Rigidity results for Riemannian spin^c manifolds with foliated boundary

TL;DR

This paper establishes integral inequalities for solutions of the basic Dirac equation on Riemannian spin^c manifolds with foliated boundaries, characterizing cases of equality in specific geometric contexts like Kähler-Einstein manifolds.

Contribution

It introduces new integral inequalities for the basic Dirac operator on manifolds with foliated boundaries and characterizes the equality cases in certain geometric settings.

Findings

Derived integral inequalities involving mean curvature and O'Neill tensor.

Characterized equality cases for manifolds in Kähler-Einstein and product geometries.

Abstract

Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a K\"ahler-Einstein manifold or a Riemannian product of a K\"ahler-Einstein manifold with R (or with the circle S^1).