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26. CJM 2006 (vol 58 pp. 381)

Jakobson, Dmitry; Nadirashvili, Nikolai; Polterovich, Iosif
 Extremal Metric for the First Eigenvalue on a Klein Bottle The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere ${\mathbb S}^4$ by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawson's $\tau_{3,1}$-torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for $\lambda_1$ on a Klein bottle of a given area. We present numerical evidence and prove the first results towards this conjecture. Keywords:Laplacian, eigenvalue, Klein bottleCategories:58J50, 53C42

27. CJM 2005 (vol 57 pp. 1314)

Zhitomirskii, M.
 Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra In 1999 V. Arnol'd introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a $1$-form to a singular submanifold. We prove that a germ of any subset $N$ of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to $N$ of the contact structure. This is a generalization of the Darboux-Givental' theorem for smooth submanifolds of a contact manifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V.~Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold $N$ in terms of the algebraic restriction of the contact structure to $N$. In particular, the algebraic restriction is zero if and only if $N$ is contained in a smooth Legendrian submanifold of $M$. Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curvesCategories:53D10, 14B05, 58K50

28. CJM 2005 (vol 57 pp. 961)

Borwein, Jonathan M.; Wang, Xianfu
 Cone-Monotone Functions: Differentiability and Continuity We provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone $K$ with non-empty interior. We also show that the set of nowhere $K$-monotone functions has a $\sigma$-porous complement in the space of continuous functions endowed with the uniform metric. Keywords:Cone-monotone functions, Aronszajn null set, directionally porous, sets, GÃ¢teaux differentiability, separable spaceCategories:26B05, 58C20

29. CJM 2005 (vol 57 pp. 771)

Schrohe, E.; Seiler, J.
 The Resolvent of Closed Extensions of Cone Differential Operators We study closed extensions $\underline A$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted $L_p$-space. Under suitable conditions we show that the resolvent $(\lambda-\underline A)^{-1}$ exists in a sector of the complex plane and decays like $1/|\lambda|$ as $|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underline A$. As an application we treat the Laplace--Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$. Keywords:Manifolds with conical singularities, resolvent, maximal regularityCategories:35J70, 47A10, 58J40

30. CJM 2005 (vol 57 pp. 871)

Zhang, Xi
 Hermitian Yang-_Mills--Higgs Metrics on\\Complete KÃ¤hler Manifolds In this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact K\"ahler manifolds. Keywords:vortex equation, Hermitian Yang--Mills--Higgs metric,, holomorphic vector bundle, KÃ¤hler manifoldsCategories:58E15, 53C07

31. CJM 2005 (vol 57 pp. 506)

Gross, Leonard; Grothaus, Martin
 Reverse Hypercontractivity for Subharmonic Functions Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, $e^{-tA}$, can be bounded {\it below} from $L^p$ to $L^q$ when $p,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions. Keywords:Reverse hypercontractivity, subharmonicCategories:58J35, 47D03, 47D07, 32Q99, 60J35

32. CJM 2005 (vol 57 pp. 251)

Cocos, M.
 Some New Results on $L^2$ Cohomology of Negatively Curved Riemannian Manifolds The present paper is concerned with the study of the $L^2$ cohomology spaces of negatively curved manifolds. The first half presents a finiteness and vanishing result obtained under some curvature assumptions, while the second half identifies a class of metrics having non-trivial $L^2$ cohomology for degree equal to the half dimension of the space. For the second part we rely on the existence and regularity properties of the solution for the heat equation for forms. Category:58J50

33. CJM 2005 (vol 57 pp. 225)

Booss-Bavnbek, Bernhelm; Lesch, Matthias; Phillips, John
 Unbounded Fredholm Operators and Spectral Flow We study the gap (= projection norm'' = graph distance'') topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary. Categories:58J30, 47A53, 19K56, 58J32

34. CJM 2005 (vol 57 pp. 17)

Bédos, Erik; Conti, Roberto; Tuset, Lars
 On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C$^{*}$-categories. Keywords:quantum group, amenabilityCategories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32

35. CJM 2005 (vol 57 pp. 99)

Fegan, H. D.; Steer, B.
 Second Order Operators on a Compact Lie Group We describe the structure of the space of second order elliptic differential operators on a homogenous bundle over a compact Lie group. Subject to a technical condition, these operators are homotopic to the Laplacian. The technical condition is further investigated, with examples given where it holds and others where it does not. Since many spectral invariants are also homotopy invariants, these results provide information about the invariants of these operators. Categories:58J70, 43A77

36. CJM 2004 (vol 56 pp. 963)

Ishiwata, Satoshi
 A Berry-Esseen Type Theorem on Nilpotent Covering Graphs We prove an estimate for the speed of convergence of the transition probability for a symmetric random walk on a nilpotent covering graph. To obtain this estimate, we give a complete proof of the Gaussian bound for the gradient of the Markov kernel. Categories:22E25, 60J15, 58G32

37. CJM 2004 (vol 56 pp. 926)

 K-Homology of the Rotation Algebras $A_{\theta}$ We study the K-homology of the rotation algebras $A_{\theta}$ using the six-term cyclic sequence for the K-homology of a crossed product by ${\bf Z}$. In the case that $\theta$ is irrational, we use Pimsner and Voiculescu's work on AF-embeddings of the $A_{\theta}$ to search for the missing generator of the even K-homology. Categories:58B34, 19K33, 46L

