Integral pinching results for manifolds with boundary

Catino, Giovanni; Ndiaye, Cheikh Birahim

Catino, Giovanni ; Ndiaye, Cheikh Birahim

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 785-813.

Résumé

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.

MR 2789475 | Zbl 1246.53062

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},
     pages = {785--813},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {4},
     year = {2010},
     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, Tome 9 (2010) no. 4, pp. 785-813. http://www.numdam.org/item/ASNSP_2010_5_9_4_785_0/

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