On Harnack inequalities and singularities of admissible metrics in the Yamabe problem |
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Authors: | Neil S Trudinger and Xu-Jia Wang |
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Institution: | (1) Centre for Mathematics and its Applications, Australian National University, Canberra, ACT, 0200, Australia |
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Abstract: | In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g
0) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g
0) is not conformally equivalent to the unit sphere S
n
and that the set of all such metrics is compact. When (M, g
0) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence
theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the
solvability of the k-Yamabe problem for k > n/2.
This work was supported by the Australian Research Council. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 53C21 58J05 |
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