The volume of a k-dimensional foliation F in a Riemannian manifold Mn is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing the construction by Gluck and Ziller in [4], “singular” foliations by 3-spheres are constructed on round spheres S, as well as a singular foliation by 7-spheres on S, which minimize v...

We prove that the Schrödinger map initial-value problem { ∂ts = s×∆xs on R × [−1, 1]; s(0) = s0 is locally well-posed for small data s0 ∈ H σ0 Q (R ; S), σ0 > (d+ 1)/2, Q ∈ S.

We consider open sets of maps in a manifold M exhibiting non-uniform expanding behaviour in some domain S ⊂ M . Assuming that there is a forward invariant region containing S where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuously in the L1-norm with the map. As a main application we show that the open class of maps introdu...

Let N and P be smooth manifolds of dimensions n and p (n ≥ p ≥ 2) respectively. Let Ω (N, P ) denote an open subspace of J∞(N, P ) which consists of all Boardman submanifolds Σ (N, P ) of symbols J with J ≤ I. An Ω -regular map f : N → P refers to a smooth map having only singularities in Ω (N, P ) and satisfying transversality condition. We will prove what is called the homotopy principle for ...

For a closed orientable surface S, any map f : S→S whose n-th power is homotopic to the identity, is homotopic to a homeomorphism g of S of order n. This famous theorem of Nielsen is known to fail in general for aspherical manifolds. In this paper, for model aspherical manifolds M associated to a finitely extendable set of data, we, however, present a weaker version of Nielsen’s result. We show...

Let S be a closed, orientable surface of genus at least 2. The space TH ×ML, where TH is the “hyperbolic” Teichmüller space of S and ML is the space of measured geodesic laminations on S, is naturally a real symplectic manifold. The space CP of complex projective structures on S is a complex symplectic manifold. A relation between these spaces is provided by Thurston’s grafting map Gr. We prove...

The following definition is an example of defining things by mapping properties, that is, by the way the object relates to other objects, rather than by internal structure. The first proposition, which says that there is at most one such thing, is typical, as is its proof. Let R be a commutative ring with 1. Let S be a set. A free R-moduleM on generators S is an R-module M and a set map i : S →...

In 1966, Michael [11] introduced the concept of compact-covering maps. Since many important kinds of maps are compact-covering, such as closed maps on paracompact spaces, much work has been done to seek the characterizations of metric spaces under various compact-covering maps, for example, compact-covering (open) s-maps, pseudosequence-covering (quotient) s-maps, sequence-covering (quotient) s...

Map display is basic function and spatial reference for vehicle navigation system. Based on object-oriented programming (OOP) and C++ computer language, the paper provides and practices an approach of parsing main map data of Kiwi format and translating them into MapInfo’s MIF format, which is open and used widely for spatial data sharing. Finally, by a case study, it extracts necessary road at...