The Geometric Complexity of Special Lagrangian T 2-cones

We prove a number of results on the geometric complexity of special Lagrangian (SLG) T -cones in C. Every SLG T -cone has a fundamental integer invariant, its spectral curve genus. We prove that the spectral curve genus of an SLG T -cone gives a lower bound for its geometric complexity, i.e. the area, the stability index and the Legendrian index of any SLG T -cone are all bounded below by explicit linearly growing functions of the spectral curve genus. We prove that the cone on the Clifford torus (which has spectral curve genus zero) in S is the unique SLG T -cone with the smallest possible Legendrian index and hence that it is the unique stable SLG T -cone. This leads to a classification of all rigid “index 1” SLG cone types in dimension three. For cones with spectral curve genus two we give refined lower bounds for the area, the Legendrian index and the stability index. One consequence of these bounds is that there exist S-invariant SLG torus cones of arbitrarily large area, Legendrian and stability indices. We explain some consequences of our results for the programme (due to Joyce) to understand the “most common” three-dimensional isolated singularities of generic families of SLG submanifolds in almost Calabi-Yau manifolds.