U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions
This is the third in a series of three papers math.DG/0111324, math.DG/0111326 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surve...
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| Formaat: | Journal article |
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2002
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| author | Joyce, D |
| author_facet | Joyce, D |
| author_sort | Joyce, D |
| collection | OXFORD |
| description | This is the third in a series of three papers math.DG/0111324, math.DG/0111326 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. Let N be such a U(1)-invariant SL 3-fold. Then |z_1|^2-|z_2|^2=2a on N for some real a. Locally, N can be written as a kind of graph of functions u,v : R^2 --> R satisfying a nonlinear Cauchy-Riemann equation depending on a. When a is nonzero, u,v are smooth and N is nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. The first paper math.DG/0111324 studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy-Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The second paper math.DG/0111326 extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicity n>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in C^3, including singular fibres. |
| first_indexed | 2024-03-06T19:16:44Z |
| format | Journal article |
| id | oxford-uuid:18a65560-c4be-40ac-b28d-4a0a392a57c0 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:16:44Z |
| publishDate | 2002 |
| record_format | dspace |
| spelling | oxford-uuid:18a65560-c4be-40ac-b28d-4a0a392a57c02022-03-26T10:44:21ZU(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:18a65560-c4be-40ac-b28d-4a0a392a57c0Symplectic Elements at Oxford2002Joyce, DThis is the third in a series of three papers math.DG/0111324, math.DG/0111326 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. Let N be such a U(1)-invariant SL 3-fold. Then |z_1|^2-|z_2|^2=2a on N for some real a. Locally, N can be written as a kind of graph of functions u,v : R^2 --> R satisfying a nonlinear Cauchy-Riemann equation depending on a. When a is nonzero, u,v are smooth and N is nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. The first paper math.DG/0111324 studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy-Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The second paper math.DG/0111326 extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicity n>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in C^3, including singular fibres. |
| spellingShingle | Joyce, D U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions |
| title | U(1)-invariant special Lagrangian 3-folds. III. Properties of singular
solutions |
| title_full | U(1)-invariant special Lagrangian 3-folds. III. Properties of singular
solutions |
| title_fullStr | U(1)-invariant special Lagrangian 3-folds. III. Properties of singular
solutions |
| title_full_unstemmed | U(1)-invariant special Lagrangian 3-folds. III. Properties of singular
solutions |
| title_short | U(1)-invariant special Lagrangian 3-folds. III. Properties of singular
solutions |
| title_sort | u 1 invariant special lagrangian 3 folds iii properties of singular solutions |
| work_keys_str_mv | AT joyced u1invariantspeciallagrangian3foldsiiipropertiesofsingularsolutions |