Pseudoholomorphic Research Articles

The topology of the space ofJ–holomorphic maps to ℂP2

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 range of dimensions tending to infinity with degree. The proof involves comparing the scanning map of topological chiral homology [Lur09], [And10], [Mil12]) with analytic gluing maps for J-holomorphic curves ([MS94], [Sik03]). This is an extension of the work done in [Mil11] regarding genus zero case.

Morse homotopy and topological conformal field theory

By studying spaces of flow graphs in a closed oriented manifold, we equip the Morse complex with the operations of an open topological conformal field theory. This complements previous constructions due to R. Cohen et al., K. Costello, K. Fukaya and M. Kontsevich and is also the Morse theoretic counterpart to a conjectural construction of operations on the chain complex of the Lagrangian Floer homology of the zero section of a cotangent bundle, obtained by studying uncompactified moduli spaces of higher genus pseudoholomorphic curves.

Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

We prove that every C 1 -smooth, area preserving dieomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic C 1 -smooth dieomorphisms. In particular, every smooth irrational pseudo-rotation can be C 0 -approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.

J-Holomorphic Curves in a Nef Class

Taubes established fundamental properties of $J-$holomorphic subvarieties in dimension 4 in \cite{T1}. In this paper, we further investigate properties of reducible $J-$holomorphic subvarieties. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is $J-$nef. For a spherical class, it has particularly strong consequences. It is shown that, for any tamed $J$, each irreducible component is a smooth rational curve. We also completely classify configurations of maximal dimension. To prove these results we treat subvarieties as weighted graphs and introduce several combinatorial moves.

Potential theory on almost complex manifolds

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their dual objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the classical definition. In the second theorem, using our viscosity definitions, the Dirichlet problem is solved for the complex Monge-Ampere equation in both the homogeneous and inhomogeneous forms. These two results are based on theorems found in two recent papers of the authors. Finally, it is shown that the plurisubharmonic functions considered here agree with the plurisubharmonic distributions. In particular, this proves a conjecture of Nefton Pali.

A theorem on the removal of boundary singularities of pseudo-holomorphic curves

A theorem on the removal of boundary singularities of pseudo-holomorphic curves

Tikhonov’s theorem on passage to the limit and pseudoholomorphic solutions of singularly perturbed problems

Tikhonov’s theorem on passage to the limit and pseudoholomorphic solutions of singularly perturbed problems

A reverse isoperimetric inequality for J-holomorphic curves

We prove that the length of the boundary of a $J$-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real $J$-holomorphic curve. The infimum over $J$ of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be $2\pi$ for the Lagrangian submanifold $\mathbb R P^n \subset \mathbb C P^n.$ We apply our result to prove compactness of moduli of $J$-holomorphic maps to non-compact target spaces that are asymptotically exact. In a different direction, our result implies the adic convergence of the superpotential.

Pseudo-holomorphic curves in nearly Kähler manifolds

We study pseudo-holomorphic curves in general nearly Kähler manifolds. For that purpose, we first introduce the fundamental equations of submanifold geometry in terms of the characteristic connection of the nearly Kähler structure. Then we classify pseudo-holomorphic curves with parallel second fundamental form in Chern-flat nearly Kähler manifolds. Moreover, we give a new Simons type identity. As an application of this identity, we show that the closed pseudo-holomorphic curves in Chern-flat nearly Kähler manifolds with a second fundamental form of controlled growth are totally geodesic.

Pseudo-rotations with sufficiently Liouvillean rotation number are $$C^0$$ C 0 -rigid

It is an open question in smooth ergodic theory whether there exists a Hamiltonian disk map with zero topological entropy and (strong) mixing dynamics. Weak mixing has been known since Anosov and Katok first constructed examples in 1970. Currently all known examples with weak mixing are irrational pseudo-rotations with Liouvillean rotation number on the boundary. Our main result however implies that for a dense subset of Liouville numbers (strong) mixing cannot occur. Our approach involves approximating the flow of a suspension of the given disk map by pseudoholomorphic curves. Ellipticity of the Cauchy–Riemann equation allows quantitative \(L^2\)-estimates to be converted into \(C^0\)-estimates between the pseudoholomorphic curves and the trajectories of the flow on growing time scales. Arithmetic properties of the rotation number enter through these estimates.

Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications

We establish a quantization result for the angular part of the energy of solutions to elliptic linear systems of Schrodinger type with antisymmetric potentials in two dimensions. This quantization is a consequence of uniform Lorentz‐Wente type estimates in degenerating annuli. Moreover this result is optimal in the sense that we exhibit a sequence of functions satisfying our hypothesis whose radial part of the energy is not quantized. We derive from this angular quantization the full energy quantization for general critical points to functionals which are conformally invariant or also for pseudoholomorphic curves on degenerating Riemann surfaces.

Symplectic embeddings of polydisks

We show that the polydisk $$P(1,2)$$ , the product of disks of areas $$1$$ and $$2$$ , can be symplectically embedded in a ball $$B(R)$$ of capacity $$R$$ if and only if $$R \ge 3$$ . Hence, the inclusion map gives the optimal embedding and neither the Embedded Contact Homology nor Ekeland–Hofer capacities give sharp obstructions in this situation. Our proof applies the theory of pseudoholomorphic curves in manifolds with cylindrical ends, and in particular finite energy foliations.

Removal of singularities and Gromov compactness for symplectic vortices

We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compact- ication by polystable vortices. This extends results of Mundet i Riera for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vor- tices that allows us to apply techniques used by McDu and Salamon in their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for symplectic vortices.

The projective plane, J-holomorphic curves and Desarguesʼ theorem

By a theorem of Gromov, for an almost complex structure J on CP2 tamed by the standard symplectic structure, the J-holomorphic curves representing the positive generator of homology form a projective plane. We show that this satisfies the Theorem of Desargues if and only if J is isomorphic to the standard complex structure. This answers a question of Ghys.

Thin instantons in $G_2$-manifolds and Seiberg-Witten invariants

For two nearby disjoint coassociative submanifolds $C$ and $C'$ in a $G_2$-manifold, we construct thin instantons with boundaries lying on $C$ and $C'$ from regular J-holomorphic curves in $C$. We explain their relationship with the Seiberg-Witten invariants for $C$.

A hierarchy of local symplectic filling obstructions for contact 3-manifolds

We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact type embeddings into any closed symplectic 4-manifold, and has vanishing contact invariant in Embedded Contact Homology, and we give examples of contact manifolds that have planar k-torsion for any $k \ge 2$ but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness and compactness theorems for certain classes of J-holomorphic curves in blown up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains. The results are applied further in followup papers on weak symplectic fillings (arXiv:1003.3923, joint with K. Niederkrueger), non-exact symplectic cobordisms (arXiv:1008.2456) and Symplectic Field Theory (arXiv:1009.3262, joint with J. Latschev).

Pseudoholomorphic Curves on Nearly Kähler Manifolds

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curves on a 6-dimensional sphere with the standard (G_2-invariant) almost complex structure.

Global properties of tight Reeb flows with applications to Finsler geodesic flows on S2

AbstractWe show that if a Finsler metric on S2 with reversibility r has flag curvatures K satisfying (r/(r+1))2 < K ≤ 1, then closed geodesics with specific contact-topological properties cannot exist, in particular there are no closed geodesics with precisely one transverse self-intersection point. This is a special case of a more general phenomenon, and other closed geodesics with many self-intersections are also excluded. We provide examples of Randers type, obtained by suitably modifying the metrics constructed by Katok [21], proving that this pinching condition is sharp. Our methods are borrowed from the theory of pseudo-holomorphic curves in symplectizations. Finally, we study global dynamical aspects of 3-dimensional energy levels C2-close to S3

Associative submanifolds of the 7-sphere

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the nearly K\"ahler 6-sphere. By classifying the associative group orbits, we exhibit the first known explicit example of an associative 3-fold in S^7 which does not arise from other geometries. We then study associative 3-folds satisfying the curvature constraint known as Chen's equality, which is equivalent to a natural pointwise condition on the second fundamental form, and describe them using a new family of pseudoholomorphic curves in the Grassmannian of 2-planes in R^8 and isotropic minimal surfaces in S^6. We also prove that associative 3-folds which are ruled by geodesic circles, like minimal surfaces in space forms, admit families of local isometric deformations. Finally, we construct associative 3-folds satisfying Chen's equality which have an S^1-family of global isometric deformations using harmonic 2-spheres in S^6.

Dynamische Systeme

This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the ”Moser-Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry related to Hamiltonian dynamics. Highlights were new results on Arnold diffusion and a new approach to the study of Hamiltonian systems based on pseudoholomorphic curve methods.