Existence of complete conformal metrics of negative Ricci curvature on manifolds with boundary
Existence of complete conformal metrics of negative Ricci curvature on manifolds with boundary
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DOI：
10.1007/s00526-010-0352-0
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发表时间：
2009-07
影响因子：
2.1
通讯作者：
M. Gursky;J. Streets;M. Warren
中科院分区：
文献类型：
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作者：
M. Gursky;J. Streets;M. Warren
我们证明，在有边界的紧黎曼流形上，存在 u|∂M== 0 且 u 解决了 σk-Ricci 问题。在 k=n 的情况下，度量具有负 Ricci 曲率。此外，我们证明了内部解决 σk-Ricci 问题时存在完整的共形相关度量。通过采用（Mazzeo and Pacard, Pacific J. Math. 212(1), 169–185 (2003)）的结果，我们展示了我们构建的完整度量与庞加莱-爱因斯坦度量的存在之间的有趣关系。最后对正曲率情况下的相应问题进行简要讨论。
We show that on a compact Riemannian manifold with boundary there existssuch that,u|∂M≡ 0 andusolves theσk-Ricci problem. In the casek=nthe metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the interior solving theσk-Ricci problem. By adopting results of (Mazzeo and Pacard, Pacific J. Math. 212(1), 169–185 (2003)), we show an interesting relationship between the complete metrics we construct and the existence of Poincaré–Einstein metrics. Finally we give a brief discussion of the corresponding questions in the case of positive curvature.