*Published Paper*

**Inserted:** 24 nov 2008

**Last Updated:** 28 dec 2010

**Journal:** Ann. Sc. Norm. Super. Pisa Cl. Sci.

**Volume:** 9

**Pages:** 785-813

**Year:** 2010

**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.

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