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Folded Holomorphic Maps
Prof. Jens von Bergmann ( University of Notre Dame, USA )

2008 - 06 - 03 (Tue.)
11:00 - 12:10
308, Mathematics Research Center Building (ori. New Math. Bldg.)

The theory of folded holomorphic maps is a program to extend the tools of $J$--holomorphic curves to all oriented 4--manifolds by utilizing folded symplectic structures.
A folded symplectic structure on a manifold is a closed 2-form that is non-degenerate away from a real codimension 1 hypersurface ("fold"), on which its kernel gives a 1-dimensional foliation. Every oriented 4--manifold admits such a structure. Folded holomorphic maps are pairs of $J$--holomorphic maps from two halfs of a Riemann surfaces. The halfs are separated by a codimension 1 submanifold, which has image on the fold and is satisfying an appropriate boundary condition. This boundary condition constitutes the heart of the theory and can be visualized by "tunneling" in the fold, allowing maps to exit at a location that is different from where they enter the fold. We prove that under the simplifying assumption that the fold is "circle-invariant" this leads to a Fredholm problem. Such folds occur frequently. Under suitable genericity assumptions, moduli spaces of folded holomorphic maps have the expected dimension. In the case of pseudoconvex folds we show that they possess a natural compactification with boundary stratum of codimension at least 2. We will explain the theory and the mentioned results and give examples of moduli spaces of folded holomorphic maps, including moduli spaces of maps into the the 4-sphere.

2007-10-02 (Tue.)		Isoperimetric Problem in Heisenberg group
2007-10-09 (Tue.)		Isoperimetric Problem in Heisenberg group
2007-10-16 (Tue.)		Proof of positive mass theorem II (after Schoen and Yau)
2007-10-23 (Tue.)		Proof of positive mass theorem II (after Schoen and Yau)
2007-10-30 (Tue.)		Proof of positive mass theorem II(after Schoen and Yau)
2007-10-30 (Tue.)		Rigidity Theorem of Higher Dimensional Black
2007-11-13 (Tue.)		Proof of positive mass theorem II (after Schoen and Yau)
2007-11-20 (Tue.)		The Mass and the Paneitz operator in CR geometry
2007-11-27 (Tue.)		A Remark on Degenerate Singularity in three Dimensional Ricci Flow
2007-12-04 (Tue.)		The balance between existence and nonexistence theorems via an index theory
2007-12-04 (Tue.)		Concentration phenomena for some singularly perturbed elliptic PDEs
2008-01-08 (Tue.)		Tautness and Algebraicity
2008-03-11 (Tue.)		The Strong Oka’s Lemma, bounded plurisubharmonic functions and the -Neumann problem
2008-03-25 (Tue.)		Vafa-Witten theory for non-simply laced gauge groups
2008-04-08 (Tue.)		The Cauchy problem of the Ward equation I
2008-04-15 (Tue.)		The Cauchy problem of the Ward equation II.
2008-04-22 (Tue.)		Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature
2008-04-29 (Tue.)		Geometric aspects of random walks on a crystal lattice
2008-05-01 (Thu.)		Harmonic forms and Weight Poincar\'{e} inequality
2008-05-06 (Tue.)		Toward the QMMP via A + B model
2008-06-03 (Tue.)		Folded Holomorphic Maps
2008-06-10 (Tue.)		On Path Integrals
2008-06-17 (Tue.)		Constant Mean Curvature Foliation of Isolated Physical Systems
2008-06-17 (Tue.)		The positivity of quasi-local mass
2008-06-24 (Tue.)		A Torelli type theorem of birational equivalence
2008-07-18 (Fri.)		Classification of gradient shrinking solitons
2008-07-31 (Thu.)		The Bernstein problem on the Heisenbeg Groups
2008-09-23 (Tue.)		The Tian-Yau-Zelditch asymptotic expansion
2008-10-01 (Wed.)		Introducing a complexity to spatial graphs
2008-10-07 (Tue.)		On the singularities of the Szego projection for ''0,q'' forms
2008-10-14 (Tue.)		A resent result of Phong on Kähler-Ricci flow
2008-10-21 (Tue.)		On the moduli space of flat bundles over a 2-manifold
2008-10-28 (Tue.)		A note of weighted Poincaré inequality
2008-11-04 (Tue.)		CR Obata’s Theorem
2008-11-04 (Tue.)		Augmented Spinors
2008-11-11 (Tue.)		Energies of Knots and the Related Variational Problems
2008-11-18 (Tue.)		Immersed equations of the Hyperbolic Surfaces
2008-11-25 (Tue.)		Removable singularities and uniqueness of standard solution of Ricci flow
2008-11-25 (Tue.)		Lagrangian Submanifolds Foliated by (n-1)-spheres in R^2n
2008-12-02 (Tue.)		Interplays between Symmetries and Optimalities
2008-12-16 (Tue.)		A generalization of Lagrangian mean curvature flow
2008-12-16 (Tue.)		Toric Sasaki-Einstein manifolds
2009-02-17 (Tue.)		On Lagrangian submanifolds in complex hyperquadrics obtained from isoparametric hypersurfaces
2009-02-17 (Tue.)		Aronson–Benilan estimates for porous medium and fast diffusion equations.
2009-02-24 (Tue.)		Classification of isoparametric hypersurfaces
2009-03-03 (Tue.)		A Generalization of Hamilton’s Differential Harnack Inequality for the Ricci Flow.
2009-03-17 (Tue.)		Classification of Isoparametric Hypersurfaces II
2009-03-17 (Tue.)		The Dirac Operator on Contact Manifolds (I)
2009-03-24 (Tue.)		On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds
2009-03-31 (Tue.)		The Dirac Operator on Contact Manifolds (II)
2009-04-07 (Tue.)		Polynomial Growth Harmonic Functions on Complete Riemannian Manifolds
2009-04-14 (Tue.)		The CR Volume Comparison
2009-04-28 (Tue.)		Some recent developments in Hardy $H^p$ theory
2009-05-12 (Tue.)		Sub-elliptic Operator and Sub-Riemannian Geometry
2009-05-26 (Tue.)		Ends of locally symmetric spaces
2009-06-02 (Tue.)		The Poisson Kernel for Heisenberg Polynomials on the Disk
2009-06-09 (Tue.)		Differential forms, conservation laws, and monotonicity formulas in p-
2009-06-16 (Tue.)		p-Harmonic geometry, submanifolds of constant mean curvature, and Yang-Mills-Born-Infeld Theory
2009-07-30 (Thu.)		Graphic Bernstein type results and Mean Curvature flow in Higher Co-dimension
2009-09-10 (Thu.)		Introduction to the Toda lattice 1
2009-09-14 (Mon.)		Introduction to the Toda lattice 2
2009-09-17 (Thu.)		The Lee Conjecture on Pseudo-Einstein Metrics
2009-10-01 (Thu.)		The Refined Analytic Torsion
2009-10-08 (Thu.)		Geometry of surfaces on the Heisenberg group from the point of view of analysis
2009-10-15 (Thu.)		Quantum differential equation of weighted projective spaces
2009-10-22 (Thu.)		Massey Product and Chern-Simons Theory
2009-12-10 (Thu.)		On a generalization of Calabi-Yau equation
2010-03-11 (Thu.)		On a simple maximum principle technique on the circle
2010-03-18 (Thu.)		An Approach to Higher Order Linking Invariants Through Holonomy and Curvature
2010-03-25 (Thu.)		On heat kernels
2010-05-27 (Thu.)		Harmonic Polynomials of the Heisenberg sub-Laplacian and Dimension Analysis