Abstract: Lagrangian Mean Curvature Flow (LMCF) provides a natural way to find special Lagrangian. In this talk, we discuss papers by Lotay-Oliveira that investigates the LMCF in very special ambient spaces, which are HyperKahler 4-manifolds generated by Gibbon-Hawking Ansatz. Notably, these spaces include the Multi-Eguchi-Hanson (ALE) and Multi-Taub-NUT (ALF). Moreover, HyperKahler 4-manifold in Gibbon-Hawking Ansatz admits a U(1)-symmetry which makes studying the U(1)-invariant Lagrangian surfaces and U(1)-invariant special Lagrangian surfaces very accessible. In particular, a U(1)-invariant LMCF in Gibbon-Hawking Ansatz will reduce to modified curve shortening flow on a plane by the HyperKahler moment map. This reduction provides us opportunities to understand various statements in the field of LMCF such as the notion of stability in Thomas-Yau conjecture and neck-pinch phenomenon in Joyce conjecture.

 

References:

Lotay, J.D. and Oliveira, G., 2020. Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz. arXiv preprint arXiv:2002.10391.

Lotay, J.D. and Oliveira, G., 2023. Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry. arXiv preprint arXiv:2305.05744.