zbMATH — the first resource for mathematics

Conformally bending three-manifolds with boundary. (English) Zbl 1239.53047
Conformal deformations of Riemannian metrics have interesting aspects which attracted and attract many geometers and analysts. The goal is always to shape the deformation in such a way that nicer properties result in the end. The conditions for this goal mostly boil down to non-linear PDE’s on the conformal factor. For example, the Yamabe equation governs the problem to find a new conformal metric with constant scalar curvature. Cf. the solution for this famous problem in the compact case by Yamabe, Trudinger, Aubin, and Schoen, synoptically discussed in [J. M. Lee and T. H. Parker, Bull. Am. Math. Soc., New Ser. 17, 37–91 (1987; Zbl 0633.53062)].
The present paper belongs to this category in that one aims to equip a given compact Riemannian manifold with boundary of dimension \( 3 \) with a complete metric of negative scalar curvature in its interior. The dimension three comes in by the relation between the sectional curvature and the Einstein tensor. The main result would be implied by a former paper of the same authors [Calc. Var. Partial Differ. Equ. 41, No. 1-2, 21–43 (2011; Zbl 1214.53035)] where they solved, in arbitrary dimensions, an analogous problem for the elementary functions of the eigenvalues of the Ricci tensor. However, it is possible to impose in the present context an additional pinching property for the sectional curvatures, rendering the curvature almost negative. (Negative sectional curvature implies the ellipticity of the original PDE.) On the other hand, the PDE offers the difficulty of missing \( C^{2} \) estimates. The authors overcome this by the perturbation method, as usual in various other applications. The proof then proceeds by the construction of subsolutions and gradient estimates (at the boundary and in the interior), first \( C^{1} \) and then \( C^{2} \), and finally performing the limiting process of the perturbation parameter.
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
Full Text: DOI EuDML
[1] Aviles, Patricio; McOwen, Robert C., Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. J., 56, 2, 395-398, (1988) · Zbl 0645.53023
[2] Bavard, C., Courbure presque nĂ©gative en dimension \(3,\) Compositio Math., 63, 2, 223-236, (1987) · Zbl 0642.53047
[3] Buser, Peter; Gromoll, Detlef, Geometry and analysis on manifolds (Katata/Kyoto, 1987), 1339, On the almost negatively curved \(3\)-sphere, 78-85, (1988), Springer, Berlin · Zbl 0651.53032
[4] Evans, Lawrence C., Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35, 3, 333-363, (1982) · Zbl 0469.35022
[5] Gursky, Matthew; Streets, Jeffrey; Warren, Micah, Complete conformal metrics of negative Ricci curvature on manifolds with boundary · Zbl 1214.53035
[6] Krylov, N. V., Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47, 1, 75-108, (1983) · Zbl 0578.35024
[7] Loewner, Charles; Nirenberg, Louis, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Partial differential equations invariant under conformal or projective transformations, 245-272, (1974), Academic Press, New York · Zbl 0298.35018
[8] Lohkamp, Joachim, Negative bending of open manifolds, J. Differential Geom., 40, 3, 461-474, (1994) · Zbl 0840.53023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.