Convex hulls of indicatrices and singularities of the transitivity zone in {{\mathbb{R}}^3}

We consider a control system on two- and three-dimensional manifolds, the indicatrices of which are the images of smooth mappings, and present a survey on classification of the generic singularities of the boundary of the local transitivity set.     

p-Harmonic obstacle problems

Summary We develop an interior partial regularity theory for vector valued Sobolev functions which locally minimize degenerate variational integrals under the additional side condition that all comparison maps take their values in the closure of a smooth region of the target space. Our results apply to the case of penergy minimizing mappings X Y between Riemannian manifolds including target manifolds Y with nonvoid boundary.     

Relatively unstable invariant sets of nonlinear operators

The note contains accurate proofs of Lipschitz versions of a collection of results from the theory of invariant manifolds for nonlinear mappings which are well known in the case of smooth mappings. Bibliography: 1 title. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 131–137.     

On global singularities of Sobolev mappings

We define various invariants for Sobolev mappings defined between manifolds which are stable under perturbation with respect to the strong Sobolev topology. We show that these invariants classify various types of ``global singularities" of Sobolev maps. These invariants are used to give a simple characterization of the strong closure of the set of smooth maps in the Sobolev space.     

Generalized quasi-isometries on smooth Riemannian manifolds

Mathematical Notes - The boundary behavior of finitely bi-Lipschitz mappings on smooth Riemannian manifolds is studied.     

The homological singularities of maps in trace spaces between manifolds

We deal with mappings defined between Riemannian manifolds that belong to a trace space of Sobolev functions. The homological singularities of any such map are represented by a current defined in terms of the boundary of its graph. Under suitable topological assumptions on the domain and target manifolds, we show that the non triviality of the singular current is the only obstruction to the strong density of smooth maps. Moreover, we obtain an upper bound for the minimal integral connection of the singular current that depends on the fractional norm of the mapping.     

A Coarea Formula for Smooth Contact Mappings of Carnot–Carathéodory Spaces

We prove the coarea formula for sufficiently smooth contact mappings of Carnot manifolds to Carnot–Carathéodory spaces. In particular, we investigate level surfaces of these mappings, and compare Riemannian and sub-Riemannian measures on them. Our main tool is the sharp asymptotic behavior of the Riemannian measure of the intersection of a tangent plane to a level surface and a sub-Riemannian ball. This calculation in particular implies that the sub-Riemannian measure of the set of characteristic points (i.e., the points at which the sub-Riemannian differential is degenerate) equals zero on almost every level set.     

Conformal mappings onto Einstein spaces

We study conformal mappings of Riemannian manifolds onto Einstein manifolds under minimal condition on the differentiability class of manifolds in question. We establish under what conditions the linear equations obtained by J. Mike?, M. L. Gavril’chenko and E. I. Gladyscheva that define such mappings.     

Slow mappings of finite distortion

We examine mappings of finite distortion from Euclidean spaces into Riemannian manifolds. We use integral type isoperimetric inequalities to obtain Liouville type growth results under mild assumptions on the distortion of the mappings and the geometry of the manifolds.     

Holomorphic Isometry from a Kähler Manifold into a Product of Complex Projective Manifolds

We study the global property of local holomorphic isometric mappings from a class of Kähler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative.     

Mappings of Bounded Mean Distortion and Cohomology

We obtain a quantitative cohomological boundedness theorem for closed manifolds receiving entire mappings of bounded mean distortion and finite lower order. We also prove an equidistribution theorem for mappings of finite distortion.     

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.     

Inverse and implicit function theorems on carnot manifolds

We obtain some analogs of the Euclidean inverse and implicit function theorems for continuously hc-differentiable mappings on Carnot manifolds.     

A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric

We first prove stochastic representation formulae for space–time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Space–time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations.     

Properties of Minimal Surfaces over Depth 2 Carnot Manifolds

Siberian Mathematical Journal - We derive necessary and sufficient conditions for the minimality of the graph surfaces for the classes of contact mappings of depth 2 Carnot manifolds into...     

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ ^N are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂ ^N are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.     

Monotone and Accretive Vector Fields on Riemannian Manifolds

The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities of Lipschitz continuous, strongly monotone mappings. We also establish the equivalence between the strong convexity of functions and the strong monotonicity of its subdifferentials on Riemannian manifolds. These results are then applied to solve the minimization of convex functions on Riemannian manifolds.     

Infinite entropy is generic in Hölder and Sobolev spaces

In 1980, Yano showed that on smooth compact manifolds, for endomorphisms in dimension one or above and homeomorphisms in dimensions greater than one, topological entropy is generically infinite. It had earlier been shown that, for Lipschitz endomorphisms on such spaces, topological entropy is always finite. In this article, we investigate what occurs between $C^0$-regularity and Lipschitz regularity, focussing on two cases: Hölder mappings and Sobolev mappings.     

利用Dirac理论进行Poisson流形及Jacobi流形的约化(英文)

通过将可约的Dirac以及Jacobi-Dirac结构分别分为两种类型,给出对应于Poisson流形和Jacobi流形的约化定理.这些约化定理的证明只需要进行一些直接的计算,而不需要借助于矩映射或者相容函数等复杂概念的引入.另外,给出了一些相应的例子和应用.     

Bump-Functions on Certain Infinite Dimensional Manifolds

Let A be a unitary commutative complete locally m-convex C ^*-algebra. We prove that the projective finitely generated A-modules admit differentiable A-valued bump-functions. Then we consider manifolds modelled on such modules and we prove that locally defined differentiable mappings and sections on these manifolds extend to global ones. This revised version was published online in June 2006 with corrections to the Cover Date.