A Poincar\'e formula for differential forms and applications

TL;DR

This paper establishes new Poincaré-type inequalities for differential forms on compact Riemannian manifolds with boundary, linking geometric curvature properties and characterizing limiting cases, with applications to inequalities involving mean and scalar curvatures.

Contribution

It introduces a novel Poincaré-type inequality for differential forms on manifolds with boundary and derives a new inequality involving boundary curvatures, including a characterization of limiting cases.

Findings

New Poincaré-type inequality for differential forms

Inequality involving mean and scalar curvatures of boundary

Characterization of limiting cases in codimension one

Abstract

We prove a new general Poincar\'e-type inequality for differential forms on compact Riemannian manifolds with nonempty boundary. When the boundary is isometrically immersed in Euclidean space, we derive a new inequality involving mean and scalar curvatures of the boundary only and characterize its limiting case in codimension one. A new Ros-type inequality for differential forms is also derived assuming the existence of a nonzero parallel form on the manifold.