Horospherical geomery in hyperbolic space Izumiya S. (Japan) Hokkaido University izumiya@math.sci.hokudai.ac.jp Recently we discovered a new geometry on submanifolds in hyperbolic space[2–8] which is called the horospherical geometry. In this talk we explain the outline of this geometry. In the previous theory of surfaces in hyperbolic space, there appeared two kinds of curvatures. One is called the extrinsic Gauss curvature Ke and another is the intrinsic Gauss curvature KI. The intrinsic Gauss curvature is nothing but the sectional curvature defined by the induced Riemannian metric on the surface. The relation between these curvatures is known that Ke = KI + 1. Of course the Gauss-Bonnet type theorem holds for the intrinsic Gauss curvature by the Chern-Weil theory. In [2] we defined a curvature Kh called a hyperbolic curvature of hypersurfaces by using the hyperbolic Gauss indicatrix. For surfaces in hyperbolic 3-space, we have the relation Kh = 2 - 2H + KI, where H is the mean curvature of the surface. Therefore Kh is an extrinsic hyperbolic invariant. In [4] we have modified the hyperbolic Gauss curvature into the horospherical Gauss curvature Kh and shown that the Gauss-Bonnet type theorem holds. This curvature is not a hyperbolic invariant but it is invariant under the canonical action of SO(n). However, the total curvature is a topological invariant. By definition, Kh(p) = if and only if Kh(p) = 0. Therefore the horospherical flatness is a hyperbolic invariant. Totally umbilical and horospherical flat hypersurfaces are hyperhorospheres. We call the geometry related to this curvature the horospherical geometry. For a general submanifolds in hyperbolic n-space, we have defined the horospherical Lipschitz-Killing curvature on the unit normal bundle and shown that the Chern-Lashof type theorem[7]. As corllaries, we have the Fenchel type theorem and the Milnor-Fary type theorem on space curves.

For surfaces in 3-dimensional hyperbolic space, there is an important class of surfaces called linear Weingarten surfaces which satisfy the relation aKI + b(2H - 2) = 0. In [1] the Weierstrass-Bryant type representation formula for such surfaces with a + b = 0 (called, the Bryant type) has been shown. This class of surfaces contains inrinsic flat surfaces (i.e., a = 0, b = 0) and CMC-1 surfaces (a = 0, b = 0). However, the horospher ical flat surface is the exceptional case (the non-Bryant type : a + b = 0).

Therefore the horospherical flat surfaces are also importnat subjects in hyperbolic geometry. We will describe the geometry of horospherical flat surfaces and singularities. As consequences, some important examples of non-singular complete surfaces with constant principal curvatures appear as horo-flat surfaces. Moreover, the cuspidal beaks appear as a generic singularities of singular horo-flat surfaces. We remark that the cuspidal beaks is a non-generic Legendrian singular point.

Finally, we will try to describe geometric information on the asymptotic (or, horo-asymptotic) directions of surfaces in the hyperbolic 3-space as an application of singularity theory to various kinds of projections [7].

References [1] Glvez J. A., Martnez A. and Miln F. Complete linear Weingarten surfaces of Bryant type. A plateau problem at infinity. // Trans.

A.M. S. –2004. 356. p. 3405 – 3428.

[2] Izumiya S., Pei D-H. and Sano T. Singularities of hyperbolic Gauss maps. // Proc. London Math. Soc. –2003. 86. p. 485 – 512.

[3] Izumiya S., Pei D-H., Romero Fuster M. C. and Takahashi M. The horospherical geometry of submanifolds in hyperbolic space. // J.

London Math. Soc. (2) –2005. 71. p. 779 – 800.

[4] Izumiya S. and Romero Fuster M. C. The horospherical GaussBonnet type theorem in hyperbolic space. // J. Math. Soc. Japan –2006. 58. p. 965 – 984.

[5] Izumiya S., Pei D-H. and Romero Fuster M. C. The horospherical geometry of surfaces in Hyperbolic 4-space. // Israel J. Math. – 2006. 154. p. 361 – 379.

