Positivity of Riemannian metric and the existence theorem of $L^2$ estimates for $d$ operator

Abstract

Apply the $L^2$ technique developed by Deng-Ning-Wang-Zhou,we give a new characterization of Nakano $q$-positivity for Riemannian flat vector bundle in the sense of $L^2$ estimate for the $d$ operator. We also give a new characterization of the positivity of the curvature operator associated to the Riemannian metric on manifold in a local version in the sense of $L^2$ estimate. Then we prove two results parallel to Liu-Yang-Zhou's answer to Lempert's question to Hermitian holomorphic vector bundle. We also study the boundary convexity of relative compact domain in certain Riemannian manifold with convexity restrictions.

arXiv commentSome typos are corrected