Characterization of Curvature positivity of Riemannian metrics on flat vector bundles

Abstract

We give a characterization of Nakano positivity of Riemannian flat vector bundles over bounded domains $D\subset {\mathbb{R}}^n$ in terms of solvability of the $d$ equation with certain good $L^2$ estimate condition. As an application, we give an alternative proof of the matrix-valued Prekopa's theorem that is originally proved by Raufi. Our methods are inspired by the recent works of Deng-Ning-Wang-Zhou on characterization of Nakano positivity of Hermitian holomorphic vector bundles and positivity of direct image sheaves associated to holomorphic fibrations.

Comments are welcome

References in corpus (2)

• Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar{\partial }$
• Curvature positivity of invariant direct images of Hermitian vector bundles