Integral pinching results for manifolds with boundary

TL;DR

This paper establishes integral pinching conditions under which certain manifolds with boundary are spherical space forms, and explores related spectral properties and conformal metrics in conformal geometry.

Contribution

It provides new integral pinching criteria that classify manifolds with boundary as spherical space forms and analyzes associated conformal invariants and operators.

Findings

3D manifolds with boundary are spherical space forms under integral pinching.

4D manifolds with boundary are spherical space forms under integral pinching.

A conformally invariant operator has trivial kernel and is non-negative under certain conditions.

Abstract

We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the $L^2$-norm of their scalar curvature and the $L^2$-norm of their Ricci tensor are spherical space forms with totally geodesic boundary. Moreover, we prove also that 4-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe invariant and an explicit integral pinching between the total integral of their $(Q,T)$-curvature and the $L^2$-norm of their Weyl curvature are spherical space forms with totally geodesic boundary. As a consequence of our work, we show that a certain conformally invariant operator which plays an important role in Conformal Geometry has a trivial kernel and is non-negative if the Yamabe invariant is positive and verifies a pinching condition together with the total integral of the $(Q,T)$-curvature. As an application of the latter spectral analysis, we show the existence of conformal metrics with constant $Q$-curvature, constant $T$-curvature, and zero mean curvature under the latter assumptions.