1,229 research outputs found
Formal conserved quantities for isothermic surfaces
Isothermic surfaces in are characterised by the existence of a pencil
of flat connections. Such a surface is special of type if there
is a family of -parallel sections whose dependence on the
spectral parameter is polynomial of degree . We prove that any
isothermic surface admits a family of -parallel sections which is a
formal Laurent series in . As an application, we give conformally invariant
conditions for an isothermic surface in to be special.Comment: 13 page
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
Harmonic maps in unfashionable geometries
We describe some general constructions on a real smooth projective 4-quadric
which provide analogues of the Willmore functional and conformal Gauss map in
both Lie sphere and projective differential geometry. Extrema of these
functionals are characterized by harmonicity of this Gauss map.Comment: plain TeX, uses bbmsl for blackboard bold, 20 page
Kaehler submanifolds with parallel pluri-mean curvature
We investigate the local geometry of a class of K\"ahler submanifolds which generalize surfaces of constant mean curvature. The role of
the mean curvature vector is played by the -part (i.e. the -components) of the second fundamental form , which we call the
pluri-mean curvature. We show that these K\"ahler submanifolds are
characterized by the existence of an associated family of isometric
submanifolds with rotated second fundamental form. Of particular interest is
the isotropic case where this associated family is trivial. We also investigate
the properties of the corresponding Gauss map which is pluriharmonic.Comment: Plain TeX, 21 page
- …