Singular riemannian foliations with sections,transnormal maps and basic forms

A singular riemannian foliation on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold Σ that meets every leaf of orthogonally and whose dimension is the codimension of the regular leaves of . We prove that the algebra of basic forms of M relative to is isomorphic to the algebra of those differential forms on Σ that are invariant under the generalized Weyl pseudogroup of Σ. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos. Marcos M. Alexandrino and Claudio Gorodski have been partially supported by FAPESP and CNPq.     

A Bernstein type theorem for minimal volume preserving maps

We show that any minimal volume preserving map from the Euclidean plane into itself is a linear diffeomorphism. We derive this from a similar result on minimal diffeomorphisms. We also show that the classical Bernstein theorem on minimal graphs is a corollary of our result.

     

Extension and convergence theorems for families of normal maps in several complex variables

Let represent the family of holomorphic (continuous) maps from a complex (topological) space to a complex (topological) space , and let be the Alexandroff one-point compactification of if is not compact, if is compact. We say that is uniformly normal if , is relatively compact in (with the compact-open topology) for each complex manifold . We show that normal maps as defined and studied by authors in various settings are, as singleton sets, uniformly normal families, and prove extension and convergence theorems for uniformly normal families. These theorems include (1) extension theorems of big Picard type for such families - defined on complex manifolds having divisors with normal crossings - which encompass results of Järvi, Kiernan, Kobayashi, and Kwack as special cases, and (2) generalizations to such families of an extension-convergence theorem due to Noguchi.

     

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

Let be a nonempty closed convex subset of a real Banach space and be a Lipschitz pseudocontractive self-map of with . An iterative sequence is constructed for which as . If, in addition, is assumed to be bounded, this conclusion still holds without the requirement that Moreover, if, in addition, has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then the sequence converges strongly to a fixed point of . Our iteration method is of independent interest.

     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

Fixed point and rigidity theorems for harmonic maps into NPC spaces

We show that harmonic maps from 2-dimensional Euclidean polyhedra to arbitrary NPC spaces are totally geodesic or constant depending on a geometric and combinatorial condition of the links of the 0-dimensional skeleton. Our method is based on a monotonicity formula rather than a codimension estimate of the singular set as developed by Gromov–Schoen or the mollification technique of Korevaar–Schoen.      

Twisted longitudinal index theorem for foliations and wrong way functoriality

For a Lie groupoid G with a twisting σ (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid.For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of the Connes–Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai, Melrose, and Singer.We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map f:W→M/F and a twisting σ on the holonomy groupoid M/F, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes and Skandalis when the twisting is trivial and of Carey and Wang for manifolds.     

A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.     

A topological invariant for pairs of maps

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝⁿ, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.     

Martingale approach to limit theorems for jump processes

We consider the weak convergence of laws of càdiàg processes determined by a sequence of operators with singularly perturbed terms. We study the problem in the martingale approach, which was formulated to establish weak limit theorems for continuous processes by Papanicolaou, Stroock and Varadhan. However, in this paper, limit processes are not necessarily continuous but càdiàg. In particular, we consider a homogenization problem of càdiàg processes in the framework of martingale problem.     

A numerical approach to some basic theorems in singularity theory

In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of singularity theory: the inverse, implicit and rank theorems for Lipschitz mappings, the splitting lemma and the Morse lemma, the density and openness of Morse functions. We expect that the results will make singularities more applicable and will be useful for numerical analysis and some fields of computing.     

Berry–Esseen theorem and local limit theorem for non uniformly expanding maps

In Young towers with sufficiently small tails, the Birkhoff sums of Hölder continuous functions satisfy a central limit theorem with speed , and a local limit theorem. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.     

A Bayesian approach to the estimation of maps between Riemannian manifolds

Let Θ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E ^s, and let γ be a smooth map of Θ into a Riemannian manifold Λ. An unknown state θ ∈ Θ is observed via X = θ + εξ, where ε > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on Θ and smooth estimators g(X) of the map γ we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θ and Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γ is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.      

Morse指数与含Grushin算子的Liouville型定理

本文研究退化椭圆型方程-Δxu-(α+1)2|x|~(2α)Δyu=|u|~(p-1)u,(x,y)∈Rm×Rk和方程-Δxu-(α+1)2|x|~(2α)Δyu=|u|~(p-1)u,(x,y)∈Π的Liouville型定理,其中-Δx-(α+1)2|x|~(2α)Δy是Grushin算子,Π={(x,y)∈Rm×Rk:x10}或{(x,y)∈Rm×Rk:y10}.本文将证明,当1p(Q+2)/(Q-2)时,上述方程Morse指数有限的有界解只有零解,其中Q=m+(α+1)k为齐次空间的维数,因此,本文将Laplace方程的结果推广到含Grushin算子的方程.     

A fixed point index theory for nowhere normal-outward compact maps and applications

A fixed point index theory is developed for a class of nowhere normal-outward compact maps defined on a cone which do not necessarily take values in the cone. This class depends on the retractions on the cone and contains self-maps for any retractions, and weakly inward maps and generalized inward maps when the retraction is a continuous metric projection. The new index coincides with the previous fixed point index theories for compact self-maps and generalized inward compact maps. New fixed point theorems are obtained for nowhere normal-outward compact maps and applied to treat some abstract boundary value problems and Sturm-Liouville boundary value problems with nonlinearities changing signs.     

Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, Lp or ?p spaces, 1<p<∞), and K a nonempty closed convex (not necessarily bounded) subset of E. Let be a k-strictly asymptotically pseudocontractive map with a nonempty fixed-point set. It is proved that (I−T) is demiclosed at 0. Furthermore, weak and strong convergence of an averaging iteration method to a fixed point of T are proved.