The σk-Loewner-Nirenberg problem on Riemannian manifolds for k < n2
Let (Mn , g0) be a smooth compact Riemannian manifold of dimension n ≥ 3 with non-empty boundary ∂M. Let Γ ⊂ R n be a symmetric convex cone and f a symmetric defining function for Γ satisfying standard assumptions. Under an algebraic condition on Γ, which is satisfied for example by the G˚arding con...
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| Format: | Journal article |
| Language: | English |
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Mathematical Sciences Publishers
2024
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| _version_ | 1817931254904913920 |
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| author | Duncan, JAJ Nguyen, LUC |
| author_facet | Duncan, JAJ Nguyen, LUC |
| author_sort | Duncan, JAJ |
| collection | OXFORD |
| description | Let (Mn , g0) be a smooth compact Riemannian manifold of dimension n ≥ 3 with non-empty boundary ∂M. Let Γ ⊂ R n be a symmetric convex cone and f a symmetric defining function for Γ satisfying standard assumptions. Under an algebraic condition on Γ, which is satisfied for example by the G˚arding cones Γ+ k when k < n 2 , we prove the existence of a locally Lipschitz viscosity solution gu = e 2u g0 to the fully nonlinear Loewner-Nirenberg problem associated to (f, Γ), ( f(λ(−g −1 u Agu )) = 1, λ(−g −1 u Agu ) ∈ Γ on M\∂M u(x) → +∞ as distg0 (x, ∂M) → 0, where Agu is the Schouten tensor of gu. Previous results on Euclidean domains show that, in general, u is not differentiable. The solution u is obtained as the limit of smooth solutions to a sequence of fully nonlinear Loewner-Nirenberg problems on approximating cones containing (1, 0, . . . , 0), for which we also have uniqueness. In the process, we obtain an existence and uniqueness result for the corresponding Dirichlet boundary value problem with finite boundary data, which is also of independent interest. An important feature of our paper is that the existence of a conformal metric g satisfying λ(−g −1Ag) ∈ Γ on M is a consequence of our results, rather than an assumption. |
| first_indexed | 2024-12-09T03:19:06Z |
| format | Journal article |
| id | oxford-uuid:be61d655-2a65-43f8-a5d2-51388340daf4 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:19:06Z |
| publishDate | 2024 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | oxford-uuid:be61d655-2a65-43f8-a5d2-51388340daf42024-11-01T12:19:12ZThe σk-Loewner-Nirenberg problem on Riemannian manifolds for k < n2Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:be61d655-2a65-43f8-a5d2-51388340daf4EnglishSymplectic ElementsMathematical Sciences Publishers2024Duncan, JAJNguyen, LUCLet (Mn , g0) be a smooth compact Riemannian manifold of dimension n ≥ 3 with non-empty boundary ∂M. Let Γ ⊂ R n be a symmetric convex cone and f a symmetric defining function for Γ satisfying standard assumptions. Under an algebraic condition on Γ, which is satisfied for example by the G˚arding cones Γ+ k when k < n 2 , we prove the existence of a locally Lipschitz viscosity solution gu = e 2u g0 to the fully nonlinear Loewner-Nirenberg problem associated to (f, Γ), ( f(λ(−g −1 u Agu )) = 1, λ(−g −1 u Agu ) ∈ Γ on M\∂M u(x) → +∞ as distg0 (x, ∂M) → 0, where Agu is the Schouten tensor of gu. Previous results on Euclidean domains show that, in general, u is not differentiable. The solution u is obtained as the limit of smooth solutions to a sequence of fully nonlinear Loewner-Nirenberg problems on approximating cones containing (1, 0, . . . , 0), for which we also have uniqueness. In the process, we obtain an existence and uniqueness result for the corresponding Dirichlet boundary value problem with finite boundary data, which is also of independent interest. An important feature of our paper is that the existence of a conformal metric g satisfying λ(−g −1Ag) ∈ Γ on M is a consequence of our results, rather than an assumption. |
| spellingShingle | Duncan, JAJ Nguyen, LUC The σk-Loewner-Nirenberg problem on Riemannian manifolds for k < n2 |
| title | The σk-Loewner-Nirenberg problem on Riemannian manifolds for k < n2 |
| title_full | The σk-Loewner-Nirenberg problem on Riemannian manifolds for k < n2 |
| title_fullStr | The σk-Loewner-Nirenberg problem on Riemannian manifolds for k < n2 |
| title_full_unstemmed | The σk-Loewner-Nirenberg problem on Riemannian manifolds for k < n2 |
| title_short | The σk-Loewner-Nirenberg problem on Riemannian manifolds for k < n2 |
| title_sort | σk loewner nirenberg problem on riemannian manifolds for k n2 |
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