A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.     

On the minimality of the p-harmonic ma p $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E _f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E _f,p among the maps in which coincide with on .      

Flat nearly Kähler manifolds

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form $\zeta \in \Lambda^3 (\mathbb{C}^m)^*We classify flat strict nearly K?hler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K?hler factor of maximal dimension and a strict flat nearly K?hler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form . The first nontrivial example occurs in dimension 4m = 12.      

Schatten-von Neumann Properties in the Weyl Calculus, and Calculus of Metrics on Symplectic Vector Spaces

Let s ^w p be the set of all a ∈ ? such that a ^w (x, D) is Schatten p-operator on L ². Then we prove the following:

• $S(m,g)\subseteq s_p^wLet s ^w p be the set of all a ∈ ℓ such that a ^w (x, D) is Schatten p-operator on L ². Then we prove the following:

  •	iff . Furthermore, when . Consequently, when ;
  •	if , then is symplectically invariantly defined. Moreover, if and is slowly varying (and σ-temperate), then the same is true for G;
  •	a generalization of sharp G?rding's inequality.

  Mathematics Subject Classifications (2000) Primary: 35S05, 47B10, 47L15 Secondary: 32F45, 16W80     

Weak Riemannian manifolds from finite index subfactors

Let N ⊂ M be a finite Jones’ index inclusion of II₁ factors and denote by U _N ⊂ U _M their unitary groups. In this article, we study the homogeneous space U _M /U _N , which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit of the Jones projection of the inclusion. We endow with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete); therefore, is a weak Riemannian manifold. We show that enjoys certain properties similar to classic Hilbert–Riemann manifolds. Among them are metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point p ₁ of , there is a ball (of uniform radius r) of the usual norm of M, such that any point p ₂ in the ball is joined to p ₁ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion , where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and .      

A local rigidity theorem for minimal surfaces in Minkowski 3-space of Randers type

Let be a Minkowski 3-space of Randers type with , where is the Euclidean metric and . We consider minimal surfaces in and prove that if a connected surface M in is minimal with respect to both the Busemann–Hausdorff volume form and the Holmes–Thompson volume form, then up to a parallel translation of , M is either a piece of plane or a piece of helicoid which is generated by lines screwing about the x ³-axis.      

Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have . Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have . We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] ^d ) is d. B. Burgstaller was supported by the Austrian Schr?dinger stipend J2471-N12.     

Edge-boundary problems with singular trace conditions

The ellipticity of boundary value problems on a smooth manifold with boundary relies on a two-component principal symbolic structure , consisting of interior and boundary symbols. In the case of a smooth edge on manifolds with boundary, we have a third symbolic component, namely, the edge symbol , referring to extra conditions on the edge, analogously as boundary conditions. Apart from such conditions ‘in integral form’ there may exist singular trace conditions, investigated in Kapanadze et al., Internal Equations and Operator Theory, 61, 241–279, 2008 on ‘closed’ manifolds with edge. Here, we concentrate on the phenomena in combination with boundary conditions and edge problem.     

On the expansion of the resolvent for elliptic boundary contact problems

Let A be an elliptic operator on a compact manifold with boundary , and let be a covering map, where Y is a closed manifold. Let A _C be a realization of A subject to a coupling condition C that is elliptic with parameter in the sector Λ. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace for N sufficiently large has a complete asymptotic expansion as . In particular, the heat trace has a complete asymptotic expansion as , and the -function has a meromorphic extension to .      

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-invariants of Riemannian foliations

For a Riemannian foliation on a closed manifold M, we define L ²-spectral sequence Betti numbers and spectral sequence Novikov–Shubin invariants. The spectral sequence of the lift of to the universal covering of M is used in the definitions. These invariants are natural extensions of the L ²-Betti numbers and the Novikov–Shubin invariants of differentiable manifolds. It is shown that these numbers are invariant by foliated homotopy equivalences, and they are computed for several examples.      

Conformally Einstein products and nearly Kähler manifolds

In the first part of this note we study compact Riemannian manifolds (M, g) whose Riemannian product with is conformally Einstein. We then consider 6-dimensional almost Hermitian manifolds of type W ₁ + W ₄ in the Gray–Hervella classification admitting a parallel vector field and show that (under some mild assumption) they are obtained as Riemannian cylinders over compact Sasaki–Einstein 5-dimensional manifolds.      

<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

Local boundedness of the number of solutions of Plateau’s problem for polygonal boundary curves

In the present article, the author proves two generalizations of his “finiteness-result” (I.H.P. Anal. Non-lineaire, 2006, accepted) which states for any extreme simple closed polygon that every immersed, stable disc-type minimal surface spanning Γ is an isolated point of the set of all disc-type minimal surfaces spanning Γ w.r.t. the C ⁰-topology. First, it is proved that this statement holds true for any simple closed polygon in , provided it bounds only minimal surfaces without boundary branch points. Also requiring that the interior angles at the vertices of such a polygon Γ have to be different from the author proves the existence of some neighborhood O of Γ in and of some integer , depending only on Γ, such that the number of immersed, stable disc-type minimal surfaces spanning any simple closed polygon contained in O, with the same number of vertices as Γ, is bounded by .      

Operator space tensor products of <Emphasis Type="Italic">C</Emphasis>*-algebras

For C*-algebras A and B, the identity map from into A _λ B is shown to be injective. Next, we deduce that the center of the completion of the tensor product A⊗B of two C*-algebras A and B with centers Z(A) and Z(B) under operator space projective norm is equal to . A characterization of isometric automorphisms of and A _h B is also obtained. Dedicated to Ajit Iqbal Singh on her 65th birthday.     

Dynamically convex Finsler metrics and <Emphasis Type="Italic">J</Emphasis>-holomorphic embedding of asymptotic cylinders

We explore the relationship between contact forms on defined by Finsler metrics on and the theory developed by H. Hofer, K. Wysocki and E. Zehnder (Hofer etal. Ann. Math. 148, 197–289, 1998; Ann. Math. 157, 125–255, 2003). We show that a Finsler metric on with curvature K ≥ 1 and with all geodesic loops of length > π is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct J-holomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on with K = 1, thus complementing the results obtained in Harris and Wysocki (Trans. Am. Math. Soc., to appear).      

Harmonic morphisms from the classical compact semisimple Lie groups

In this article, we introduce a new method for manufacturing harmonic morphisms from semi-Riemannian manifolds. This is employed to yield a variety of new examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their standard Riemannian metrics. We develop a duality principle and show how this can be used to construct the first known examples of harmonic morphisms from the non-compact Lie groups , SU ^*(2n), , SO ^*(2n), SO(p, q), SU(p, q) and Sp(p, q) equipped with their standard dual semi-Riemannian metrics.      

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

Curvature homogeneous Lorentzian three-manifolds

We study three-dimensional curvature homogeneous Lorentzian manifolds. We prove that for all Segre types of the Ricci operator, there exist examples of nonhomogeneous curvature homogeneous Lorentzian metrics in .      

Ground state alternative for <Emphasis Type="Italic">p</Emphasis>-Laplacian with potential term

Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u _k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.