SPECIAL LAGRANGIAN SUBMANIFOLDS OF LOG CALABI-YAU MANIFOLDS

We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kahler metric constructed by Tian and Yau. We prove that if X is a Tian-Yau manifold and if the compact Calabi-Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D is an element of vertical bar-K-Y vertical bar is a smooth divisor with D-2 = d, then X = Y\D admits a special Lagrangian torus fibration, as conjectured by Strominger-Yau-Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y = P-2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kahler rotation, X can be compactified to the complement of a Kodaira type I-d fiber appearing as a singular fiber in a rational elliptic surface (pi) over cap : (Y) over cap -> P-1.