U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions
This is the first of three papers math.DG/0111326, math.DG/0204343 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....
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| Format: | Journal article |
| Language: | English |
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2001
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| author | Joyce, D |
| author_facet | Joyce, D |
| author_sort | Joyce, D |
| collection | OXFORD |
| description | This is the first of three papers math.DG/0111326, math.DG/0204343 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 number 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, so that u+iv is like a holomorphic function of x+iy. When a is nonzero, u,v are always smooth and N is always nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, which correspond to singular points of N. This paper focusses on the nonsingular case, when a is nonzero. We prove analogues for our nonlinear Cauchy-Riemann equation of well-known results in complex analysis. In particular, we prove existence and uniqueness for solutions of two Dirichlet problems derived from it. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with two kinds of boundary conditions. In the sequels we extend these results to the singular case a=0. The next paper math.DG/0111326 proves existence and uniqueness of continuous weak solutions to the two Dirichlet problems when a=0. This gives existence and uniqueness of a large class of singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The final paper math.DG/0204343 studies the nature of the singularities that arise, and constructs U(1)-invariant special Lagrangian fibrations of open sets in C^3. |
| first_indexed | 2024-03-07T05:19:35Z |
| format | Journal article |
| id | oxford-uuid:de6955ef-f4b0-4e69-8486-85246fb16e73 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T05:19:35Z |
| publishDate | 2001 |
| record_format | dspace |
| spelling | oxford-uuid:de6955ef-f4b0-4e69-8486-85246fb16e732022-03-27T09:32:05ZU(1)-invariant special Lagrangian 3-folds I. Nonsingular solutionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:de6955ef-f4b0-4e69-8486-85246fb16e73EnglishSymplectic Elements at Oxford2001Joyce, DThis is the first of three papers math.DG/0111326, math.DG/0204343 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 number 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, so that u+iv is like a holomorphic function of x+iy. When a is nonzero, u,v are always smooth and N is always nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, which correspond to singular points of N. This paper focusses on the nonsingular case, when a is nonzero. We prove analogues for our nonlinear Cauchy-Riemann equation of well-known results in complex analysis. In particular, we prove existence and uniqueness for solutions of two Dirichlet problems derived from it. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with two kinds of boundary conditions. In the sequels we extend these results to the singular case a=0. The next paper math.DG/0111326 proves existence and uniqueness of continuous weak solutions to the two Dirichlet problems when a=0. This gives existence and uniqueness of a large class of singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The final paper math.DG/0204343 studies the nature of the singularities that arise, and constructs U(1)-invariant special Lagrangian fibrations of open sets in C^3. |
| spellingShingle | Joyce, D U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions |
| title | U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions |
| title_full | U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions |
| title_fullStr | U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions |
| title_full_unstemmed | U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions |
| title_short | U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions |
| title_sort | u 1 invariant special lagrangian 3 folds i nonsingular solutions |
| work_keys_str_mv | AT joyced u1invariantspeciallagrangian3foldsinonsingularsolutions |