38. CJM 2004 (vol 56 pp. 590)

Ni, Yilong
 The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary We study the Riemannian Laplace-Beltrami operator $L$ on a Riemannian manifold with Heisenberg group $H_1$ as boundary. We calculate the heat kernel and Green's function for $L$, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of $H_1$. We also restrict $L$ to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary. Categories:35H20, 58J99, 53C17

39. CJM 2004 (vol 56 pp. 638)

Śniatycki, Jędrzej
 Multisymplectic Reduction for Proper Actions We consider symmetries of the Dedonder equation arising from variational problems with partial derivatives. Assuming a proper action of the symmetry group, we identify a set of reduced equations on an open dense subset of the domain of definition of the fields under consideration. By continuity, the Dedonder equation is satisfied whenever the reduced equations are satisfied. Keywords:Dedonder equation, multisymplectic structure, reduction,, symmetries, variational problemsCategories:58J70, 35A30

40. CJM 2003 (vol 55 pp. 1100)

Khesin, Boris; Rosly, Alexei
 Polar Homology For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar $k$-chains are subvarieties of complex dimension $k$ with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincar\'e residue on it. One can also define the corresponding analogues for the intersection and linking numbers of complex submanifolds, which have the properties similar to those of the corresponding topological notions. Keywords:Poincar\' e residue, holomorphic linkingCategories:14C10, 14F10, 58A14

41. CJM 2003 (vol 55 pp. 266)

Kogan, Irina A.
 Two Algorithms for a Moving Frame Construction The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup. Categories:53A55, 58D19, 68U10

42. CJM 2003 (vol 55 pp. 247)

Cushman, Richard; Śniatycki, Jędrzej
 Differential Structure of Orbit Spaces: Erratum This note signals an error in the above paper by giving a counter-example. Categories:37J15, 58A40, 58D19, 70H33

43. CJM 2002 (vol 54 pp. 1187)

Cobo, Milton; Gutierrez, Carlos; Llibre, Jaume
 On the Injectivity of $C^1$ Maps of the Real Plane Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of (complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists $\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then $X$ is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed. Categories:34D05, 54H20, 58F10, 58F21

44. CJM 2002 (vol 54 pp. 1086)

Polterovich, Iosif
 Combinatorics of the Heat Trace on Spheres We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D.~Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in [P]. Categories:05A19, 58J35

45. CJM 2001 (vol 53 pp. 715)

Cushman, Richard; Śniatycki, Jędrzej
 Differential Structure of Orbit Spaces We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applied to analyze the differential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds. Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with singular reduced spaces. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed. Keywords:accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifoldsCategories:37J15, 58A40, 58D19, 70H33

46. CJM 2001 (vol 53 pp. 780)

Nicolaescu, Liviu I.
 Seiberg-Witten Invariants of Lens Spaces We show that the Seiberg-Witten invariants of a lens space determine and are determined by its Casson-Walker invariant and its Reidemeister-Turaev torsion. Keywords:lens spaces, Seifert manifolds, Seiberg-Witten invariants, Casson-Walker invariant, Reidemeister torsion, eta invariants, Dedekind-Rademacher sumsCategories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25

47. CJM 2001 (vol 53 pp. 278)

Helminck, G. F.; van de Leur, J. W.
 Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\KP$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau$-functions is given. Keywords:KP hierarchy, Darboux transformation, Grassmann manifoldCategories:22E65, 22E70, 35Q53, 35Q58, 58B25

48. CJM 2001 (vol 53 pp. 73)

Fukui, Toshizumi; Paunescu, Laurentiu
 Stratification Theory from the Weighted Point of View In this paper, we investigate stratification theory in terms of the defining equations of strata and maps (without tube systems), offering a concrete approach to show that some given family is topologically trivial. In this approach, we consider a weighted version of $(w)$-regularity condition and Kuo's ratio test condition. Categories:32B99, 14P25, 32Cxx, 58A35

49. CJM 2000 (vol 52 pp. 1235)

Hurtubise, J. C.; Jeffrey, L. C.
 Representations with Weighted Frames and Framed Parabolic Bundles There is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and extended by Bhosle and Ramanathan to other groups), between the symplectic variety $M_h$ of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group~$G$, with fixed conjugacy classes $h$ at the punctures, and a complex variety ${\cal M}_h$ of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For $G = \SU(2)$, we build a symplectic variety $P$ of pairs (representations of the fundamental group into $G$, weighted frame'' at the puncture points), and a corresponding complex variety ${\cal P}$ of moduli of framed parabolic bundles'', which encompass respectively all of the spaces $M_h$, ${\cal M}_h$, in the sense that one can obtain $M_h$ from $P$ by symplectic reduction, and ${\cal M}_h$ from ${\cal P}$ by a complex quotient. This allows us to explain certain features of the toric geometry of the $\SU(2)$ moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system. Categories:58F05, 14D20

50. CJM 2000 (vol 52 pp. 1057)

Urakawa, Hajime
 The Spectrum of an Infinite Graph In this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs. Keywords:infinite graph, discrete Laplacian, spectrum, essential spectrumCategories:05C50, 58G25
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