The geometry of SO(p) × SO(q)-invariant special Lagrangian cones

摘要

SO(p) × SO(q)-invariant special Lagrangian cones in ?~(p+q)(equivalently, SO(p) × SO(q)-invariant special Legendrians in S~(2(p+q)-1)) are an important family of special Lagrangians (SL) whose basic features were studied in our previous paper [13]. In some ways, they play a role analogous to that of Delaunay surfaces in the geometry of CMC surfaces in ?~3; in particular, they are natural building blocks for our gluing constructions of higher-dimensional SL cones [9, 10, 12]. In this article, we study in detail their geometry paying special attention to features needed in our gluing constructions. In particular, we classify them up to congruence; we determine their full group of symmetries (including various discrete symmetries) in all cases; we prove that many of them are closed and embedded; and finally understand the limiting singular geometry with detailed asymptotics. In understanding the detailed asymptotics a fundamental role is played by a certain conserved quantity (a component of the torque) considered in [13].