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## On the uniqueness of solutions of a certain characteristic Cauchy problem

### Annals of Global Analysis and Geometry (1984-06-01) 2: 153-162 , June 01, 1984

The question of uniqueness of solutions of the global Cauchy problem (1)–(2) below is discussed. We assume that there exists a complex constant c such that the modified equation $$\frac{{\partial ^2 u}}{{\partial t^2 }} = c_{\left| \alpha \right| \leqq 2} \sum a_\alpha (x) D_x^\alpha u$$ becomes hyperbolic. Under this and some other additional conditions (See Condition A in §2) we prove the uniqueness of solutions of the Cauchy problem within the class of functions u(t, x) such that $$|u(t,x)| \leqq C exp(a|x|^2 ) ,$$ C and a being positive constants

## Killing and twistor spinors with torsion

### Annals of Global Analysis and Geometry (2016-03-01) 49: 105-141 , March 01, 2016

We study twistor spinors (with torsion) on Riemannian spin manifolds
$$(M^{n}, g, T)$$
carrying metric connections with totally skew-symmetric torsion. We consider the characteristic connection
$$\nabla ^{c}=\nabla ^{g}+\frac{1}{2}T$$
and under the condition
$$\nabla ^{c}T=0$$
, we show that the twistor equation with torsion w.r.t. the family
$$\nabla ^{s}=\nabla ^{g}+2sT$$
can be viewed as a parallelism condition under a suitable connection on the bundle
$$\Sigma \oplus \Sigma $$
, where
$$\Sigma $$
is the associated spinor bundle. Consequently, we prove that a twistor spinor with torsion has isolated zero points. Next we study a special class of twistor spinors with torsion, namely these which are *T*-eigenspinors and parallel under the characteristic connection; we show that the existence of such a spinor for some
$$s\ne 1/4$$
implies that
$$(M^{n}, g, T)$$
is both Einstein and
$$\nabla ^{c}$$
-Einstein, in particular the equation
$${{\mathrm{Ric}}}^{s}=\frac{{{\mathrm{Scal}}}^{s}}{n}g$$
holds for any
$$s\in \mathbb {R}$$
. In fact, for
$$\nabla ^{c}$$
-parallel spinors we provide a correspondence between the Killing spinor equation with torsion and the Riemannian Killing spinor equation. This allows us to describe 1-parameter families of non-trivial Killing spinors with torsion on nearly Kähler manifolds and nearly parallel
$${{\mathrm{G}}}_2$$
-manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere
$${{\mathrm{S}}}^{3}$$
. We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.

## Quaternionic Kähler reductions of Wolf spaces

### Annals of Global Analysis and Geometry (2007-10-01) 32: 225-252 , October 01, 2007

The main purpose of the following article is to introduce a *Lie theoretical* approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called *generalized Wolf spaces*.

## Eta Invariant and Conformal Cobordism

### Annals of Global Analysis and Geometry (2005-06-01) 27: 333-340 , June 01, 2005

In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an Abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound conformally flat structures in a conformally invariant way. The eta invariant gives rise to a homomorphism from this group to the circle group, which can be highly nontrivial. It remains an interesting question of how to compute this group.

## Polynomial growth harmonic functions on finitely generated abelian groups

### Annals of Global Analysis and Geometry (2013-12-01) 44: 417-432 , December 01, 2013

In the present paper, we develop geometric analysis techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We provide a geometric analysis proof of the classical Heilbronn theorem (Heilbronn in Proc Camb Philos Soc 45:194–206, 1949) and the recent Nayar theorem (Nayar in Bull Pol Acad Sci Math 57:231–242, 2009) on polynomial growth harmonic functions on lattices $$\mathbb Z ^n$$ that does not use a representation formula for harmonic functions. In the abelian group case, by Yau’s gradient estimate, we actually give a simplified proof of a general polynomial growth harmonic function theorem of (Alexopoulos in Ann Probab 30:723–801, 2002). We calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups by linear algebra, rather than by Floquet theory Kuchment and Pinchover (Trans Am Math Soc 359:5777–5815, 2007). While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself. Moreover, we also calculate the dimension of solutions to higher order Laplace operators.

