C⁰-Contact Geometry of Surfaces in 3-Manifolds

2025/09/02 by Serraille, Baptiste, Stokić, Maksim · 1 citation
#53D10 #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2509.02430

Abstract

We prove that contact homeomorphisms preserve characteristic foliations on surfaces in contact 3-manifolds. More precisely, since the characteristic foliation is a singular 1-dimensional foliation, we show that singular points are mapped to singular points, and that the image of every 1-dimensional leaf is again a 1-dimensional leaf in the image surface. As a consequence, regular coisotropic surfaces are C⁰-rigid. In contrast, we show that contact convexity is C⁰-flexible by constructing a contact homeomorphism that sends a convex 2-torus to a non-convex one.

Citations

• C⁰-limits of Legendrian knots and contact non-squeezing
• C⁰-rigidity of characteristics in symplectic geometry

Cited by

• C⁰-rigidity of Legendrians and coisotropics via sheaf quantization

Related

• Contact structures on 3-manifolds are deformations of foliations
• Morse-Smale characteristic foliations and convexity in contact manifolds
• Topological methods in 3-dimensional contact geometry
• On Legendrian foliations in contact manifold I: Singularities and neighborhood theorems
• Foliations, contact structures and their interactions in dimension three