U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions
| dc.creator | Joyce, Dominic | |
| dc.date | 2002-04-29 | |
| dc.date | 2004-02-20 | |
| dc.date.accessioned | 2026-07-07T04:48:08Z | |
| dc.date.available | 2026-07-07T04:48:08Z | |
| dc.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. | |
| dc.description | 45 pages, LaTeX (v3) Some rewriting and improved proofs | |
| dc.identifier | https://arxiv.org/abs/math/0204343 | |
| dc.identifier | http://arxiv.org/abs/math/0204343 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63932 | |
| dc.subject | Differential Geometry | |
| dc.title | U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions | |
| dc.type | text |