On non-homogeneous Cauchy–Fueter equations and Hartogs’ phenomenon in several quaternionic variables

The Cauchy–Fueter complex is the counterpart of the Dolbeault complex in the theory of several quaternionic variables. By using the fundamental solution to the Laplacian operators of fourth order associated to this differential complex on Hⁿ${\mathbb{H}}^n$, we can solve the system of non-homogeneous Cauchy–Fueter equations and prove the Hartogs’ extension phenomenon for quaternionic regular functions on any domain. The quaternionic version of Bochner–Martinelli integral representation formula for H$\mathbb{H}$-valued functions is also given.     

HyperKähler and quaternionic Kähler manifolds with S -symmetries

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S¹$S^1$ can be constructed from another hyperKähler manifold (of lower dimension) with an action of S¹$S^1$ that fixes one complex structure and rotates the other two and vice versa. We also study the corresponding quaternionic Kähler manifolds equipped with a quaternionic Kähler action of the circle. In particular we show that any positive quaternionic Kähler manifolds with S¹$S^1$-symmetry admits a Kähler metric on an open everywhere dense subset.     

Global supersymmetry on curved spaces in various dimensions

We propose methods towards a systematic determination of d  -dimensional curved spaces where Euclidean field theories with rigid supersymmetry can be defined. The analysis is carried out from a group theory as well as from a supergravity point of view. In particular, by using appropriate gauged supergravities in various dimensions we show that supersymmetry can be defined in conformally flat spaces, such as non-compact hyperboloids Hⁿ⁺¹${\mathbb{H}}^{n+1}$ and compact spheres Sⁿ${\mathbb{S}}^n$ or – by turning on appropriate Wilson lines corresponding to R-symmetry vector fields – on S¹×Sⁿ${\mathbb{S}}^1\times {\mathbb{S}}^n$, with n<6$n<6$. By group theory arguments we show that Euclidean field theories with rigid supersymmetry cannot be consistently defined on round spheres S^d${\mathbb{S}}^d$ if d>5$d>5$ (despite the existence of Killing spinors). We also show that distorted spheres and certain orbifolds are also allowed by the group theory classification.     

Higher trace and Berezinian of matrices over a Clifford algebra

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded commutative associative algebra A$A$. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded matrices of degree 0$0$ is polynomial in its entries. In the case of the algebra A=H$A=\mathbb{H}$ of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded version of Liouville’s formula.     

Curvatures,volumes and norms of derivatives for curves in Riemannian manifolds

The well-known formulas express the curvature and the torsion of a curve in R³${\mathbb{R}}^3$ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in arbitrary Riemannian manifolds. Our motivation comes from physics. It follows that regular curves in Rⁿ${\mathbb{R}}^n$ are determined up to isometry by the norms of their n$n$ consecutive derivatives. We extend this fact to two-point homogeneous spaces.     

Topological and affine classification of complete noncompact flat 4-manifolds

In this paper we give topological and affine classification of complete noncompact flat 4-manifolds. In particular, we show that the number of diffeomorphism classes of them is equal to 44. The affine classification uses the results of [M. Sadowski, Affinely equivalent complete flat manifolds, Cent. Eur. J. Math. 2 (2) (2004) 332–338]. The affine and the topological equivalence classes are the same for flat manifolds not homotopy equivalent to S¹,T²$S^1,T^2$ or the Klein bottle.     

Killing initial data on totally umbilical & compact hypersurfaces

In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry non-trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces (Mⁿ,g,cg)$(M^n,g,cg)$ endowed with a Riemannian metric g$g$ and a second fundamental form cg$cg$ (where c∈C^∞(M)$c\in C^{\infty }\left(M\right)$ a priori) have constant mean curvature and are isometric to one of the following manifolds:

(i)

Sⁿ${\mathbb{S}}^n$ the standard sphere,     

Force free projective motions of the sphere

Suppose that the sphere Sⁿ$S^n$ has initially a homogeneous distribution of mass and let G$G$ be the Lie group of orientation preserving projective diffeomorphisms of Sⁿ$S^n$. A projective motion of the sphere, that is, a smooth curve in G$G$, is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of Sⁿ$S^n$ and, more generally, examples of subgroups H$H$ of G$G$ such that a force free motion initially tangent to H$H$ remains in H$H$ for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H=S_On+1$H=S{O}_{n+1}$). The main tool is a Riemannian metric on G$G$, which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy.     

