ncku-talk2022-09-15

Colloquium

DATE 2022-09-15　16:10-17:00

PLACE 數學系館 1F3174教室

SPEAKER 蘇瑋栢博士（中央研究院數學研究所）

TITLE Immersed Special Lagrangians and Lagrangian Mean Curvature Flow

ABSTRACT Over the last two decades, the existence of special Lagrangian submanifolds in Calabi-Yau manifolds has attracted much attention from both physics and mathematics. On the physics side, special Lagrangians are 'supersymmetric cycles' in String Theory, while on the mathematics side, they are 'volume-minimizers' in its homology class. It follows that, naively, one would try to use the mean curvature flow, the negative gradient flow of volume, to study the existence problem of special Lagrangians.
Motivated by 'Mirror Symmetry', the special Lagrangians are conjectured to be the generators of 'semi-stable objects' in the derived Fukaya category. Based on this idea, Thomas, Yau, and Joyce further conjectured that the mean curvature flow starting from a 'stable' Lagrangian submanifold would eventually converge to a unique special Lagrangian in its Fukaya isomorphism class.
Due to the higher-codimension nature of the mean curvature flow of Lagrangian submanifolds, the embeddedness is not preserved along the flow. I will explain recent joint work with Chung-Jun Tsai and Albert Wood on constructing a long-time solution with a non-smooth convergence behaviour of Lagrangian mean curvature flow to a smooth, immersed, special Lagrangian. This shows the necessity of including immersed Lagrangians as objects of the Fukaya category defined by Fukaya-Oh-Ohta-Ono. Our construction is a geometric 'parabolic glueing method' based on the construction of ancient solutions of the Ricci flow by Brendle-Kapouleas.

Colloquium

DATE	2022-09-15　16:10-17:00

PLACE	數學系館 1F3174教室

SPEAKER	蘇瑋栢博士（中央研究院數學研究所）

TITLE	Immersed Special Lagrangians and Lagrangian Mean Curvature Flow

ABSTRACT	Over the last two decades, the existence of special Lagrangian submanifolds in Calabi-Yau manifolds has attracted much attention from both physics and mathematics. On the physics side, special Lagrangians are 'supersymmetric cycles' in String Theory, while on the mathematics side, they are 'volume-minimizers' in its homology class. It follows that, naively, one would try to use the mean curvature flow, the negative gradient flow of volume, to study the existence problem of special Lagrangians. Motivated by 'Mirror Symmetry', the special Lagrangians are conjectured to be the generators of 'semi-stable objects' in the derived Fukaya category. Based on this idea, Thomas, Yau, and Joyce further conjectured that the mean curvature flow starting from a 'stable' Lagrangian submanifold would eventually converge to a unique special Lagrangian in its Fukaya isomorphism class. Due to the higher-codimension nature of the mean curvature flow of Lagrangian submanifolds, the embeddedness is not preserved along the flow. I will explain recent joint work with Chung-Jun Tsai and Albert Wood on constructing a long-time solution with a non-smooth convergence behaviour of Lagrangian mean curvature flow to a smooth, immersed, special Lagrangian. This shows the necessity of including immersed Lagrangians as objects of the Fukaya category defined by Fukaya-Oh-Ohta-Ono. Our construction is a geometric 'parabolic glueing method' based on the construction of ancient solutions of the Ricci flow by Brendle-Kapouleas.