234 research outputs found
Sequences of Willmore surfaces
In this paper we develop the theory of Willmore sequences for Willmore
surfaces in the 4-sphere. We show that under appropriate conditions this
sequence has to terminate. In this case the Willmore surface either is the
twistor projection of a holomorphic curve into complex projective space or the
inversion of a minimal surface with planar ends in 4-space. These results give
a unified explanation of previous work on the characterization of Willmore
spheres and Willmore tori with non-trivial normal bundles by various authors.Comment: 10 page
Quaternionic holomorphic geometry: Pluecker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori
The paper develops the fundamentals of quaternionic holomorphic curve theory.
The holomorphic functions in this theory are conformal maps from a Riemann
surface into the 4-sphere, i.e., the quaternionic projective line. Basic
results such as the Riemann-Roch Theorem for quaternionic holomorphic vector
bundles, the Kodaira embedding and the Pluecker relations for linear systems
are proven. Interpretations of these results in terms of the differential
geometry of surfaces in 3- and 4-space are hinted at throughout the paper.
Applications to estimates of the Willmore functional on constant mean curvature
tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue
estimates on Riemannian spin bundles in dimension 2 are given.Comment: 70 pages, 1 figur
Envelopes and osculates of Willmore surfaces
We view conformal surfaces in the 4--sphere as quaternionic holomorphic
curves in quaternionic projective space. By constructing enveloping and
osculating curves, we obtain new holomorphic curves in quaternionic projective
space and thus new conformal surfaces. Applying these constructions to Willmore
surfaces, we show that the osculating and enveloping curves of Willmore spheres
remain Willmore.Comment: 12 pages, 2 figures; v2: improved definition of Frenet curves, minor
changes in presentatio
Ballistic transport in random magnetic fields with anisotropic long-ranged correlations
We present exact theoretical results about energetic and dynamic properties
of a spinless charged quantum particle on the Euclidean plane subjected to a
perpendicular random magnetic field of Gaussian type with non-zero mean. Our
results refer to the simplifying but remarkably illuminating limiting case of
an infinite correlation length along one direction and a finite but strictly
positive correlation length along the perpendicular direction in the plane.
They are therefore ``random analogs'' of results first obtained by A. Iwatsuka
in 1985 and by J. E. M\"uller in 1992, which are greatly esteemed, in
particular for providing a basic understanding of transport properties in
certain quasi-two-dimensional semiconductor heterostructures subjected to
non-random inhomogeneous magnetic fields
Conformal Geometry of Surfaces in S4 and Quaternions
The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given
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