Nearly Kähler Submanifolds of a Space Form

In this article, we study isometric immersions of nearly Kähler manifolds into a space form (especially Euclidean space) and show that every nearly Kähler submanifold of a space form has an umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover, using this foliation we show that there is no non-homogeneous 6-dimensional nearly Kähler submanifold of a space form. We prove some results towards a classification of nearly Kähler hypersurfaces in standard space forms.     

To the question about the maximum principle for manifolds over local algebras

We consider manifolds over a local algebra A. We study basis functions of the canonical foliation which represent the real parts of A-differentiable functions. We prove that these are constant functions. We find the form of A-differentiable functions on some manifolds over local algebras, in particular, on compact manifolds. We obtain an estimate for the dimension of some spaces of 1-forms and analogs of the above results for the projective mappings of foliations.     

Partial Harmonicity of Continuous Maximal Plurisubharmonic Functions

If u is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of u has constant rank, it is known that there exists a foliation by complex manifolds, such that u is harmonic along the leaves of the foliation. In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension (1, 1) such that the function is harmonic along the currents.     

3-Quasi-Sasakian manifolds

We correct the results in section 6 of [B. Cappelletti Montano, A. De Nicola, G. Dileo, 3-Quasi-Sasakian manifolds, Ann. Global Anal. Geom. 33 (2008), 397–409], concerning the corrected energy of the Reeb distribution of a compact 3-quasi-Sasakian manifold. The results are slightly different than what was originally claimed and they are obtained by using results in [B. Cappelletti Montano, A. De Nicola, G. Dileo, The geometry of a 3-quasi-Sasakian manifold, Int. J. Math., to appear, arXiv:0801.1818], where the geometry of these manifolds is more deeply investigated. The online version of the original article can be found under doi:.     

Foliations, topology and geometry of 3-manifolds: R-covered foliations and transverse pseudo-Anosov flows

We analyse the topological and geometrical behavior of foliations on 3-manifolds. We consider the transverse structure of an R-covered foliation in a 3-manifold, where R-covered means that in the universal cover the leaf space of the foliation is Hausdorff. If the manifold is aspherical we prove that either there is an incompressible torus in the manifold; or there is a transverse pseudo-Anosov flow. It follows that manifolds with R-covered foliations satisfy the weak hyperbolization conjecture.     

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non-zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.     

A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds

In this paper,we discuss a Kazdan-Warner typed equation on certain non-compact Rie- mannian manifolds.As an application,we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundles over certain non-compact K(?)hler manifolds.     

Inverting the local geodesic X-ray transform on tensors

We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n ≥ 3. We also present an inversion formula. Under the condition that the manifold can be foliated with a continuous family of strictly convex surfaces, we prove a global result which also implies a lens rigidity result near such a metric. The class of manifolds satisfying the foliation condition includes manifolds with no focal points, and does not exclude existence of conjugate points.     

Vrănceanu connections and foliations with bundle-like metrics

We show that the Vrănceanu connection which was initially introduced on non-holonomic manifolds can be used to study the geometry of foliated manifolds. We prove that a foliation is totally geodesic with bundle-like metric if and only if this connection is a metric one. We introduce the notion of a foliated Riemannian manifold of constant transversal Vrănceanu curvature and the notion of a transversal Einstein foliated Riemannian manifold. The geometry of these two classes of manifolds is studied and the relationship between them is determined.     

Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications

In this paper, maximal element theorem on Hadamard manifolds is established. First, we prove the existence of solutions for maximal element theorem on Hadamard manifolds. Further, we prove that most of problems in maximal element theorem on Hadamard manifolds (in the sense of Baire category) are essential and that, for any problem in maximal element theorem on Hadamard manifolds, there exists at least one essential component of its solution set. As applications, we study existence and stability of solutions for variational relation problems on Hadamard manifolds, and existence and stability of weakly Pareto-Nash equilibrium points for n-person multi-objective games on Hadamard manifolds.     

A note on doubly warped product contact C R-submanifolds in trans-Sasakian manifolds

Warped product C R-submanifolds in Kählerian manifolds were intensively studied only since 2001 after the impulse given by B. Y. Chen in [3], [4]. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kählerian geometry, was developed, namely warped product contact C R-submanifolds in Sasakian manifolds (cf. [7], [8]). In this note we prove that there exists no proper doubly warped product contact C R-submanifolds in trans-Sasakian manifolds.     

环流形上的极值度量——存在性和K-稳定性

本文主要研究环流形上的极值度量的存在性和K-稳定性.本文将Donaldson关于环流形上有关常数量曲率度量的稳定性概念的约化推广到一般的极值度量的情形.通过这个约化,本文证明环流形上极值度量的存在性可以推出流形对于环形变的相对K-稳定性.在不知道是否存在极值度量的情形下,本文还给出环流形相对K-稳定的一个充分性条件.对环曲面的情形,基于Arrezo-Pacard-Singer的工作,本文证明任意一个环曲面上存在含有极值度量的Ka¨hler类,并给出一些环曲面上有不存在极值度量的K¨ahler类的例子.关于一般的环流形上的极值度量的存在性,本文用变分方法研究其弱解,证明在能量泛函逆紧性假设下,存在弱极小化子.     

Four-Dimensional Almost Kähler Einstein and *-Einstein Manifolds

In the framework of studying the integrability of almost Kähler manifolds, we prove that a four-dimensional almost Kähler Einstein and -Einstein manifold is a Kähler manifold. Further, we estimate the *-scalar curvature of a four-dimensional compact almost Kähler Einstein and weakly *-Einstein manifold with negative scalar curvature.     

Hilbert schemes of surfaces are dense in moduli

We prove that the locus of Hilbert schemes of n points on a projective K 3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.     

The dual foliation in open manifolds with nonnegative sectional curvature

We study several properties of the Sharafutdinov dual foliation in open manifolds with nonnegative sectional curvature.

     

Concerning Nikodym-type sets in 3-dimensional curved spaces

We investigate maximal functions involving averages over geodesics in three-dimensional Riemannian manifolds. We first show that one can easily extend the Euclidean results of Bourgain and Wolff if one assumes constant curvature. These results need not hold if this assumption is dropped. Nonetheless, we formulate a generic geometric condition which allows favorable estimates. Curiously, this condition ensures that one is in some sense as far as possible from the constant curvature case. Assuming this, one can prove dimensional estimates for Nikodym-type sets which are essentially optimal. Optimal estimates for the related maximal functions are still open though.

     

Numerical calculation of invariant manifolds for maps

Many processes in the sciences and in engineering are modelled by dynamical systems and—in discretized version—by nonlinear maps. To understand the often complicated dynamical behaviour it is a well established tool to use the concept of invariant manifolds of the system. In this way it is often possible to reduce the dimension of the system considerably. In this paper we propose a new method to calculate numerically invariant manifolds near fixed points of maps. We prove convergence of our procedure and provide an error estimation. Finally, the application of the method is illustrated by examples.     

The geometry of a bi-Lagrangian manifold

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian distributions). We show that many different geometric structures can be attached to these manifolds and we carefully analyze the associated connections. Moreover, we introduce the problem of the intersection of the two leaves, one of each foliation, through a point and show a lot of significative examples.