454 research outputs found
Dressing preserving the fundamental group
In this note we consider the relationship between the dressing action and the
holonomy representation in the context of constant mean curvature surfaces. We
characterize dressing elements that preserve the topology of a surface and
discuss dressing by simple factors as a means of adding bubbles to a class of
non finite type cylinders.Comment: 36 pages, 1 figur
Generalized DPW method and an application to isometric immersions of space forms
Let be a complex Lie group and denote the group of maps from
the unit circle into , of a suitable class. A differentiable
map from a manifold into , is said to be of \emph{connection
order } if the Fourier expansion in the loop parameter of the
-family of Maurer-Cartan forms for , namely F_\lambda^{-1}
\dd F_\lambda, is of the form . Most
integrable systems in geometry are associated to such a map. Roughly speaking,
the DPW method used a Birkhoff type splitting to reduce a harmonic map into a
symmetric space, which can be represented by a certain order map,
into a pair of simpler maps of order and respectively.
Conversely, one could construct such a harmonic map from any pair of
and maps. This allowed a Weierstrass type description
of harmonic maps into symmetric spaces. We extend this method to show that, for
a large class of loop groups, a connection order map, for ,
splits uniquely into a pair of and maps. As an
application, we show that constant non-zero curvature submanifolds with flat
normal bundle of a sphere or hyperbolic space split into pairs of flat
submanifolds, reducing the problem (at least locally) to the flat case. To
extend the DPW method sufficiently to handle this problem requires a more
general Iwasawa type splitting of the loop group, which we prove always holds
at least locally.Comment: Some typographical correction
Isothermic submanifolds of symmetric -spaces
We extend the classical theory of isothermic surfaces in conformal 3-space,
due to Bour, Christoffel, Darboux, Bianchi and others, to the more general
context of submanifolds of symmetric -spaces with essentially no loss of
integrable structure.Comment: 35 pages, 3 figures. v2: typos and other infelicities corrected
Discrete -nets and Guichard nets
We provide a convincing discretisation of Demoulin's -surfaces along
with their specialisations to Guichard and isothermic surfaces with no loss of
integrable structure.Comment: 39 A4 page
New constructions of twistor lifts for harmonic maps
We show that given a harmonic map from a Riemann surface to a
classical compact simply connected inner symmetric space, there is a
-holomorphic twistor lift of (or its negative) if and only if it
is nilconformal. In the case of harmonic maps of finite uniton number, we give
algebraic formulae in terms of holomorphic data which describes their extended
solutions. In particular, this gives explicit formulae for the twistor lifts of
all harmonic maps of finite uniton number from a surface to the above symmetric
spaces.Comment: Some minor changes and a correction of Example 8.
Constructing solutions to the Bj\"orling problem for isothermic surfaces by structure preserving discretization
In this article, we study an analog of the Bj\"orling problem for isothermic
surfaces (that are more general than minimal surfaces): given a real analytic
curve in , and two analytic non-vanishing orthogonal
vector fields and along , find an isothermic surface that is
tangent to and that has and as principal directions of
curvature. We prove that solutions to that problem can be obtained by
constructing a family of discrete isothermic surfaces (in the sense of Bobenko
and Pinkall) from data that is sampled along , and passing to the limit
of vanishing mesh size. The proof relies on a rephrasing of the
Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its
discretization which is induced from the geometry of discrete isothermic
surfaces. The discrete-to-continuous limit is carried out for the Christoffel
and the Darboux transformations as well.Comment: 29 pages, some figure
Some fundamental algebraic tools for the semantics of computation: Part 3. indexed categories
AbstractThis paper presents indexed categories which model uniformly defined families of categories, and suggests that they are a useful tool for the working computer scientist. An indexed category gives rise to a single flattened category as a disjoint union of its component categories plus some additional morphisms. Similarly, an indexed functor (which is a uniform family of functors between the components categories) induces a flattened functor between the corresponding flattened categories. Under certain assumptions, flattened categories are (co)complete if all their components are, and flattened functors have left adjoints if all their components do. Several examples are given. Although this paper is Part 3 of the series “Some fundamental algebraic tools for the semantics of computation”, it is entirely independent of Parts 1 and 2
Willmore Surfaces of Constant Moebius Curvature
We study Willmore surfaces of constant Moebius curvature in . It is
proved that such a surface in must be part of a minimal surface in
or the Clifford torus. Another result in this paper is that an isotropic
surface (hence also Willmore) in of constant could only be part of a
complex curve in or the Veronese 2-sphere in . It is
conjectured that they are the only examples possible. The main ingredients of
the proofs are over-determined systems and isoparametric functions.Comment: 16 pages. Mistakes occured in the proof to the main theorem (Thm 3.6)
has been correcte
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