[6] Izumiya S., Saji K. and Takahashi, M. Horospherical flat surfaces in hyperbolic 3-space. // preprint –2007.

[7] Izumiya S. and Tari F. Projections of surfaces in the hyperbolic space to hyperhorospheres and hyperplanes, preprint –2007.

[8] Buosi M., Izumiya S. and Soares Ruas M. A. Bounds for total absolute horospherical curvature of submanifolds in hyperbolic space.

in preparation.

Invariant Mbius measure and Gauss–Kuzmin face distribution Karpenkov O. N. (Netherlands) Mathematisch Instituut, Universiteit Leiden karpenk@mccme.ru Introduction. A famous statement on distribution of integers in of ordinary continued fractions was originally formulated by K. F. Gauss in 1800 and further proved by R. O. Kuzmin in 1928. One year later the result was proved one more time by P. Lvy. In 1989 V. I. Arnold generalized statistical problems to the case of one-dimensional and multidimensional continued fractions in the sense of Klein, see for instance in [1] and [2].

The one-dimensional case was studied in details by M. O. Avdeeva and V. A. Bykovskii a few years ago.

There exists a unique up to multiplication by a constant form of the highest dimension on the manifold of n-dimensional continued fractions, such that the form is invariant under the natural action of the group P GL(n+1). A measure corresponding to the integral of such form is called a Mbius measure. In the present talk we show an explicit formula to calculate invariant forms in special coordinates. In the one-dimensional case the Mbius measure is induced by the relativistic measure of threedimensional de Sitter world. The author is grateful to V. I. Arnold for constant attention to this work.

Definitions. Consider an n-dimensional real vector space with lattice of integer points in it and a collection of n straight lines in the space passing throw the origin in general position. Take any positive cone of any n vectors, lying on the distinct chosen lines (by positive cone we mean the set of all linear combinations of the vectors with real positive coefficients).

The boundary of the convex hull of all integer points contained inside the cone is called the sail of the cone. The set of all sails for such cones is called the (n-1)-dimensional continued fraction in the sense of Klein.

In this talk we study frequencies of faces of multidimensional continued fractions. Denote the sets of all ordered collections of n+1 independent straight lines by F CFn. We say that F CFn is a space of n-dimensional framed continued fractions.

Explicit formula for the Mbius form. Consider an n + 1-dimensional real vector space with the standard metrics on it. Let be an arbitrary hyperplane of the space with chosen Euclidean coordinates OX1... Xn, let also does not pass through the origin. By the chart F CFn, of F CFn we denote the set of all collections of n+ordered straight lines such that any of them intersects. Let the intersection of with i-th plane is a point with coordinates (x1,i,..., xn,i) at the plane. For an arbitrary tetrahedron A1... An+1 in the plane we denote by V(A1,..., An+1) its oriented Euclidean volume in the coordinates OX1,1... Xn,1X1,2... Xn,n+1 of the chart F CFn,.

Denote by |v| the Euclidean length of the vector v in the coordinates OX1,1... Xn,1X1,2... Xn,n+1 of the chart F CFn,. Note that the map F CFn, is everywhere dense in F CFn.

Proposition. The restriction of any Mbius measure to F CFn, is proportional to the measure defined by the integration of the form n+1 n dxj,i i=1 j=.

V(A1,..., An+1)n+Table. Some results of calculations of relative frequencies N face lS µ2 N face lS µI 3 1.3990 · 10-2 VI 7 3.1558 · 10-II 5 1.5001 · 10-3 VII 11 3.4440 · 10-III 7 3.0782 · 10-4 VIII 7 5.6828 · 10-IV 9 9.4173 · 10-5 IX 7 1.1865 · 10-V 11 3.6391 · 10-5 X 6 9.9275 · 10-In Table we show the results of relative frequencies calculations for integer-linear types of faces (on unit integer distance to the origin). In a column “face” we draw a picture of a face type; in a column “lS” we write integer areas of the faces; in a column “µ2” we show the approximate relative frequency for the corresponding face type.

