Kai Cieliebak, Ana Rita Gaio, Dietmar A. Salamon
We outline the construction of invariants of Hamiltonian group actions on symplectic manifolds. These invariants can be viewed as an equivariant version of Gromov-Witten invariants. They are derived from solutions of a PDE involving the Cauchy-Riemann operator, the curvature of a connection, and the moment map.
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H-holomorphic maps are a parameter version of J-holomorphic maps into contact manifolds. They have arisen in efforts to prove the existence of higher--genus holomorphic open book decompositions and efforts to prove the existence of finite energy foliations and the Weinstein conjecture, as well as in folded holomorphic maps. For all these applications it is essential to understand the compactness properties of the space of H-holomorphic maps. We prove that the space of H-holomorphic maps with bounded periods into a manifold with stable Hamiltonian structure possesses a natural compactification.…
The Runge approximation theorem for holomorphic maps (U -> C) is a fundamental result in complex analysis. The aim of this article is to prove such a result for (pseudo-)holomorphic maps from a compact Riemann surface to a compact (almost-)complex manifold M under certain assumptions. Though the setting is definitively that of pseudo-holomorphic maps it also covers some complex varieties.
We provide a short proof that an $L^2_1$ and $J$-holomorphic curve is in fact smooth. As an application, we deduce a removal of singularity theorem for curves of finite energy.
We prove the exponential decay of the derivative of the gluing maps with respect to the gluing parameter.
For a smooth map $f:X^4\toΣ^2$ that is locally modeled by holomorphic maps, the domain is shown to admit a symplectic structure that is symplectic on some regular fiber, if and only if $f^*[Σ]\ne0$. If so, the space of symplectic forms on $X$ that are symplectic on all fibers is nonempty and contractible. The cohomology classes of these forms vary with the maximum possible freedom on the reducible fibers, subject to the obvious constraints. The above results are derived via an analogous theorem for locally holomorphic maps $f:X^{2n}\to Y^{2n-2}$ with $Y$ symplectic.
These notes present a systematic treatment of local properties of J-holomorphic maps and of Gromov's convergence for sequences of such maps, specifying the assumptions needed for all statements. In particular, only one auxiliary statement depends on the manifold being symplectic. The content of these notes roughly corresponds to Chapters 2 and 4 of McDuff-Salamon's book on the subject.
Every oriented 4-manifold admits a folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface ("fold") in a controlled fashion. We define folded holomorphic maps, i.e. pseudo-holomorphic maps that are discontinuous across the fold. The boundary values on the fold are mediated by tunneling maps which are punctured H-holomorphic maps into the folding hypersurface with prescribed asymptotics on closed characteristics. Our main result is that the linearized operator of this boundary value pro…
We investigate nicely embedded H--holomorphic maps into stable Hamiltonian three--manifolds. In particular we prove that such maps locally foliate and satisfy a no--first--intersection property. Using the compactness results of arXiv:0904.1603 we show that connected components of the space of such maps can be compactified if they contain a global surface of section. As an application we prove that any contact structure on a 3--manifold admits and H--holomorphic open book decomposition. This work is motivated by the program laid out by Abbas, Cieliebak and Hofer to give a proof to the Weinstein…
The purpose of this paper is to generalize a theorem of Segal from [Seg79] proving that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding space of continuous maps through a range of dimensions increasing with degree. We will address if a similar result holds when other almost complex structures are put on a projective space. For any compatible almost complex structure J on CP^2, we prove that the inclusion map from the space of J-holomorphic maps to the space of continuous maps induces a homology surjection through a …
We prove that every finite energy $J$-holomorphic map $u(s,t):\mathbb R\times S^1 \rightarrow {\mathbb R} \times \widetilde{M}$ exponentially converges to a periodic orbit of Reeb vector field of $\widetilde M,$ as $s\to \infty.$
We show that conformally compact, globally hyperbolic, Lorentzian Einstein-Weyl 3-manifolds are in natural one-to-one correspondence with orientation-reversing diffeomorphisms of the 2-sphere. The proof hinges on a holomorphic-disk analog of Hitchin's mini-twistor correspondence.
Dirac-harmonic maps are critical points of a fermionic action functional, generalizing the Dirichlet energy for harmonic maps. We consider the case where the source manifold is a closed Riemann surface with the canonical Spin^c-structure determined by the complex structure and the target space is a Kaehler manifold. If the underlying map f is a J-holomorphic curve, we determine a space of spinors on the Riemann surface which form Dirac-harmonic maps together with f. For suitable complex structures on the target manifold the tangent bundle to the moduli space of J-holomorphic curves consists of…
The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: are the intersections non-empty? are they smooth? what are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of $J$-holomorphic maps. As an application, we use our framework to give…
We construct an unwrapped Floer theory for bundles of Liouville sectors. In particular, we construct a compatible collection of unwrapped Fukaya categories of fibers of a Liouville bundle, and prove that the two natural constructions of continuation maps in this setting behave compatibly. These constructions are exploited in [OT19] to construct homotopically coherent actions of Lie groups on wrapped Fukaya categories, thereby proving a conjecture from Teleman's 2014 ICM address.
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