On the negative case of the Singular Yamabe Problem |
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Authors: | David L Finn |
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Institution: | (1) Department of Mathematics, Rose-Hulman Institute of Technology, 5500 Wabash Avenue, 47803 Terre Haute, IN |
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Abstract: | Let (M, g) be a compact Riemannian manifold of dimension n ≥3, and let Γ be a nonempty closed subset of M. The negative case
of the Singular Yamabe Problem concerns the existence and behavior of a complete metric g on M∖Γ that has constant negative
scalar curvature and is pointwise conformally related to the smooth metric g. Previous results have shown that when Γ is a
smooth submanifold (with or without boundary) of dimension d, there exists such a metric if and only if
. In this paper, we consider a more general class of closed sets and show the existence of a complete conformai metric ĝ with
constant negative scalar curvature which depends on the dimension of the tangent cone to Γ at every point. Specifically, provided
Γ admits a nice tangent cone at p, we show that when the dimension of the tangent cone to Γ at p is less than
then there cannot exist a negative Singular Yamabe metric ĝ on M∖Γ. |
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Keywords: | Math Subject Classifications" target="_blank">Math Subject Classifications 58G30 53C21 35J60 35B40 |
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