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Integral pinching results for manifolds with boundary. (English) Zbl 1246.53062

Summary: We prove that Riemannian manifolds with boundary satisfying an explicit integral pinching condition are spherical space-forms. More precisely, we show that three-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the \(L^2\)-norm of the scalar curvature and the \(L^2\)-norm of the Ricci tensor are spherical space-forms with totally geodesic boundary. Moreover, we also prove that four-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe invariant and an explicit integral pinching between the total integral of the (\(Q, T \))-curvature and the \(L^2\)-norm of the Weyl curvature are spherical space-forms with totally geodesic boundary. As a consequence, we show that a certain conformally invariant operator, which plays an important role in conformal geometry, is non-negative and has trivial kernel 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.

MSC:

53C24 Rigidity results
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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