146 research outputs found
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
Curved Flats, Pluriharmonic Maps and Constant Curvature Immersions into Pseudo-Riemannian Space Forms
We study two aspects of the loop group formulation for isometric immersions
with flat normal bundle of space forms. The first aspect is to examine the loop
group maps along different ranges of the loop parameter. This leads to various
equivalences between global isometric immersion problems among different space
forms and pseudo-Riemannian space forms. As a corollary, we obtain a
non-immersibility theorem for spheres into certain pseudo-Riemannian spheres
and hyperbolic spaces.
The second aspect pursued is to clarify the relationship between the loop
group formulation of isometric immersions of space forms and that of
pluriharmonic maps into symmetric spaces. We show that the objects in the first
class are, in the real analytic case, extended pluriharmonic maps into certain
symmetric spaces which satisfy an extra reality condition along a totally real
submanifold. We show how to construct such pluriharmonic maps for general
symmetric spaces from curved flats, using a generalised DPW method.Comment: 21 Pages, reference adde
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
Clifford algebras and new singular Riemannian foliations in spheres
Using representations of Clifford algebras we construct indecomposable
singular Riemannian foliations on round spheres, most of which are
non-homogeneous. This generalizes the construction of non-homogeneous
isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.Comment: 21 pages. Construction of foliations in the Cayley plane added.
Proofs simplified and presentation improved, according to referee's
suggestions. To appear in Geom. Funct. Ana
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
A note on isoparametric polynomials
We show that any homogeneous polynomial solution of |\nabla
F(x)|^2=m^2|x|^(2m-2), m>1, is either a radially symmetric polynomial F(x)=\pm
|x|^m (for even m's) or it is a composition of a Chebychev polynomial and a
Cartan-M\"unzner polynomial.Comment: 6 page
New examples of Willmore submanifolds in the unit sphere via isoparametric functions,II
This paper is a continuation of a paper with the same title of the last two
authors. In the first part of the present paper, we give a unified geometric
proof that both focal submanifolds of every isoparametric hypersurface in
spheres with four distinct principal curvatures are Willmore. In the second
part, we completely determine which focal submanifolds are Einstein except one
case.Comment: 19 pages,to appear in Annals of Global Analysis and Geometr
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
Polar foliations and isoparametric maps
A singular Riemannian foliation on a complete Riemannian manifold is
called a polar foliation if, for each regular point , there is an immersed
submanifold , called section, that passes through and that meets
all the leaves and always perpendicularly. A typical example of a polar
foliation is the partition of into the orbits of a polar action, i.e., an
isometric action with sections. In this work we prove that the leaves of
coincide with the level sets of a smooth map if is simply
connected. In particular, we have that the orbits of a polar action on a simply
connected space are level sets of an isoparametric map. This result extends
previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter
and West, and Terng.Comment: 9 pages; The final publication is available at springerlink.com
http://www.springerlink.com/content/c72g4q5350g513n1
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