The intersection of complete geodesics on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> with positive curvature

Let M be a Riemannian manifold. A complete geodesic on M means that :(-,+)M is a normalized geodesic. In this paper, we prove that on (S²,g) with positive curvature, any two complete geodesics must intersect an infinite number of times, and a complete geodesic must self-intersect an infinite number of times. Mathematics Subject Classification (2000) 53C40 (53C22)     

Holomorphic Torus Actions on Compact Locally Conformal Kähler Manifolds

Given a torus action (T ², M) on a smooth manifold, the orbit map ev _x(t)=t·xfor each xMinduces a homomorphism ev _*: ²H ₁(M;). The action is said to be Rank-kif the image of ev _*has rank k(2) for each point of M. In particular, if ev _*is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kähler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kähler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kähler if a lift of the metric to the universal covering space is conformal to a Kähler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kähler manifold Mis Rank-1 provided that Mhas no Kähler structure.     

A global pinching theorem for compact surfaces inS 3 with constant mean curvature

LetM be a compact minimal surface inS ³. Y. J. Hsu^[5] proved that if S₂22, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·₂ denotes theL ²-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS ³ with constant mean curvature.Supported by NSFH.     

Contact metric manifolds

In this paper we study contact metric manifoldsM ²ⁿ⁺¹(, , ,g) with characteristic vector field belonging to thek-nullity distribution. Moreover we prove that there exist i) nonK-contact, contact metric manifolds of dimension greater than 3 with Ricci operator commuting with and ii) 3-dimensional contact metric manifolds with non-zero constant -sectional curvature.     

The structure of an infinite,complete, convex hypersurface inE 4 with bounded mean curvature

The following theorem is proved. THEOREM. If on an infinite, complete, convex hypersurface F in E⁴ the mean curvature is 1 – H 1, where 0 10^–11, then F is a cylindrical hypersurface.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 8–9, 1991.     

A metric space of directions in Minkowski space

A new angular measure in a d-dimensional Minkowski space M was introduced recently. It determines the lengths of rectifiable curves in the (d - 1)-dimensional topological sphere S of all directions in M. Thus, a length structure appears on S. This results in the appropriate intrinsic metric in S. The paper deals with some properties of the length structure and the resulting metric space S. In particular, it shows that diam S 2 .     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.     

Functional central limit theorems for a large network in which customers join the shortest of several queues

We consider N single server infinite buffer queues with service rate . Customers arrive at rate N, choose L queues uniformly, and join the shortest. We study the processes for large N, where R^N_t(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data R^N₀ satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the R^N in equilibrium under the stability condition <, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits lim _N lim _t= lim _t lim _N by a compactness-uniqueness method. We deduce a posteriori the CLT for R^N₀ under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit.Mathematics Subject Classification (2000):Primary: 60K35; Secondary: 60K25, 60B12, 60F05, 37C75, 37A30     

An Integral Geometry Problem in a Nonconvex Domain

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary M is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if M is convex and there is no geodesic of infinite length in M. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of M we assume that M is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for     

Absolut stetige Maße auf lokalkompakten Halbgruppen

LetS be a locally compact semigroup. It is shown that if a measure is absolutely continuous and ifS is cancellative, then the measure concentrated on a Borel subsetB ofS (i. e. =(B.)) is also absolutely continuous. Other properties of absolutely continuous measures will be obtained. Moreover we will answer the question when absolutely continuous probability measures exist. This is the case ifS admits an invariant integral on the space of all continuous functions onS with compact support. Another result is the following: If the compact semigroupS has a connected kernel then there exist absolutely continuous probability measures if and only ifS is amenable.     

Rigidity of compact minimal submanifolds in a unit sphere

LetM be ann-dimensional compact minimal submanifold of a unit sphereS ^n+p (p2); and letS be a square of the length of the second fundamental form. IfS2/3n everywhere onM, thenM must be totally geodesic or a Veronese surface.     

Some Criteria for Multivalently Starlikeness

Let S_n(p)(p, n N) be the class of functions f() = ^p + a_p+n^p+n + which are p-valently starlike in the unit disk. Some sufficient conditions for a function f() to be in the class S_n(p) are given.AMS Subject Classification (2000): primary 30C45     

Intermediate Asymptotics for Inhomogeneous Nonlinear Heat Conduction

We consider the initial-value problem for a quasilinear heat-conduction or diffusion equation with variable density decreasing at infinity. We show that the asymptotic behavior of the given process is self-similar. Indeed, as t the solution of the problem approaches a self-similar solution of a certain singular limit equation. The limit solution has compact support for any t > 0 and cusp-type shape at the space origin.     

