We propose a general method that parameterizes general surfaces with complex (possible branching) topology using Riemann surface structure. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. We can then automatically partition the surface using a critical graph that connects zero points in the global conformal structure on the surface. The trajectories of iso-parametric curves canonically partition a surface into patches. Each of these patches is either a topological disk or a cylinder and can be conformally mapped to a parallelogram by integrating a holomorphic I-form defined on the surface. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. For surfaces with similar topology

A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical -operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.

The notion of a generalized CRF-structure on a smooth manifold was recently introduced and studied by Vaisman (2008)[6]. An important class of generalized CRF-structures on an odd dimensional manifold M consists of CRF-structures having complementary frames of the form , where is a vector field and is a 1-form on M with ()= 1. It turns out that these kinds of CRF-structures give rise to a special class of what we called strong generalized contact structures in Poon and Wade [5]. More precisely, we show that to any CRF-structures with complementary frames of the form , there corresponds a canonical Lie bialgebroid. Finally, we explain the relationship between generalized contact structures and another generalization of the notion of a CauchyRiemann structure on a manifold.

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the (n−2)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points x ∈ X^{(k)} with k ≤ n−2, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of x ∈ X.

In this companion piece to [9], some variations on the main results there are sketched. In particular, • the recursions in [9], which we interpreted as the quantum Lefschetz, is re- formulated in terms of Givental’s quantization formalism, or equivalently, a summation of finitely many graphs; • varietiesofmodificationoftheauxilliaryspaces(masterspaces)forthefixed point localization are given, leading to different (looking) recursions; • applications of this circle of ideas to derive (apparently) new relations of Gromov–Witten invariants.