References [1] V. I. Arnold, Preface, Pseudoperiodic topology, Amer. Math. Soc.

Transl., v. 197(2), (1999), pp. ix–xii.

[2] V. I. Arnold, Higher Dimensional Continued Fractions, Regular and Chaotic Dynamics, v. 3(1998), n. 3, pp. 10–17.

KP hierarchiy for Hodge integrals Kazarian M. (Russia) Steklov Mathematical Institute, Moscow kazarian@mccme.ru Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of such known results on Hodge integrals as Witten conjecture, Virasoro constrains, Faber’s lg-conjecture etc. Among other results we show that the properly arranged generating function for Hodge integrals satisfies equations of the KP hierarchy.

Parshin’s symbols and logarithmic functional Khovanskii A. (Canada, Russia) Department of Mathematics, University of Toronto Independent University of Moscow Institute for System Studies askold@math.toronto.edu Ten years ago at a conference in Toronto dedicated to V.I. Arnold’s sixth anniversary I presented an explicit formula for product in the group (C)n of roots of a system of algebraic equations with general enough set of Newton polyhedra [1]. This formula (with is multi-dimensional generalization of Vieta formula) uses Parshin’s symbols. Its proof however is based on simple geometry and combinatorics and does not use Parshin’s reciprocity laws related to his symbols. This proof convinced me that over the complex numbers there should be an intuitive geometric explanation of Parshin’s symbols and reciprocity laws. I will discuss such an explanation based on a logarithmic functional, whose argument is an (n - 1)dimensional cycle in the group (C)n. It generalizes the usual logarithm (which can be considered as the zero-dimensional logarithmic functional) and inherits its main properties.

References [1] Khovanskii A. Newton polyhedrons, a new formula for mixed volume, product of roots of a system of equations. In Proceed. of a Conf. in Honor of V. I. Arnold, Fields Inst. Comm. vol. 24, Amer.

Math. Soc., pages 325–364, USA, 1999.

[2] Parshin A. N. Local class field theory, Trudy Mat. Inst. Steklov, volume 165. 1984.

[3] Parshin A. N. Galois cohomology and Brauer group of local fields, Trudy Mat. Inst. Steklov, volume 183. 1984.

One-round dynamics near a homoclinic orbit to a reversible saddle-center Koltsova O. Yu. (Russia) Nizhny Novgorod University koltsova@uic.nnov.ru We studied some elements of global behavior of reversible and Hamiltonian dynamical systems with a homoclinic orbit to a saddle-center.

We considered a reversible vector field with a symmetric homoclinic orbit to a singular point p of the saddle-center type. Under some generic condition we proved the existence of a two-dimensional manifold filled by symmetric homoclinic orbits to the center manifold. We also established the existence of a countable set of two-dimensional manifolds accumulating to. These manifolds consist of one parameter families of symmetric periodic orbits.

If we consider a Hamiltonian reversible vector field then all obtained manifolds are foliated by Hamiltonian level sets such that:

- There exists a countable set of symmetric periodic orbits on the level where p is located. These orbits are accumulated to.

- There exist two symmetric homoclinic orbits to each periodic orbit on c W and a countable set of symmetric periodic orbits accumulated to these homoclinic ones.

- There are a finite number of periodic orbits on all other levels of Hamiltonian.

Comparison of the results for reversible systems with those obtained in the Hamiltonian category has already led to a number of observations of differences, most notably the occurrence of non-symmetric heteroclinic cycles.

This research was supported by Russian Foundation of Basic Research (grant 07-01-00715a), program of supporting Russian scientific schools (grant 9686.2006.1) and Royal Society.

Three theorems on perturbed KdV equation on a circle Kuksin S. B. (UK) Department of Mathematics, Heriot-Watt University kuksin@ma.hw.ac.uk I will discuss a KAM-theory for perturbed KdV equation, an averaging theory for its Hamiltonian perturbation and an averaging theory for random perturbation of the equation.

