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Rationally Convex Surfaces with hyperbolic complex tangencies

Georgios Dimitroglou Rizell, Mark G. Lawrence

TL;DR

The paper constructs the first examples of rationally convex surfaces in $\mathbb{C}^2$ with only hyperbolic complex tangencies, presenting both fillable and non-fillable families. It develops an ambient Legendrian-surgery framework on a Clifford torus to generate genus $g$ surfaces with $2(g-1)$ hyperbolic tangencies and analyzes the moduli space of holomorphic discs to obtain holomorphic fillings that yield rational convexity. It provides a local Lagrangian model for hyperbolic tangencies and proves a symplectic criterion for rational convexity, then extends these methods to produce knotted and non-fillable examples via Lin’s singular exact Lagrangians and knotted Lagrangian cobordisms. The results illuminate the interaction between polynomial hulls, Lagrangian/symplectic geometry, and complex tangencies in $\mathbb{C}^2$, revealing a rich landscape of embeddings with controlled convexity properties and distinct smooth isotopy types.

Abstract

We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by handle-bodies, and those that do not have any compact Riemann surfaces attached at all. The fillable examples all live in the round sphere and are unknotted, while the non-fillable examples can moreover be produced in several different smooth isotopy classes.

Rationally Convex Surfaces with hyperbolic complex tangencies

TL;DR

The paper constructs the first examples of rationally convex surfaces in with only hyperbolic complex tangencies, presenting both fillable and non-fillable families. It develops an ambient Legendrian-surgery framework on a Clifford torus to generate genus surfaces with hyperbolic tangencies and analyzes the moduli space of holomorphic discs to obtain holomorphic fillings that yield rational convexity. It provides a local Lagrangian model for hyperbolic tangencies and proves a symplectic criterion for rational convexity, then extends these methods to produce knotted and non-fillable examples via Lin’s singular exact Lagrangians and knotted Lagrangian cobordisms. The results illuminate the interaction between polynomial hulls, Lagrangian/symplectic geometry, and complex tangencies in , revealing a rich landscape of embeddings with controlled convexity properties and distinct smooth isotopy types.

Abstract

We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by handle-bodies, and those that do not have any compact Riemann surfaces attached at all. The fillable examples all live in the round sphere and are unknotted, while the non-fillable examples can moreover be produced in several different smooth isotopy classes.

Paper Structure

This paper contains 13 sections, 23 theorems, 36 equations, 11 figures.

Table of Contents

  1. Introduction and results
  2. Fillable rationally convex surfaces with only hyperbolic tangencies
  3. Holomorphic fillings of a surgery on the Clifford torus
  4. Graph structure of the moduli space of discs
  5. Holomorphic fillings of a surgery on a knotted rationally convex torus
  6. Rational convexity of $T^g$ (Proof of Theorem \ref{['thm:ratconvexS3']})
  7. Symplectic condition for rational convexity
  8. The converse statement: rationally convex are singular Lagrangians
  9. Lagrangian model of a complex hyperbolic tangency
  10. Local symplectic description of the hyperbolic tangency
  11. Digression: Results about Lagrangian fillings of $\Lambda_{\pm1,-2}$.
  12. Non-fillable rationally convex surfaces with only hyperbolic tangencies
  13. Constructing knotted rationally convex surfaces

Key Result

Theorem 1.1

There exist rationally convex oriented surfaces of genus $g \ge 1$ contained in $S^3 \subset \mathbb{C}^2$, whose complex tangencies consist of a number $2(g-1)$ of hyperbolic points. Furthermore, for each $g \ge 1$, such surfaces can be realised in several smooth isotopy classes of embeddings in $S

Figures (11)

  • Figure 1: The characteristic distribution $\xi \cap TC_\epsilon$ with the stable and unstable manifolds of $h_\pm$ shown in blue. Here $\theta$ is an $S^1$-valued coordinate on the cylinder, which can be taken to coincide with the ambient coordinate $\pm y$ near $h_\pm$.
  • Figure 2: The surface $\Sigma_i^j$ is a sphere with four discs removed. The filling produced by Bedford--Klingenberg yields the handle-body that bounds the surface in the picture, which contains the parts of the disc families $\mathcal{T}^\pm_i \subset \mathcal{T}_i$.
  • Figure 3: The model a single incoming edge at a hyperbolic point, with two outgoing edges. Note that the hyperbolic point is negative here. (Using the terminology from BedfordKlingenberg, the discs approach the hyperbolic point "from the outside.")
  • Figure 4: The a priori possibilities of the graphs that corresponds to the moduli space of the discs produced by BedfordKlingenberg and that fill a component $\Sigma_i^j \subset \Sigma_i$. Configuration (a) is shown on the left, where the disc family $\mathcal{T}^{j,+}_i$ form the two incoming edges at $h_+$, while the disc family $\mathcal{T}^{j,-}_i$ form the two outgoing edges at $h_-$. Configuration (b) is shown on the right, where one of the two disc families in $\mathcal{T}^{j,+}_i$ forms the unique incoming edge at $h_-$, while one of the two disc families in $\mathcal{T}^{j,-}_i$ forms the unique outgoing edge at $h_+$.
  • Figure 5: The filling of $T^g_{\operatorname{Cl}}$ obtained by gluing together the partial fillings shown on the left in Figure \ref{['fig:moduli']}.
  • ...and 6 more figures

Theorems & Definitions (45)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3: Theorem \ref{['thm:exactsurfaces']}, Corollary \ref{['cor:rationallyconvex']}, and Theorem \ref{['thm:knotted']}
  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • ...and 35 more