Integral pinching results for manifolds with boundary

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.

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.

Classification : 53C24, 53C20, 53C21, 53C25

[1] A. L. Besse, “Einstein Manifolds”, Springer Verlag, Berlin, 1987. | MR | Zbl

[2] T. P. Branson, “The Functional Determinant”, Global Analysis Research Center, Lecture Notes Series, Vol. 4, Seoul National University, 1993. | MR | Zbl

[3] T. P. Branson, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293–345. | EuDML | MR | Zbl

[4] T. P. Branson and B. Oersted, Explicit functional determinants in four dimensions, Proc. Amer. Math. Soc. 113 (1991), 669–682. | MR | Zbl

[5] M. Berger, Les variétés riemanniennes $\frac{1}{4}$-pincées. Ann. Scuola Norm. Sup. Pisa 14 (1960), 161–170. | EuDML | Numdam | MR | Zbl

[6] S. Brendle and R. M. Schoen, Classification of manifolds with weakly $\frac{1}{4}$-pinched curvatures, Acta Math. 200 (2008), 1–13. | MR | Zbl

[7] C. Bohm and B. Wilking, Manifolds with positive curvature operator are space-forms, Ann. of Math. 167 (2008) 1079–1097. | MR | Zbl

[8] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261–301. | MR | Zbl

[9] G. Catino and Z. Djadli, Conformal deformations of integral pinched 3-manifolds, Adv. Math. 223 (2010), 393–404. | MR | Zbl

[10] G. Catino, Z. Djadli and C. B. Ndiaye, A sphere theorem on locally conformally flat even–dimensional manifolds, 2008, preprint. | MR | Zbl

[11] S. S. Chen, Local estimates for some fully nonlinear elliptic equations, Int. Math. Res. Not. 55 (2005), 3403–3425. | MR | Zbl

[12] S. S. Chen, Conformal deformation on manifolds with boundary, Geom. Funct. Anal. 19 (2009), 1024–1064. | MR

[13] S. Y. A. Chang and J. Qing, The zeta functional determinants on manifolds with boundary. I. The formula, J. Funct. Anal. 147 (1997), 327–362. | MR | Zbl

[14] S. Y. A. Chang and J. Qing, The zeta functional determinants on manifolds with boundary. II. Extremal metrics and compactness of isospectral set, J. Funct. Anal. 147 (1997), 363–399. | MR | Zbl

[15] S. Y. A. Chang, M. J. Gursky and P. C. Yang, A conformally invariant sphere theorem in four dimensions, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 105–143. | EuDML | Numdam | MR | Zbl

[16] S. Y. A. Chang and P. C. Yang, On a fourth order curvature invariant, In: “Comtemporary Mathematics”, Vol. 237, Spectral Problems in Geometry and Arithmetic, Ed. T. Branson, AMS, 1999, 9–28. | MR

[17] Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant $Q$-curvature, Ann. of Math. 168 (2008), 813–858. | MR | Zbl

[18] J. F. Escobar, The Yamabe problem on manifolds with boundary, J. Differential Geom. 35 (1992), 21–84. | MR | Zbl

[19] L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math. 35 (1982), 333–363. | MR | Zbl

[20] C. Fefferman and C. R. Graham, $Q$-curvature and Poincaré metrics, Math. Res. Lett. 9 (2002), 139–151. | MR | Zbl

[21] C. Fefferman and C. Graham, Conformal invariants, In: “Élie Cartan et les mathématiques d’aujourd’hui”, Asterisque, 1985, 95–116. | Numdam | MR | Zbl

[22] R. S. Hamilton, Three manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306. | MR | Zbl

[23] Y. Ge, C.-S. Lin and G. Wang, On the ${\sigma }_2$-scalar curvature, J. Differential Geom. 84 (2010), 45–86. | MR

[24] D. Gilbar and N. Trudinger, “Elliptic Partial Differential Equations of Second Order”, 2^nd edition, Springer-Verlag, 1983. | MR

[25] C. R. Graham, R. Jenne, L. Mason and G. Sparling, Conformally invariant powers of the Laplacian, I: existence, J. London Math. Soc. 46 (1992), 557–565. | MR | Zbl

[26] M. J. Gursky and J.A. Viaclovsky, A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom. 63 (2003), 131–154. | MR | Zbl

[27] W. Klingenberg, Über Riemannsche Mannigfaltigkeiten mit nach oben beschränkter Krümmung. Ann. Mat. Pura Appl. 60 (1962), 49–59. | MR | Zbl

[28] N.V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 75–108. | MR

[29] P. L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic Hamilton-Jacobi-Bellman equation, Math. Z. 191 (1986), 11–15. | EuDML | MR | Zbl

[30] C. Margerin, A sharp characterization of the smooth 4-sphere in curvature terms, Comm. Anal. Geom. 6 (1998), 21–65. | MR | Zbl

[31] C. B. Ndiaye, Conformal metrics with constant $Q$-curvature for manifolds with boundary, Comm. Anal. Geom. 16 (2008), 1049–1124. | MR | Zbl

[32] C. B. Ndiaye, Constant $T$-curvature conformal metric on 4-manifolds with boundary, Pacific J. Math. 240 (2009), 151–184. | Zbl

[33] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), 3 pages. | EuDML | Zbl

[34] S. Paneitz, Essential unitarization of symplectics and applications to field quantization, J. Funct. Anal. 48 (1982), 310–359. | Zbl

[35] H. E. Rauch, A contribution to differential geometry in the large, Ann. of Math. 54 (1951), 38–55. | Zbl

[36] J. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J. 101 (2000), 283–316. | Zbl

[37] Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 (1996), 203–221. | Zbl

[38] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275. | Zbl