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In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit of complex integrals over Riemannian…
Following earlier work of Loftin-McIntosh, we study minimal Lagrangian immersions of the universal cover of a closed surface (of genus at least 2) into CH2, with prescribed data of a conformal structure plus a holomorphic cubic…
In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of…
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation…
The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex triples and holomorphic symplectic 2-forms on manifolds…
We show that each cluster seed in the augmentation variety is inhabited by an embedded exact Lagrangian filling. This resolves the matter of surjectivity of the map from Lagrangian fillings to cluster seeds. The main new technique to…
We enlarge the class of open Riemann surfaces known to be holomorphically embeddable into the plane by allowing them to have additional isolated punctures compared to the known embedding results.
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding…
A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of…
We consider embeddings of a finite complex in a sphere. We give a homotopy theoretic classification of such embeddings in a wide range.
We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of…
Given a sequence of automorphisms of the polydisk, we show that the associated composition semigroup homomorphisms on the ball of bounded holomorphic functions on the polydisk admit a universal inner function if a certain condition on the…
We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating…
This supplementary part of the paper gr-qc 9312038 contains the necessary proofs of the claims stated in the main part.
We prove an existence result for exact lagrangian cobordisms between closed legendrians.
In this article, we discuss some properties of holomorphic fibrations in the complex analytic setting.
The geometry of a toroidal scheme over a DVR is encoded in a $\mathbb{Z}$-PL space known as the dual polyhedral complex. Any such dual complex is a skeleton, i.e. a nonarchimedean analytic retract, and admits a combinatorial divisor theory…
We prove several interpolation results for holomorphic Legendrian curves lying in an odd dimensional complex Euclidean space with the standard contact structure. In particular, we show that an arbitrary countable set of points in…
Over the complex numbers, the complement of a collection of hyperplanes is a widely-studied object; the cohomology ring, in particular, is known to have a structure depending only on the combinatorial properties of the intersection of…
In the present paper continuing our previous work we prove an extension theorem for matrices with entries in the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of…