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Regularity of viscosity solutions of the $\sigma_k$-Yamabe-type Problem for $k>n/2$
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abstract
We study the regularity of Lipschitz viscosity solutions to the $\sigma_k$ Yamabe problem in the negative cone case. If either $k=n$ or the manifold is conformally flat and $k>n/2$, we prove that all Lipschitz viscosity solutions are smooth away from a closed set of measure zero. For the general $k>n/2$ case, under certain assumptions, we prove the existence of a Lipschitz viscosity solution that is smooth away from a closed set of measure zero.
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The $\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\frac{n}{2}$ and beyond
The fully nonlinear Loewner-Nirenberg problem is shown to admit solutions when mu_Gamma^+>1-delta, in particular for sigma_k with k<=n/2, and whenever any admissible conformal metric exists.
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