13 research outputs found
A Berger type normal holonomy theorem for complex submanifolds
We prove a kind of Berger-Simons' Theorem for the normal holonomy group of a complex submanifold of the projective spac
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
Convergence of vector bundles with metrics of Sasaki-type
If a sequence of Riemannian manifolds, , converges in the pointed
Gromov-Hausdorff sense to a limit space, , and if are vector
bundles over endowed with metrics of Sasaki-type with a uniform upper
bound on rank, then a subsequence of the converges in the pointed
Gromov-Hausdorff sense to a metric space, . The projection maps
converge to a limit submetry and the fibers converge to
its fibers; the latter may no longer be vector spaces but are homeomorphic to
, where is a closed subgroup of ---called the {\em wane
group}--- that depends on the basepoint and that is defined using the holonomy
groups on the vector bundles. The norms converges to a map
compatible with the re-scaling in and the -action
on converges to an action on compatible with the
limiting norm.
In the special case when the sequence of vector bundles has a uniform lower
bound on holonomy radius (as in a sequence of collapsing flat tori to a
circle), the limit fibers are vector spaces. Under the opposite extreme, e.g.
when a single compact -dimensional manifold is re-scaled to a point, the
limit fiber is where is the closure of the holonomy group of the
compact manifold considered.
An appropriate notion of parallelism is given to the limiting spaces by
considering curves whose length is unchanged under the projection. The class of
such curves is invariant under the -action and each such curve preserves
norms. The existence of parallel translation along rectifiable curves with
arbitrary initial conditions is also exhibited. Uniqueness is not true in
general, but a necessary condition is given in terms of the aforementioned wane
groups .Comment: 44 pages, 1 figure, in V.2 added Theorem E and Section 4 on
parallelism in the limit space
Dressing with Control: using integrability to generate desired solutions to Einstein's equations
21 pages, no figures21 pages, no figures21 pages, no figures21 pages, no figuresMotivated by integrability of the sine-Gordon equation, we investigate a technique for constructing desired solutions to Einstein's equations by combining a dressing technique with a control-theory approach. After reviewing classical integrability, we recall two well-known Killing field reductions of Einstein's equations, unify them using a harmonic map formulation, and state two results on the integrability of the equations and solvability of the dressing system. The resulting algorithm is then combined with an asymptotic analysis to produce constraints on the degrees of freedom arising in the solution-generation mechanism. The approach is carried out explicitly for the Einstein vacuum equations. Applications of the technique to other geometric field theories are also discussed
A five-gene signature and clinical outcome in non-small-cell lung cancer
BACKGROUND: Current staging methods are inadequate for predicting the outcome of treatment of non-small-cell lung cancer (NSCLC). We developed a five-gene signature that is closely associated with survival of patients with NSCLC. METHODS: We used computer-generated random numbers to assign 185 frozen specimens for microarray analysis, real-time reverse-transcriptase polymerase chain reaction (RT-PCR) analysis, or both. We studied gene expression in frozen specimens of lung-cancer tissue from 125 randomly selected patients who had undergone surgical resection of NSCLC and evaluated the association between the level of expression and survival. We used risk scores and decision-tree analysis to develop a gene-expression model for the prediction of the outcome of treatment of NSCLC. For validation, we used randomly assigned specimens from 60 other patients. RESULTS: Sixteen genes that correlated with survival among patients with NSCLC were identified by analyzing microarray data and risk scores. We selected five genes (DUSP6, MMD, STAT1, ERBB3, and LCK) for RT-PCR and decision-tree analysis. The five-gene signature was an independent predictor of relapse-free and overall survival. We validated the model with data from an independent cohort of 60 patients with NSCLC and with a set of published microarray data from 86 patients with NSCLC. CONCLUSIONS: Our five-gene signature is closely associated with relapse-free and overall survival among patients with NSCLC