convexity assumptions.

Findings

Proves a sharp lower bound for Tanaka-Webster holomorphic sectional curvature.

Shows $rac12$-positivity for convex hypersurfaces in K"ahler manifolds.

Answers a 2012 question on scalar curvature positivity of convex domain boundaries.

Abstract

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 K\"ahler 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 $\frac12$-positivity in the sense of Cao, Chang, and Chen, for compact "convex" real hypersurfaces in a K\"ahler 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 K\"ahler manifolds.