Conformally bending three-manifolds with boundary.

*(English)*Zbl 1239.53047Conformal 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.

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.

Reviewer: Rolf Walter (Dortmund)

##### MSC:

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.) |

##### Keywords:

manifold with boundary; conformal bending; scalar curvature; pinching of sectional curvature; almost negative curvature##### References:

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