The fundamental groups of thecomplements of affine pseudoholomorphic curves in CPKulikov V. S. (Russia) Steklov Mathematical Institute, RAS kulikov@mi.ras.ru Let H CP2 be a pseudo-holomorphic curve with respect to some -tamed almost complex structureon CP2, where is symplectic Fubini– Studiform on CP2, and L be apseudo-holomorphic line in general position with respect to H. We call H = H (CP2 \ L) an “affine” pseudoholomorphic curve. In the talk, I give a complete description of the set of fundamentalgroups of the complements of “affine” pseudoholomorphic curves in CP2 \ L in terms of theirrepresentations.

Periodic with multivariate time solutions of system of the quasilinear differential equations in partial derivative Kulzhumiyeva A. A. (Kazakhstan) Aktobe State University by Zhubanov aiman-80@mail.ru Sartabanov Zh. A. (Kazakhstan) Aktobe State University by Zhubanov aiman-80@mail.ru In report is considered system of the quasilinear differential equations of the type m x x Dax + aj(t) = P (, t)x + f(, t, x), (1) tj j=where (, t) = (, t1,..., tm) – multivariate time, a(t) = (a1(t),..., am(t)) – vector-function.

We shall consider that for vector-function a(t), n n-matrix P (, t) and n-vector-function f(, t, x) meet the condition of periodicity and of smoothness of the type (1) a(t + k) = a(t) Ct (Rm), k Zm, (2) (0,1) P ( +, t + k) = P (, t) C,t (R Rm), k Zm, (3) (0,1,1) f( +, t + k, x) = f(, t, x) C,t,x (R Rm Rn), k Zm, (4) where k = (k11,..., kmm) – multiple periods, moreover = 0, 1,..., m – rationally incommensurable constants.

We research problem about existence (,, )-periodic solutions x(, t, ) of the system (1) at condition (2), (3) and (4), where = (-, t) – characteristics of the operator Da: Da = 0, (0, t) = t.

For solution of the delivered problem we shall expect: 1) uniform system, corresponding to system (1), has not (,, )-periodic solutions, except trivial; 2) solutions of the system (1) satisfy initial condition of the (1) type x|=0 = u(t) U, where U = u(t) | u(t + k) = u(t) Ct (Rm), k Zm, | · | – sign of Evklid’s metrics.

On the strength of [1 - -3] delivered problem to equivalent problem about existence of the periodic solution of the integral equation (+,t) x(, t, x) = [X-1( +, t, ) - X-1(, t, )]-(,t) X-1(h0, h, )f(h0, h,, x(h0, h, )) ds. (5) Here we integrate along characteristics h0 = s, h = (s-0, t0) of operator Da.

We shall notice that solution (5) of the system (1) depends not only from multivariate time, but also from = (-, t) – characteristics of the operator Da. Account to such dependencies of the solution have principle importance in study quasiperiodic solutions of the common differential equations, which possible get under t = (, 0).

In space unceasing, evenly limited, (,, )-periodic n-vector-function x(, t, ) with rate x = sup |x(, t, )| shall define operator ( +,t) (T x)(, t, x) = [X-1( +, t, ) - X-1(, t, )]-(,t) X-1(h0, h, )f(h0, h,, x(h0, h, )) ds.

Hereinafter we install that operator T has a unique still point in this space and it possesses the property of smoothness.

Theorem. If are executed condition (2), (3), (4) and uniform system, corresponding to system (1) has only trivial periodic solution, that system (1) allows unique (,, )-periodic solution.

It is discussed event when period depends on.

References [1] Kulzhumiyeva A.A., Sartabanov Zh.A. // Scientific papers international conference “Differential equations and computer algebra systems”. Brest, October 5-8, 2005. – Part 1. – P. 163–165. (in Russian) [2] Kulzhumiyeva A.A., Sartabanov Zh.A. // Izdenis – Poisk. Series natural and technical sciences. – 2005. No2. – P. 194–200. (in Russian) [3] Kulzhumiyeva A.A., Sartabanov Zh.A. // International Scientific Conference “Mathematical analysis, differential equations and their applications”. Uzhgorod, September 18-23, 2006. – P. 159–160.

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