## Construction Des Applications Harmoniques Non Rigides D'un Tore Dans La Sphère

### Annals of Global Analysis and Geometry (1983-06-01) 1: 105-118 , June 01, 1983

### Zusammenfassung

Le but de Particle present est l'étude des déformations harmoniques géodésiques. f d'un tore dans la sphere avec la densité d'énergie 1/2 . Nous montrons que tout les deformations harmoniques sont isométriques, c'est-à-dire que l'application f est rigide, si et seulement si f est totalement géodésique. De plus nous calculons les espaces de deformations harmoniques pour la 3-sphère.

## Connections and Higgs fields on a principal bundle

### Annals of Global Analysis and Geometry (2008-03-01) 33: 19-46 , March 01, 2008

Let *M* be a compact connected Kähler manifold and *G* a connected linear algebraic group defined over
$${\mathbb{C}}$$
. A Higgs field on a holomorphic principal *G*-bundle ε_{G} over *M* is a holomorphic section *θ* of
$$\text{ad}(\epsilon_{G})\otimes {\Omega}^{1}_{M}$$
such that *θ*∧ *θ* = 0. Let *L*(*G*) be the Levi quotient of *G* and (ε_{G}(*L*(*G*)), *θ*_{l}) the Higgs *L*(*G*)-bundle associated with (ε_{G}, *θ*). The Higgs bundle (ε_{G}, *θ*) will be called semistable (respectively, stable) if (ε_{G}(*L*(*G*)), *θ*_{l}) is semistable (respectively, stable). A semistable Higgs *G*-bundle (ε_{G}, *θ*) will be called pseudostable if the adjoint vector bundle ad(ε_{G}(*L*(*G*))) admits a filtration by subbundles, compatible with *θ*, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat *G*-bundles over *M* and the category of pseudostable Higgs *G*-bundles over *M* with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kähler manifold. As an application, we give various equivalent conditions for a holomorphic *G*-bundle over a complex torus to admit a flat holomorphic connection.

## Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

### Annals of Global Analysis and Geometry (2011-01-01) 39: 1-12 , January 01, 2011

Studying the condition $${h(FX,Y)-h(X,FY)=g(FX,Y)\eta, 0\ne\eta\in T^\perp(M)}$$ on the almost contact structure *F* and on the second fundamental form *h* of *n*-dimensional real submanifolds *M* of complex hyperbolic space $${\mathbb {CH}^{\frac{n+p}{2}}}$$ when their maximal holomorphic tangent subspace is (*n* − 1)-dimensional, we obtain the complete classification of such submanifolds *M* and we characterize certain model spaces in complex hyperbolic space.

## Flat Pseudo-Reimannian manifolds with a nilpotent transitive group of isometries

### Annals of Global Analysis and Geometry (1992-01-01) 10: 87-101 , January 01, 1992

Flat pseudo-Riemannian manifolds with a nilpotent transitive group of isometries are shown to be complete. Also flat pseudo-Riemannian homogeneous manifolds with non-trivial holonomy are shown to contain a complete geodesic.

## A comparison theorem for the mean exit time from a domain in a Kähler manifold

### Annals of Global Analysis and Geometry (1992-01-01) 10: 73-80 , January 01, 1992

Let *M* be a Kähler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let ω be a domain in *M* with some bounds on the mean and JN-mean curvatures of its boundary ∂ω. The main result of this paper is a comparison theorem between the Mean Exit Time function defined on ω and the Mean Exit Time from a geodesic ball of the complex projective space *ℂℙ*^{n}(λ) which involves a characterization of the geodesic balls among the domain ω. In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of ω, using the Index Lemma for Submanifolds.