### A compendium of pseudoholomorphic beasts in $\mathbf{R}\times ({S}^{1}\times {S}^{2})$.

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Let $\Sigma $ be a surface with a symplectic form, let $\phi $ be a symplectomorphism of $\Sigma $, and let $Y$ be the mapping torus of $\phi $. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $\mathbb{R}\times \mathbb{Y}$, with cylindrical ends asymptotic to periodic orbits of $\phi $ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to understand...

In this paper the Nijenhuis tensor characteristic distributions on a non-integrable four-dimensional almost complex manifold is investigated for integrability, singularities and equivalence.

We prove a compactness theorem for holomorphic curves in 4-dimensional symplectizations that have embedded projections to the underlying 3-manifold. It strengthens the cylindrical case of the SFT compactness theorem [BEH+C03] by using intersection theory to show that degenerations of such sequences never give rise to multiple covers or nodes, so transversality is easily achieved. This has application to the theory of stable finite energy foliations introduced in [HWZ03], and also suggests a new...

The paper discusses some aspects of Gromov’s theory of gluing complex discs to Lagrangian manifolds.

We give in ${\mathbb{R}}^{6}$ a real analytic almost complex structure $J$, a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$, such that there is no germ of $J$-holomorphic disc $\gamma $ included in $M$ with $\gamma \left(0\right)=0$ and $\frac{\partial \gamma}{\partial x}\left(0\right)=v$, although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$, we give sufficient conditions under which there exists such a germ of disc.

Let $(M,J,\omega )$ be a manifold with an almost complex structure $J$ tamed by a symplectic form $\omega $. We suppose that $M$ has the complex dimension two, is Levi-convex and with bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of $M$ can be foliated by the boundaries of pseudoholomorphic discs.

We consider a compact almost complex manifold $(M,J,\omega )$ with smooth Levi convex boundary $\partial M$ and a symplectic tame form $\omega $. Suppose that ${S}^{2}$ is a real two-sphere, containing complex elliptic and hyperbolic points and generically embedded into $\partial M$. We prove a result on filling ${S}^{2}$ by holomorphic discs.

First, we give some characterizations of J-hyperbolic points for almost complex manifolds. We apply these characterizations to show that the hyperbolic embeddedness of an almost complex submanifold follows from relative compactness of certain spaces of continuous extensions of pseudoholomorphic curves defined on the punctured unit disc. Next, we define uniformly normal families of pseudoholomorphic curves. We prove extension-convergence theorems for these families similar to those obtained by Kobayashi,...

Let $S$ be a Riemann surface. Let ${\mathbb{H}}^{3}$ be the $3$-dimensional hyperbolic space and let ${\partial}_{\infty}{\mathbb{H}}^{3}$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $\varphi :S\to {\partial}_{\infty}{\mathbb{H}}^{3}=\widehat{\u2102}$. If $i:S\to {\mathbb{H}}^{3}$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\widehat{\u0131}$, of $i$ by $\widehat{\u0131}=N$. Let $\overrightarrow{n}:U{\mathbb{H}}^{3}\to {\partial}_{\infty}{\mathbb{H}}^{3}$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S,\varphi )$ is a convex immersion $i$ of constant Gaussian curvature equal to $k\in (0,1)$ such that the Gauss lifting $(S,\widehat{\u0131})$ is complete and $\overrightarrow{n}\circ \widehat{\u0131}=\varphi $. In this paper, we show...

We establish the Schwarz Reflection Principle for $J$-complex discs attached to a real analytic $J$-totally real submanifold of an almost complex manifold with real analytic $J$. We also prove the precise boundary regularity and derive the precise convergence in Gromov compactness theorem in ${\mathcal{C}}^{k,\alpha}$-classes.

The definition of the Kobayashi-Royden pseudo-metric for almost complex manifolds is similar to its definition for complex manifolds. We study the question of completeness of some domains for this metric. In particular, we study the completeness of the complement of submanifolds of co-dimension 1 or 2. The paper includes a discussion, with proofs, of basic facts in the theory of pseudo-holomorphic discs.

First, we give some characterizations of the Kobayashi hyperbolicity of almost complex manifolds. Next, we show that a compact almost complex manifold is hyperbolic if and only if it has the Δ*-extension property. Finally, we investigate extension-convergence theorems for pseudoholomorphic maps with values in pseudoconvex domains.