Yamabe metrics on cylindrical manifolds |
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Authors: | K Akutagawa B Botvinnik |
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Institution: | (1) Shizuoka University, Shizuoka, Japan. E-mail: smkacta@ipc.shizuoka.ac.jp, JP;(2) University of Oregon, Eugene, USA. E-mail: botvinn@math.uoregon.edu, US |
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Abstract: | We study a particular class of open manifolds. In the category of
Riemannian manifolds these are complete manifolds with cylindrical
ends. We give a natural setting for the conformal geometry on such
manifolds including an appropriate notion of the cylindrical Yamabe
constant/invariant. This leads to a corresponding version of the Yamabe
problem on cylindrical manifolds. We find a positive solution to
this Yamabe problem: we prove the existence of minimizing metrics
and analyze their singularities near infinity. These singularities turn
out to be of very particular type: either almost conical or almost cuspsingularities. We describe the supremum case, i.e., when the cylindrical
Yamabe constant is equal to the Yamabe invariant of the sphere.
We prove that in this case such a cylindrical manifold coincides conformally
with the standard sphere punctured at a finite number of
points. In the course of studying the supremum case, we establish a
Positive Mass Theorem for specific asymptotically flat manifolds with
two almost conical singularities. As a by-product, we revisit known
results on surgery and the Yamabe invariant.
Submitted: Submitted: August 2001. Revision: January 2003
RID="*"
ID="*"Partially supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 14540072. |
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