27 research outputs found
Curved flats in symmetric spaces
In this paper we study maps (curved flats) into symmetric spaces which are
tangent at each point to a flat of the symmetric space. Important examples of
such maps arise from isometric immersions of space forms into space forms via
their Gauss maps. Further examples are found in conformal geometry, e.g. the
curved flats obtained from isothermic surfaces and conformally flat 3-folds in
the 4-sphere. Curved flats admit a 1-parameter family of deformations (spectral
parameter) which enables us to make contact to integrable system theory. In
fact, we give a recipe to construct curved flats (and thus the above mentioned
geometric objects) from a hierarchy of finite dimensional algebraically
completely integrable flows.Comment: 9 pages, latex2e, no figures, also available at
http://www_sfb288.math.tu-berlin.de/preprints.htm
Isometric Immersions of Space Forms and Soliton Theory
This paper studies isometric immersions of space forms by means of a
hierarchy of finite dimensional integrable systems in Lax form on loop
algebras.Comment: 15 pages, latex2e, postscript available at
http://www_sfb288.math.tu-berlin.de/preprints.htm
Towards a constrained Willmore conjecture
We give an overview of the constrained Willmore problem and address some
conjectures arising from partial results and numerical experiments.
Ramifications of these conjectures would lead to a deeper understanding of the
Willmore functional over conformal immersions from compact surfaces.Comment: 17page
Darboux transforms and spectral curves of constant mean curvature surfaces revisited
We study the geometric properties of Darboux transforms of constant mean
curvature (CMC) surfaces and use these transforms to obtain an
algebro-geometric representation of constant mean curvature tori. We find that
the space of all Darboux transforms of a CMC torus has a natural subset which
is an algebraic curve (called the spectral curve) and that all Darboux
transforms represented by points on the spectral curve are themselves CMC tori.
The spectral curve obtained using Darboux transforms is not bi-rational to, but
has the same normalisation as, the spectral curve obtained using a more
traditional integrable systems approach.Comment: 7 figure
Schwarzian Derivatives and Flows of Surfaces
This paper goes some way in explaining how to construct an integrable
hierarchy of flows on the space of conformally immersed tori in n-space. These
flows have first occured in mathematical physics -- the Novikov-Veselov and
Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of
the Dirac operator. Later, using spinorial representations of surfaces, the
same flows were interpreted as deformations of surfaces in 3- and 4-space
preserving the Willmore energy. This last property suggest that the correct
geometric setting for this theory is Moebius invariant surface geometry. We
develop this view point in the first part of the paper where we derive the
fundamental invariants -- the Schwarzian derivative, the Hopf differential and
a normal connection -- of a conformal immersion into n-space together with
their integrability equations. To demonstrate the effectivness of our approach
we discuss and prove a variety of old and new results from conformal surface
theory. In the the second part of the paper we derive the Novikov-Veselov and
Davey-Stewartson flows on conformally immersed tori by Moebius invariant
geometric deformations. We point out the analogy to a similar derivation of the
KdV hierarchy as flows on Schwarzian's of meromorphic functions. Special
surface classes, e.g. Willmore surfaces and isothermic surfaces, are preserved
by the flows
Conformal maps from a 2-torus to the 4-sphere
We study the space of conformal immersions of a 2-torus into the 4-sphere.
The moduli space of generalized Darboux transforms of such an immersed torus
has the structure of a Riemann surface, the spectral curve. This Riemann
surface arises as the zero locus of the determinant of a holomorphic family of
Dirac type operators parameterized over the complexified dual torus. The kernel
line bundle of this family over the spectral curve describes the generalized
Darboux transforms of the conformally immersed torus. If the spectral curve has
finite genus the kernel bundle can be extended to the compactification of the
spectral curve and we obtain a linear 2-torus worth of algebraic curves in
projective 3-space. The original conformal immersion of the 2-torus is
recovered as the orbit under this family of the point at infinity on the
spectral curve projected to the 4-sphere via the twistor fibration.Comment: 27 pages, 5 figure
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