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The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics
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We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.
Forward citations
Cited by 2 Pith papers
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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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The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates
For fully nonlinear Loewner-Nirenberg equations with boundary data w=0, the hyperbolic solution is unique when μ_Γ^+>1, while for μ_Γ^+≤1 all solutions form a one-parameter family depending only on x_n.
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