Fredholm-Regularity of Holomorphic Discs in Plane Bundles over Compact Surfaces

@article{Guilfoyle2018FredholmRegularityOH,
  title={Fredholm-Regularity of Holomorphic Discs in Plane Bundles over Compact Surfaces},
  author={Brendan Guilfoyle and Wilhelm Klingenberg},
  journal={arXiv: Analysis of PDEs},
  year={2018}
}
We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with a single complex point has $C^{2,\alpha}$-close sections such that any (non-multiply covered) holomorphic disc with boundary in these sections are Fredholm regular. Fredholm regularity is also established when the complex surface is neutral Kahler… 
1 Citations
Proof of the Toponogov Conjecture on Complete Surfaces
We prove a conjecture of Toponogov on complete convex surfaces, namely that such surfaces must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an

References

SHOWING 1-10 OF 18 REFERENCES
On genericity for holomorphic curves in four-dimensional almost-complex manifolds
We consider spaces of immersed (pseudo-)holomorphic curves in an almost complex manifold of dimension four. We assume that they are either closed or compact with boundary in a fixed totally real
Pseudo holomorphic curves in symplectic manifolds
Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called
Proof of the Caratheodory Conjecture
A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. In this paper we prove this for
On Weingarten surfaces in Euclidean and Lorentzian 3-space
J-Holomorphic Curves and Symplectic Topology
The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was
The Maslov class of the Lagrange surfaces and Gromov’s pseudo-holomorphic curves
For an immersed Lagrange submanifold W c T*X, one can define a nonnegative integer topologic invariant m(W) such that the image of H1 (W; Z) under the Maslov class is equal to m(W) * Z. In this
Holomorphic curves in symplectic geometry
Introduction: Applications of pseudo-holomorphic curves to symplectic topology.- 1 Examples of problems and results in symplectic topology.- 2 Pseudo-holomorphic curves in almost complex manifolds.-
An indefinite Kähler metric on the space of oriented lines
The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space T of oriented affine lines in R3 with the tangent bundle of
Lectures on Symplectic Manifolds
Introduction Symplectic manifolds and lagrangian submanifolds, examples Lagrangian splittings, real and complex polarizations, Kahler manifolds Reduction, the calculus of canonical relations,
Symplectic hypersurfaces and transversality in Gromov-Witten theory
We use Donaldson hypersurfaces to construct pseudo-cycles which define Gromov-Witten invariants for any symplectic manifold which agree with the invariants in the cases where transversality could be
...
...