Student Analysis Seminar (Fall 2025): Lagrangian Mean Curvature Flow
Student Analysis Seminar (Fall 2025): Lagrangian Mean Curvature Flow
Organizers: Raphael Tsiamis, Yipeng Wang
Time: Wednesdays 4:30 - 6:30 pm
Location: Columbia Math Department, Room 507
This seminar will study the properties and applications of Lagrangian mean curvature flow (LMCF). The LMCF is a nonlinear parabolic PDE with links to symplectic topology, Riemannian and complex geometry, and theoretical physics. Importantly, the LMCF provides a canonical way to deform Lagrangian submanifolds in Calabi-Yau manifolds with the goal of finding special Lagrangians, which are volume-minimizing within their homology classes. Consequently, the theory of LMCF combines techniques from complex geometry and mean curvature flow in high codimension.
In this seminar, we will begin by discussing the properties of Lagrangian submanifolds, which play a central role in understanding Calabi-Yau 3-folds and mirror symmetry (SYZ conjecture), as well as certain Gromov-Witten invariants. The construction of such submanifolds can be approached using mean curvature flow techniques, thanks to influential ideas of Thomas-Yau and M-T Wang.
This seminar will develop the foundational theory of Lagrangian mean curvature flow, including its existence and singularity formation properties, as well as other building blocks from the past twenty years. We will discuss important conjectures in the field, notably due to Thomas-Yau and Joyce, which seek to describe the long-time behavior of the flow, its singularity formation, and how one may flow past singularities. We will then venture into the more recent developments in the field, which include (but are not limited to) an understanding of singularity formation for the LMCF of surfaces, translating solutions, and immortal solutions.
Joyce, D., Lee, Y.-I., Tsui, M.-P. Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom. 84(1), 127-161 (2010)
Lambert, B., Lotay, J. D., Schulze, F. Ancient solutions in Lagrangian mean curvature flow, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22(3), 1169-1205 (2021)
Lotay, J.D., Schulze, F. and Székelyhidi, G. Ancient solutions and translators of Lagrangian mean curvature flow. Publ. math. IHES 140, 1–35 (2024). https://doi.org/10.1007/s10240-023-00143-5
Neves, A. Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent. math. 168, 449–484 (2007). https://doi.org/10.1007/s00222-007-0036-3
Neves, A. Singularities of Lagrangian mean curvature flow: monotone case. Math. Res. Lett. 17(1), 109-126 (2010)
Neves, A. Finite time singularities for Lagrangian mean curvature flow. Ann. of Math. (2) 177, 1029-1076 (2013). http://dx.doi.org/10.4007/annals.2013.177.3.5
Smoczyk, K. and Wang, M.-T., Mean curvature flows of Lagrangian submanifolds with convex potentials. J. Diff. Geom. 62 (2002) 243-257
Smoczyk, K. A canonical way to deform a Lagrangian submanifold. Preprint arXiv:dg-ga/9605005 (1996). https://arxiv.org/abs/dg-ga/9605005
Smoczyk, K. Longtime existence of the Lagrangian mean curvature flow. Calc. Var. PDE 20, 25–46 (2004). https://doi.org/10.1007/s00526-003-0226-9
Thomas, R.P. and Yau, S.-T. Special Lagrangians, stable bundles and mean curvature flow. Comm. Anal. Geom. 10(5), 1075-1113 (2002)
Tsai, C.-J., Tsui, M.-P. and Wang, M.-T., Mean curvature flows of two-convex Lagrangians. J. Diff. Geom. 128 (3), 1269-1284 (2024)
Wang, M.-T., Mean Curvature Flow of Surfaces in Einstein Four-Manifolds. J. Diff. Geom. 57(2): 301-338 (2001), DOI: 10.4310/jdg/1090348113
In the first meeting, we will set out the goals of the seminar and discuss various aspects of the references, schedule, and content of the talks.
In the first week, we will introduce Lagrangian mean curvature flow and its basic properties. We first recall key definitions from symplectic and Kähler geometry, notably Calabi-Yau manifolds and Lagrangian submanifolds. We then discuss Smoczyk's foundational paper, "A canonical way to deform Lagrangian submanifolds," which initiated the theory of Lagrangian mean curvature flow.
We continue our discussion of Smoczyk's paper "A canonical way to deform Lagrangian submanifolds." We will then discuss Mu-Tao Wang's "Mean Curvature Flow of Surfaces in Einstein Four-Manifolds," which introduces the notion of almost calibrated submanifolds. This is a key notion in Lagrangian mean curvature flow, and we will study its importance as a conserved property. We will also discuss applications of these ideas, showing that the symplectic property is preserved and the flow does not develop a type I singularity.
Let be a smooth map from a domain in ℝ^n to ℝ^m, with n, m ≥ 1. Its graph defines an embedded submanifold in ℝ^{n+m}. We focus on the case where this graph is either minimal or evolving by the mean curvature flow. The situation is well-understood when m=1, but considerably less so when m>1. In this talk, I will describe a 2-area-decreasing condition on f in the higher-codimension setting, with important consequences for the associated PDEs. This condition is closely connected to the notion of 2-convexity in Lagrangian mean curvature flow.
In this talk, I will present the results of Neves (2007) on singularities of the Lagrangian mean curvature flow in the complex n-space when the initial condition is a zero-Maslov class Lagrangian. We first show that the rescaled flow near a singularity converges weakly to a finite union of area-minimizing Lagrangian cones. Then, under the additional assumption that the Lagrangian is almost calibrated and rational, we see that each connected component of the rescaled flow converges to a single area-minimizing Lagrangian cone.