We prove that some 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.

@article{ASNSP_2010_5_9_4_785_0, author = {Catino, Giovanni and Ndiaye, Cheikh Birahim}, title = {Integral pinching results for manifolds with boundary}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {4}, year = {2010}, pages = {785-813}, zbl = {1246.53062}, mrnumber = {2789475}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_785_0} }

Catino, Giovanni; Ndiaye, Cheikh Birahim. Integral pinching results for manifolds with boundary. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 4, pp. 785-813. http://www.numdam.org/item/ASNSP_2010_5_9_4_785_0/

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