On the Space of Weakly Almost Periodic Functionals and Its Amenability

Let S be a locally compact semitopological semigroup with measure algebra M(S), M₀(S) the set of all probability measures in M(S) and WF(S) the space of weakly almost periodic functionals on M(S)^*. Assuming that M₀(S) has the semiright invariant isometry property, it is shown that WF(S) has a topological left invariant mean (TLIM) whenever the center of M₀(S) is nonempty; in particular if either the center of S is nonempty or S has a left identity, then WF(S) has a TLIM. Finally if, for each M₀(S), the mapping v v * of M₀(S) into itself is surjective and the center of M₀(S) is nonempty, then WF(S) has a TLIM. We also generalize some results from discrete case to topological one.AMS Subject Classification (1991): 43A07     

Proof of two conjectures of H. Brezis and L.A. Peletier

In this paper, we consider the problem: –u=N(N–2)u ^p– , u>0 on ; u=0 on , where is a smooth and bounded domain inR ^N, N3, p= , and >0. We prove a conjecture of H. Brezis and L.A. Peletier about the asymptotic behaviour of solutions of this problem which are minimizing for the Sobolev inequality as goes to zero. We give similar results concerning the related problem: –u=N(N–2)u^p+u, u>0 on ; u=0 on , for N is larger than 4.     

Homogeneous Submanifolds of Codimension Two

We study codimension 2 homogeneous submanifolds of Euclidean space for which the index of minimum relative nullity is small. We prove that if min_xMf(x)n-5, where (x) denotes the nullity of the second fundamental form of the immersion f at the point x, then the manifold M ⁿ is either isometric to a sphere or to a product of two spheres S²×S ^n–2 or covered by the Riemannian product S ^n–1 ×R. As a consequence, we obtain a classification of compact codimension 2 homogeneous submanifolds of dimension at least 5.     

On spectral variations under bounded real matrix perturbations

Summary In this paper we investigate the set of eigenvalues of a perturbed matrix {ie509-1} whereA is given and ^{n × n}, ||< is arbitrary. We determine a lower bound for thisspectral value set which is exact for normal matricesA with well separated eigenvalues. We also investigate the behaviour of the spectral value set under similarity transformations. The results are then applied tostability radii which measure the distance of a matrixA from the set of matrices having at least one eigenvalue in a given closed instability domain _b.     

Maximal Homomorphic Group Image and Convergence of Convolution Sequences on a Semigroup

Let be a probability measure generating a locally compact semigroup S. If the convolution sequence ⁿ is tight, in particular if S is compact, S admits a closed minimal ideal K. The convergence of ⁿ is characterized in terms of convergence of a homomorphic image (~) ⁿ on a factor group of the compact group G in the Rees–Suschkewitsch decomposition of K.     

On the relation between Fourier and Leont’ev coefficients with respect to smirnov spaces

Yu. Melnik showed that the Leontev coefficients _f () in the Dirichlet series 2}}$$ " align="middle" border="0"> of a function f E ^p (D), 1 < p < , are the Fourier coefficients of some function F L ^p , ([0, 2]) and that the first modulus of continuity of F can be estimated by the first moduli and majorants in f. In the present paper, we extend his results to moduli of arbitrary order.Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 517–526, April, 2004.     

Disconnectedness of sublevel sets of some Riemannian functionals

LetM be a compact manifold of dimension greater than four. Denote byRiem(M) the space of Riemannian structures onM (i.e. of isometry classes of Riemannian metrics onM) endowed with the Gromov-Hausdorff metric. LetRiem (M) Riem(M) be its subset formed by all Riemannian structures such that vol()=1 andinj() , whereinj() denotes the injectivity radius of.We prove that for all sufficiently small positive the spaceRiem (M) is disconnected. Moreover, if is sufficiently small, thenRiem (M) is representable as the union of two non-empty subsetsA andB such that the Gromov-Hausdorff distance between any element ofA and any element ofB is greater than/9. We also prove a more general result with the following informal meaning: There exist two Riemannian structures of volume one and arbitrarily small injectivity radius onM such that any continuous path (and even any sequence of sufficiently small jumps) in the space of Riemannian structures of volume one onM connecting these Riemannian structures must pass through Riemannian structures of injectivity radius uncontrollably smaller than the injectivity radii of these two Riemannian structures.These results can be generalized for at least some four-dimensional manifolds. The technique used in this paper can also be used to prove the disconnectedness of many other subsets of the space of Riemannian structures onM formed by imposing various constraints on curvatures, volume, diameter, etc.This work was partially supported by the New York University Research Challenge Fund grant, by NSF grant DMS 9114456 and by the NSERC operating grant OGP0155879.