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Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Fanghua Lin, Tristan Rivière (1999)

Journal of the European Mathematical Society

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S 1 -valued function defined on the boundary of a bounded regular domain of R n . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology...

Complex methods in real integral geometry

Eastwood, Michael (1997)

Proceedings of the 16th Winter School "Geometry and Physics"

This is an exposition of a general machinery developed by M. G. Eastwood, T. N. Bailey, C. R. Graham which analyses some real integral transforms using complex methods. The machinery deals with double fibrations M Ω η Ω ˜ @ > τ > > X ( Ω complex manifold; M totally real, real-analytic submanifold;...

Complex sprays and complex curves.

Colum Watt (2001)

Revista Matemática Complutense

After defining what is meant by a complex spray X on a complex manifold M, we introduce the notion of a spray complex curve associated to X. Several equivalent formulations are derived and we give necessary and sufficient conditions for M to admit spray complex curves for X through each point and in each direction. Refinements of this result are then derived for some special cases, for example when X is the horizontal radial vector field associated to a complex Finsler metric.

Complex structures on S O g M

Tommaso Pacini (1999)

Bollettino dell'Unione Matematica Italiana

Data una varietà Riemanniana orientata M , g , il fibrato principale S O g M di basi ortonormali positive su M , g ha una parallelizzazione canonica dipendente dalla connessione di Levi-Civita. Questo fatto suggerisce la definizione di una classe molto naturale di strutture quasi-complesse su M , g . Dopo le necessarie definizioni, discutiamo qui l'integrabilità di queste strutture, esprimendola in termini della struttura Riemanniana g .

Complex surfaces in ℂ⁴ with recurrent shape operators

Paweł Witowicz (2007)

Annales Polonici Mathematici

We study complex affine surfaces in ℂ⁴ with the transversal bundle defined by Nomizu and Vrancken. We classify the surfaces that have recurrent shape operators and parallel transversal metric.

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