Special Lagrangian submanifolds with isolated conical singularities. I. Regularity

Abstract

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 x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m 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 C_i near x_i with all its derivatives, at rates determined by the eigenvalues of the Laplacian on the intersection of C_i 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 C^m. 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 N^t of compact, nonsingular SL m-folds in M.