Metrics with prescribed Ricci curvature near the boundary of a manifold |
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Authors: | Artem Pulemotov |
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Institution: | 1. School of Mathematics and Physics, The University of Queensland, St Lucia, QLD, 4072, Australia 2. Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL, 60637-1514, USA
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Abstract: | Suppose $M$ is a manifold with boundary. Choose a point $o\in \partial M$ . We investigate the prescribed Ricci curvature equation $\mathop {\mathrm{Ric}}\nolimits (G)=T$ in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian metric. The letter $T$ on the right-hand side denotes a (0,2)-tensor. Our main theorems address the questions of the existence and the uniqueness of solutions. We explain, among other things, how these theorems may be used to study rotationally symmetric metrics near the boundary of a solid torus $\mathcal{T }$ . The paper concludes with a brief discussion of the Einstein equation on $\mathcal{T }$ . |
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