Higher dimensional Calabi–Yau manifolds of Kummer type

Based on Cynk–Hulek method from [7] we construct complex Calabi–Yau varieties of arbitrary dimensions using elliptic curves with an automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall generalize the result of [11] to obtain arbitrarily dimensional Calabi–Yau manifolds which are Zariski in any characteristic $p\not\equiv \text{◂⋅▸}1(\mathrm{mod}12)$.     

Hermitian Yang-Mills Metrics on Higgs Bundles over Asymptotically Cylindrical Kähler Manifolds

Let V be an asymptotically cylindrical Kahler manifold with asymptotic cross-section     

Noncompact quasi‐Einstein manifolds conformal to a Euclidean space

The goal of this article is to investigate nontrivial m‐quasi‐Einstein manifolds globally conformal to an n‐dimensional Euclidean space. By considering such manifolds, whose conformal factors and potential functions are invariant under the action of an $(n?1)$‐dimensional translation group, we provide a complete classification when $\lambda =0$ and $m\ge 1$ or $m=2?n$.     

Approximate the dynamical behavior for stochastic systems by Wong–Zakai approaching

Random invariant manifolds and foliations play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. In a general way, these random objects are difficult to be visualized geometrically or computed numerically. The current work provides a perturbation approach to approximate these random invariant manifolds and foliations. After briefly discussing the existence of random invariant manifolds and foliations for a class of stochastic systems driven by additive noises, the corresponding Wong–Zakai type of convergence result in path-wise sense is established.     

Erratum to “Curvature of hyperkähler quotients”

The original version of the article was published in Central European Journal of Mathematics, 2008, 6(2), 191–203, DOI: 10.2478/s11533-008-0026-8. Unfortunately, the original version of this article contains a mistake, which we correct here. The online version of the original article can be found at     

Lower-Dimensional Volumes and Kastler-Kalau-Walze Type Theorem for Manifolds with Boundary

In this paper, we define lower-dimensional volumes of spin manifolds with boundary. We compute the lower-dimensional volume Vol（2＇2） for 5-dimensional and 6-dimensional spin manifolds with boundary and we also get the Kastler-Kalau-Walze type theorem in this case.     

A correspondence for isometric immersions into product spaces and its applications

We present a correspondence for isometric immersions that are graphs in Riemannian or semi-Riemannian warped product spaces. We use this correspondence to give several existence and non-existence theorems for hypersurfaces in Riemannian or Lorentzian spaces. In the case of surfaces, we obtain further applications regarding height estimates, harmonic representation of surfaces and the existence of holomorphic quadratic differentials in homogeneous and non-homogeneous spaces.