Integral pinching results for manifolds with boundary
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 4, p. 785-813
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 $\left(Q,T\right)$-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 $\left(Q,T\right)$-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.
Classification:  53C24,  53C20,  53C21,  53C25
@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, Série 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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