Some remarks on special Kähler geometry

Given a special Kähler manifold M$M$, we give a new, direct proof of the relationship between the quaternionic structure on T^∗M$T^{\ast }M$ and the variation of Hodge structures on T^CM$T^{\mathbb{C}}M$.     

Singularities of renormalization group flows

We discuss singularity formation in certain renormalization group flows. Special cases are the Ricci Yang–Mills and B$B$-field flows. We point out some results suggesting that topological hypotheses can make RG flows much less singular than Ricci flow. In particular we show that for rotationally symmetric initial data on S²×S¹$S^2\times S^1$ one gets long time existence and convergence of RYM flow, in stark contrast to the case for Ricci flow [S. Angenent, D. Knopf, An example of neckpinching for Ricci flow on Sⁿ⁺¹$S^{n+1}$, Math. Res. Lett. 11 (4) (2004) 493–518]. Other results are given which allow one to rule out many singularity models under strictly topological hypotheses. A conjectural picture of singularity formation for RG flow on 3-manifolds is given.     

Pre-quantization of the moduli space of flat G-bundles over a surface

For a simply connected, compact, simple Lie group G$G$, the moduli space of flat G$G$-bundles over a closed surface Σ$\Sigma$ is known to be pre-quantizable at integer levels. For non-simply connected G$G$, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction–namely a certain cohomology class in H³(G²;Z)$H^3(G^2;\mathbb{Z})$–that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G$G$.     

Resolutions of non-regular Ricci-flat Kähler cones

We present explicit constructions of complete Ricci-flat Kähler metrics that are asymptotic to cones over non-regular Sasaki–Einstein manifolds. The metrics are constructed from a complete Kähler–Einstein manifold (V,_gV)$(V,g_V)$ of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kähler metrics on the total spaces of (i) holomorphic C²/_Zp${\mathbb{C}}^2/{\mathbb{Z}}_p$ orbifold fibrations over V$V$, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP¹${\mathbb{W}\mathbb{C}\mathbb{P}}^1$, with generic fibres being the canonical complex cone over V$V$, and (iii) the canonical orbifold line bundle over a family of Fano orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kähler metrics on the total spaces of (a) rank two holomorphic vector bundles over V$V$, and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base V$V$. When V=CP¹$V={\mathbb{C}\mathbb{P}}^1$ our results give Ricci-flat Kähler orbifold metrics on various toric partial resolutions of the cone over the Sasaki–Einstein manifolds Y^p,q$Y^{p,q}$.     

On complete spacelike hypersurfaces with two distinct principal curvatures in Lorentz–Minkowski space

We investigate complete spacelike hypersurfaces in Lorentz–Minkowski space with two distinct principal curvatures and constant m$m$th mean curvature. By using Otsuki’s idea, we obtain the global classification result. As their applications, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Lorentz–Minkowski (n+1)$(n+1)$-spaces (n≥3$n\ge 3$) of nonzero constant m$m$th mean curvature (m≤n−1$m\le n-1$) with two distinct principal curvatures λ$\lambda$ and μ$\mu$ satisfying inf(λ−μ)²>0$inf{(\lambda -\mu )}^2>0$ are the hyperbolic cylinders. We also obtain a global characterization for hyperbolic cylinder Hⁿ⁻¹(c)×R${\mathbb{H}}^{n-1}\left(c\right)\times \mathbb{R}$ in terms of square length of the second fundamental form.