Special Lagrangians with multiple isolated singularities

Abstract

We extend the Caffarelli-Hardt-Simon perturbation argument for truncated regular minimal cones to the special Lagrangian setting and prove a bridge principle for regular special Lagrangian cones in the spirit of Nathan Smale. Our bridge principle yields a general existence theorem for conically singular special Lagrangian submanifolds with prescribed regular tangent cones: for any finite list of such cones in ${\mathbb{C}}^m$ having the same Lagrangian angle and suitably arranged, there exists a connected special Lagrangian submanifold with boundary and isolated conical singularities whose tangent cones at its singularities are precisely the prescribed cones. When applied to the graphical special Lagrangian cone in ${\mathbb{C}}^5$ recently discovered by Bhattacharya, Orriols, and Skorobogatova, we obtain infinite families of $C^{1,1}$ viscosity solutions to the special Lagrangian equation having any finite number of isolated singularities.

arXiv comment36 pages. Comments are welcome! v2: We added Corollary 1, in which we apply the main theorem to prove the existence of solutions to the special Lagrangian equation with multiple isolated singularities