$L^2$-estimates on flat vector bundles and Pr\'ekopa's theorem

TL;DR

This paper develops $L^2$-estimates for differential operators on flat vector bundles over convex manifolds and extends Prékopa's theorem, linking complex analysis, geometry, and convex analysis.

Contribution

It introduces new $L^2$-estimates for the operator $d$ on flat vector bundles and generalizes Prékopa's theorem to a geometric setting.

Findings

Established H"ormander's $L^2$-estimate in this context

Generalized Prékopa's theorem to convex Riemannian manifolds

Provided geometric applications of the $L^2$-estimates

Abstract

In this paper, we will construct H\"ormander's $L^2$-estimate of the operator $d$ on a flat vector bundle over a $p$-convex Riemannian manifold and discuss some geometric applications of it. In particular, we will generalize the classical Pr\'ekopa's theorem in convex analysis.