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

Generalized DPW method and an application to isometric immersions of space forms

Let $G$ be a complex Lie group and $\Lambda G$ denote the group of maps from the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $\Lambda G$, is said to be of \emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $\lambda$ of the ${\mathbb S}^1$-family of Maurer-Cartan forms for $F$, namely F_\lambda^{-1} \dd F_\lambda, is of the form $\sum_{i=a}^b \alpha_i \lambda^i$. 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 $(_{-1}^1)$ map, into a pair of simpler maps of order $(_{-1}^{-1})$ and $(_1^1)$ respectively. Conversely, one could construct such a harmonic map from any pair of $(_{-1}^{-1})$ and $(_1^1)$ 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 $(_a^b)$ map, for $a<0<b$, splits uniquely into a pair of $(_a^{-1})$ and $(_1^b)$ 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

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

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

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

Polar foliations and isoparametric maps

A singular Riemannian foliation $F$ on a complete Riemannian manifold $M$ is called a polar foliation if, for each regular point $p$, there is an immersed submanifold $\Sigma$, called section, that passes through $p$ and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of $M$ into the orbits of a polar action, i.e., an isometric action with sections. In this work we prove that the leaves of $F$ coincide with the level sets of a smooth map $H: M\to \Sigma$ if $M$ 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

Singular riemannian foliations with sections, transnormal maps and basic forms

A singular riemannian foliation F on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold (a section) that meets every leaf of F orthogonally and whose dimension is the codimension of the regular leaves of F. We prove that the algebra of basic forms of M relative to F is isomorphic to the algebra of those differential forms on a section that are invariant under the generalized Weyl pseudogroup of this section. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of F coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of F are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.Comment: Preprint IME-USP; The final publication is available at springerlink.com http://www.springerlink.com/content/q48682633730t831

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

Minimal cubic cones via Clifford algebras

We construct two infinite families of algebraic minimal cones in $R^{n}$. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any $n\ge4$, $n\ne 16k+1$, there is at least one minimal cone in $R^{n}$ given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in $R^{m^2}$, $m\ge2$, defined by an irreducible homogeneous polynomial of degree $m$. These examples provide particular answers to the questions on algebraic minimal cones posed by Wu-Yi Hsiang in the 1960's.Comment: Final version, corrects typos in Table