Singularities of Lagrangian Immersions and its Applications in Lagrangian Floer Theory

TL;DR

The paper analyzes singularities of Lagrangian immersions into Cartesian products of surfaces using jet-bundle methods to show that a locally Hamiltonian perturbation yields a generic local structure where the bisingular set is a union of bifold points and finitely many cusp points. It then develops a transversality framework to perturb Lagrangian immersions so that the 1-jet of the components is transverse to the relevant singular strata, enabling a clean global structure for the bisingular set. Building on this, the work studies Lagrangian correspondences and their compositions on closed surfaces, proving canonical identifications of intersection data and bigons under favorable transversality and covering-map assumptions. These results are applied to compare Floer complexes arising from Lagrangian correspondences and to discuss quilted Floer theory, including the impact of bifold singularities and potential bubbling phenomena. Overall, the paper provides a detailed geometric mechanism to relate Floer-theoretic data across correspondences and offers concrete constructions and conjectures for extending these relations to quilted settings.

Abstract

In this article, we study the singularities of Lagrangian immersions into Cartesian product of surfaces. After applying a Hamiltonian isotopy in the Weinstein tubular neighbourhood of the Lagrangian immersion, the singular points of the Lagrangian immersion can be expressed locally as fold points with finitely many cusp points. This result has applications in comparing two Lagrangian Floer complexes associated to curves on surfaces related by a certain Lagrangian correspondence and the quilt Floer complex induced by these three Lagrangian immersions.

Figures (7)

• Figure 1: The convex corners are marked in blue. The blue in the first quadrant corresponds to the part of the bigon near $(-1,0)$ and the blue in the third quadrant corresponds to the part of the bigon near $(1,0)$. The horizontal line is a part of $L$ and the vertical line is a part of $L'$.
• Figure 2: The figure for Example \ref{['exp:8.10']}. The intersections of the blue with the red from the top to the bottom in the lower right picture are $a,d,b,e,c,f$.
• Figure 3: The trivial Lagrangian composition in the backside is not drawn
• Figure 4: The top line is the Lagrangian composition near $x$. Notice that the blue curve in the top left of Figure 3 first meets the red curve, then meets the small fold circle, finally meets the big fold circle. So after performing the Lagrangian composition, the blue and red curves intersect only once around $x$. The bottom line is the Lagrangian composition near $y$. Notice that the blue curve in the top left of Figure 3 first meets the small fold circle, then meets the big fold circle, finally meets the big fold circle again. So after performing the Lagrangian composition, the blue and red curves do not intersect around $y$.
• Figure :

Theorems & Definitions (97)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Transversality Theorem, Theorem 4.4 golubitsky2012stable
• Remark 2.4
• Theorem 2.5: Whitney
• Remark 2.6
• Remark 2.7