2002/11/30 by Dominic Joyce
Mathematics · Physics and Astronomy · #math.DG #hep-th
published as Annals of Global Analysis and Geometry 25 (2004), 201-251. · 48 pages. (v2) new material and references. (v3) Changed notation
arxiv created 2003/03/21 · arxiv updated 2009/11/30
This is the first in a series of five papers math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x1,...,xn locally modelled on special Lagrangian cones C1,...,Cn in Cm with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272, which surveys the series, gives examples, and applies the results to prove some conjectures. This first paper lays the foundations for the series, giving definitions and proving auxiliary results in symplectic geometry and asymptotic analysis that will be needed later. We also prove results on the regularity of X near its singular points. We show that X converges to the cone Ci near xi with all its derivatives, at rates determined by the eigenvalues of the Laplacian on the intersection of Ci with the unit sphere. We show that if X is a special Lagrangian integral current with a tangent cone C at x satisfying some conditions, then X has an isolated conical singularity at x in our sense. We also prove analogues of many of our results for Asymptotically Conical SL m-folds in Cm. The sequel math.DG/0211295 studies the deformation theory of compact SL m-folds X in M with conical singularities. The third and fourth papers math.DG/0302355, math.DG/0302356 construct desingularizations of X, realizing X as a limit of a family Nt of compact, nonsingular SL m-folds in M.