Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry
Let <i>g</i><sub>0</sub> be a smooth Riemannian metric on a closed manifold <i>M<sup>n</sup></i> of dimension <i>n</i> ≥ 3. We study the existence of a smooth metric <i>g</i> conformal to <i>g</i><sub>0</sub...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Elsevier
2023
|
| _version_ | 1826311380741914624 |
|---|---|
| author | Duncan, JAJ Nguyen, L |
| author_facet | Duncan, JAJ Nguyen, L |
| author_sort | Duncan, JAJ |
| collection | OXFORD |
| description | Let <i>g</i><sub>0</sub> be a smooth Riemannian metric on a closed manifold <i>M<sup>n</sup></i> of dimension <i>n</i> ≥ 3. We study the existence of a smooth metric <i>g</i> conformal to <i>g</i><sub>0</sub> whose Schouten tensor <i>A<sub>g</sub></i> satisfies the differential inclusion λ(<i>g</i><sup>−1</sup><i>A<sub>g</sub></i>) ∈ Γ on <i>M<sup>n</sup></i>, where Γ ⊂ R<sup><i>n</i></sup> is a cone satisfying standard assumptions. Inclusions
of this type are often assumed in the existence theory for
fully nonlinear elliptic equations in conformal geometry. We
assume the existence of a continuous metric <i>g</i><sub>1</sub> conformal to <i>g</i><sub>0</sub> satisfying λ(<i>g</i><sub>1</sub><sup>−1</sup><i>A</i><sub><i>g</i></sub><sub>1</sub>) ∈ Γ̅' in the viscosity sense on <i>M<sup>n</sup></i>, together with a nondegenerate ellipticity condition, where Γ' = Γ or Γ' is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ<sub>2</sub> operator is a conformal invariant in three dimensions. We also give a
generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green’s functions for fully nonlinear Yamabe problems. |
| first_indexed | 2024-03-07T08:07:30Z |
| format | Journal article |
| id | oxford-uuid:042fcbd1-f5f3-4f17-975a-6c3fd22979ad |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:07:30Z |
| publishDate | 2023 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:042fcbd1-f5f3-4f17-975a-6c3fd22979ad2023-11-14T11:31:04ZDifferential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:042fcbd1-f5f3-4f17-975a-6c3fd22979adEnglishSymplectic ElementsElsevier2023Duncan, JAJNguyen, LLet <i>g</i><sub>0</sub> be a smooth Riemannian metric on a closed manifold <i>M<sup>n</sup></i> of dimension <i>n</i> ≥ 3. We study the existence of a smooth metric <i>g</i> conformal to <i>g</i><sub>0</sub> whose Schouten tensor <i>A<sub>g</sub></i> satisfies the differential inclusion λ(<i>g</i><sup>−1</sup><i>A<sub>g</sub></i>) ∈ Γ on <i>M<sup>n</sup></i>, where Γ ⊂ R<sup><i>n</i></sup> is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric <i>g</i><sub>1</sub> conformal to <i>g</i><sub>0</sub> satisfying λ(<i>g</i><sub>1</sub><sup>−1</sup><i>A</i><sub><i>g</i></sub><sub>1</sub>) ∈ Γ̅' in the viscosity sense on <i>M<sup>n</sup></i>, together with a nondegenerate ellipticity condition, where Γ' = Γ or Γ' is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ<sub>2</sub> operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green’s functions for fully nonlinear Yamabe problems. |
| spellingShingle | Duncan, JAJ Nguyen, L Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry |
| title | Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry |
| title_full | Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry |
| title_fullStr | Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry |
| title_full_unstemmed | Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry |
| title_short | Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry |
| title_sort | differential inclusions for the schouten tensor and nonlinear eigenvalue problems in conformal geometry |
| work_keys_str_mv | AT duncanjaj differentialinclusionsfortheschoutentensorandnonlineareigenvalueproblemsinconformalgeometry AT nguyenl differentialinclusionsfortheschoutentensorandnonlineareigenvalueproblemsinconformalgeometry |