On Harnack Inequalities and Singularities of Admissible Metrics in the Yamabe Problem
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چکیده
In this paper we study the local behaviour of admissible metrics in the kYamabe problem on compact Riemannian manifolds (M, g0) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g0) is not conformally equivalent to the unit sphere S and that the set of all such metrics is compact. When (M,g0) 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 a recent result of Gursky and Viaclovski on the solvability of the k-Yamabe problem for k > n/2.
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