[Paper Review] On the violation of Thurston-Bennequin inequality for a certain non-convex hypersurface

Atsuhide Mori|arXiv (Cornell University)|Nov 2, 2011

Geometric and Algebraic Topology6 references5 citations

TL;DR

This paper demonstrates that in contact manifolds of dimension greater than three, there exist non-convex hypersurfaces violating the Thurston-Bennequin inequality, and no smooth approximation by convex hypersurfaces exists. The construction uses a specific hypersurface in the 1-jet space $J^1(bR^n, bR)$ with a singular characteristic foliation featuring a source and sink point, whose indices violate the inequality, and proves non-approximability via convexity obstruction using symplectic topology constraints.

ABSTRACT

We show that any open subset of a contact manifold of dimension greater than three contains a certain non-convex hypersurface violating the Thurston-Bennequin inequality.

Motivation & Objective

• To investigate whether the Thurston-Bennequin inequality, valid for convex surfaces in 3D contact manifolds, extends to higher dimensions.
• To construct a specific non-convex hypersurface in $J^1(bR^n, bR)$ that violates the inequality for $n > 1$.
• To show that such a hypersurface cannot be smoothly approximated by any convex hypersurface in the same contact manifold.
• To contrast the behavior of contact hypersurfaces in 3D (where all surfaces are approximable by convex ones) with higher dimensions.

Proposed method

• Construct a hypersurface $\Sigma \subset J^1(\bbR^n, \bbR)$ using a vector field $X$ defined on a neighborhood of $\Sigma$ with prescribed contact Hamiltonian and singularities.
• Define the characteristic foliation $\mathcal{F}_\Sigma$ on $\Sigma$ via the symplectic orthogonal of $T\Sigma \cap \ker\alpha$, which exhibits isolated singular points at the origin in cylindrical coordinates.
• Compute the indices of the singular points: a source ($S_+$) and a sink ($S_-$), both with index 1, leading to violation of the Thurston-Bennequin inequality.
• Use the Eliashberg-Floer-McDuff theorem to show that the boundary of a convex symplectic manifold cannot contain a disjoint union of spheres and other components, leading to contradiction if $\Sigma$ were approximable by convex surfaces.
• Employ a covering construction and projection to analyze the foliation structure on a quarter-sphere model, identifying elliptic and hyperbolic singularities.
• Use the contact form $\beta = (2r^2 - 1)dz + r^2(r^2 - 1)d\theta + \sum_{i=1}^{n-1}(x_i dy_i - y_i dx_i)$ to define the contact structure and verify the required geometric conditions.

Experimental results

Research questions

• RQ1 Can the Thurston-Bennequin inequality be violated by a non-convex hypersurface in a contact manifold of dimension greater than three?
• RQ2 Is there a hypersurface in $J^1(\bbR^n, \bbR)$ for $n > 1$ that violates the Thurston-Bennequin inequality and cannot be smoothly approximated by a convex hypersurface?
• RQ3 What topological or geometric obstruction prevents smooth approximation of such a hypersurface by convex ones in higher dimensions?
• RQ4 How does the behavior of characteristic foliations and singularities differ between 3D and higher-dimensional contact manifolds?

Key findings

• The hypersurface $\Sigma$ constructed in $J^1(\bbR^n, \bbR)$ for $n > 1$ violates the Thurston-Bennequin inequality due to the presence of a sink point with index 1 and a source point with index 1, resulting in $\sum_{p \in S_-} \mathrm{Ind}\,p = 1 > 0$.
• The boundary $\partial\Sigma$ is contact-type, as the characteristic foliation is transverse to the boundary, satisfying the geometric conditions for the inequality's formulation.
• No convex hypersurface can smoothly approximate $\Sigma$, as assuming such an approximation leads to a contradiction with the Eliashberg-Floer-McDuff theorem on symplectic boundaries.
• The singular foliation $\mathcal{F}_\Sigma$ on $\Sigma$ has a source and sink point, and its pushforward under a covering map reveals additional elliptic and hyperbolic singularities, confirming the non-convex nature.
• The construction relies on a vector field $X$ with a contact Hamiltonian function that ensures the contact structure is preserved and the singularities are isolated and of index 1.
• The result holds in any open subset of $J^1(\bbR^n, \bbR)$ for $n > 1$, showing the phenomenon is generic in higher-dimensional contact geometry.