Characterization of Curvature positivity of Riemannian metrics on flat vector bundles

TL;DR

This paper characterizes Nakano positivity of flat Riemannian vector bundles on bounded domains using solvability of the d-equation with specific L^2 estimates, providing an alternative proof of Prekopa's theorem.

Contribution

It offers a new characterization of Nakano positivity for flat vector bundles and applies this to give an alternative proof of Prekopa's theorem.

Findings

Characterization of Nakano positivity via d-equation solvability

Alternative proof of matrix-valued Prekopa's theorem

Extension of methods from Hermitian holomorphic 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.