NDLI: On the negative case of the Singular Yamabe Problem

Content Provider	Springer Nature Link
Author	Finn, David L.
Copyright Year	1999
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 $$d > \frac{{n - 2}}{2}$$ . 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 $$\frac{{n - 2}}{2}$$ then there cannot exist a negative Singular Yamabe metric ĝ on M∖Γ.
Starting Page	73
Ending Page	92
Page Count	20
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	9
Issue Number	1
e-ISSN	1559002X
Language	English
Publisher	Springer-Verlag
Publisher Date	1999-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	conformai deformation scalar curvature singular solutions to semilinear elliptic equations tangent cones Differential Geometry Convex and Discrete Geometry Fourier Analysis Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Global Analysis and Analysis on Manifolds
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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