Special Lagrangians are an important class of minimal submanifolds in Calabi-Yau manifolds introduced by Harvey-Lawson. Apart from the interest from geometric analysis, they also play important roles in mirror symmetry, serve as the conjectural stable objects in the Fukaya category and the special Lagrangians tori are the key objects in the Strominger-Yau-Zaslow conjecture. However, there are not many examples of special Lagrangians constructed in the literature. In this series of the talk, I will explain a construction of special Lagrangians in Calabi-Yau 3-folds with a K3-fibration near the adiabatic limits. The construction is a step toward the Donaldson-Scaduto conjecture. The talks assume minimal background in analysis. This is a joint work with Shih-Kai Chiu.
Talk 1: I will introduce the notion of Calabi-Yau and special Lagrangians and general schemes of constructing special Lagrangians. In particular, I will focus on the qualitative version of deformation of special Lagrangians.
Talk 2: I will introduce the Donaldson-Scaduto conjecture and the result with Chiu, which comes from a quantitative deformation of ansatz special Lagrangians. Then I will focus on the result of Li on the geometry of Calabi-Yau manifolds with a K3-fibration near adiabatic limits and in particular the weighted norm used.
Talk 3-4: These two talks will mainly focus on how to invert the Hodge-Dirac operators on the ansatz special Lagrangian by constructing a parametrix with suitable estimates. This is the key step of the deformation argument.