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Search: MSC category 58 ( Global analysis, analysis on manifolds )

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26. CMB 2010 (vol 53 pp. 542)

Pintea, Cornel
Smooth Mappings with Higher Dimensional Critical Sets
In this paper we provide lower bounds for the dimension of various critical sets, and we point out some differential maps with high dimensional critical sets.

Categories:58K05, 57R70

27. CMB 2009 (vol 53 pp. 340)

Lusala, Tsasa; Śniatycki, Jędrzej; Watts, Jordan
Regular Points of a Subcartesian Space
We discuss properties of the regular part $S_{\operatorname{reg}}$ of a subcartesian space $S$. We show that $S_{\operatorname{reg}}$ is open and dense in $S$ and the restriction to $ S_{\operatorname{reg}}$ of the tangent bundle space of $S$ is locally trivial.

Keywords:differential structures, singular and regular points
Category:58A40

28. CMB 2009 (vol 53 pp. 122)

Mo, Xiaohuan; Zhou, Linfeng
A Class of Finsler Metrics with Bounded Cartan Torsion
In this paper, we find a class of $(\alpha,\beta)$ metrics which have a bounded Cartan torsion. This class contains all Randers metrics. Furthermore, we give some applications and obtain two corollaries about curvature of this metrics.

Keywords:Finsler manifold, $(\alpha,\beta)$ metric, Cartan torsion, R-quadratic, flag curvature
Category:58E20

29. CMB 2009 (vol 40 pp. 271)

Bergweiler, Walter
Non-real periodic points of entire functions
It is shown that if $f$ is an entire transcendental function, $l$ a straight line in the complex plane, and $n\geq 2$, then $f$ has infinitely many repelling periodic points of period $n$ that do not lie on $l$.

Categories:30D05, 58F23

30. CMB 2009 (vol 40 pp. 204)

Meyerhoff, Robert; Ouyang, Mingqing
The $\eta$-invariants of cusped hyperbolic $3$-manifolds
In this paper, we define the $\eta$-invariant for a cusped hyperbolic $3$-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.

Categories:57M50, 53C30, 58G25

31. CMB 2009 (vol 40 pp. 285)

Crawford, T. Arleigh
The space of harmonic maps from the $2$-sphere to the complex projective plane
In this paper we study the topology of the space of harmonic maps from $S^2$ to $\CP 2$. We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to $\CP n$ for $n\geq 2$. We show that the components of maps to $\CP 2$ are complex manifolds.

Categories:58E20, 58D27

32. CMB 2009 (vol 52 pp. 18)

Chinea, Domingo
Harmonicity of Holomorphic Maps Between Almost Hermitian Manifolds
In this paper we study holomorphic maps between almost Hermitian manifolds. We obtain a new criterion for the harmonicity of such holomorphic maps, and we deduce some applications to horizontally conformal holomorphic submersions.

Keywords:almost Hermitian manifolds, harmonic maps, harmonic morphism
Categories:53C15, 58E20

33. CMB 2009 (vol 52 pp. 66)

Dryden, Emily B.; Strohmaier, Alexander
Huber's Theorem for Hyperbolic Orbisurfaces
We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

Keywords:Huber's theorem, length spectrum, isospectral, orbisurfaces
Categories:58J53, 11F72

34. CMB 2008 (vol 51 pp. 467)

Wang, Yue
Coupled Vortex Equations on Complete Kähler Manifolds
In this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact K\"ahler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

Categories:58J05, 53C07

35. CMB 2008 (vol 51 pp. 249)

Mangoubi, Dan
On the Inner Radius of a Nodal Domain
Let $M$ be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$. We give upper and lower bounds on the inner radius of the type $C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincar\'{e} type inequality proved by Maz'ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

Categories:58J50, 35P15, 35P20

36. CMB 2008 (vol 51 pp. 100)

Petkov, Vesselin
Dynamical Zeta Function for Several Strictly Convex Obstacles
The behavior of the dynamical zeta function $Z_D(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function $\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re s = s_2$, where $s_2$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions $Z_D(s), \Pi(s)$ has a singularity at $s = s_2$.

Keywords:dynamical zeta function, periodic rays
Categories:11M36, 58J50

37. CMB 2007 (vol 50 pp. 447)

Śniatycki, Jędrzej
Generalizations of Frobenius' Theorem on Manifolds and Subcartesian Spaces
Let $\mathcal{F}$ be a family of vector fields on a manifold or a subcartesian space spanning a distribution $D$. We prove that an orbit $O$ of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$ and it has constant rank on $O$. This result implies Frobenius' theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

Keywords:differential spaces, generalized distributions, orbits, Frobenius' theorem, Sussmann's theorem
Categories:58A30, 58A40

38. CMB 2007 (vol 50 pp. 113)

Li, ZhenYang; Zhang, Xi
Hermitian Harmonic Maps into Convex Balls
In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary.

Keywords:Hermitian harmonic map, Hermitian manifold, convex ball
Categories:58E15, 53C07

39. CMB 2006 (vol 49 pp. 337)

Berlanga, R.
Homotopy Equivalence and Groups of Measure-Preserving Homeomorphisms
It is shown that the group of compactly supported, measure-preserving homeomorphisms of a connected, second countable manifold is locally contractible in the direct limit topology. Furthermore, this group is weakly homotopically equivalent to the more general group of compactly supported homeomorphisms.

Categories:57S05, 58F11

40. CMB 2006 (vol 49 pp. 226)

Engman, Martin
The Spectrum and Isometric Embeddings of Surfaces of Revolution
A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This leads to generalizations of some classical results in the theory of surfaces.

Categories:58J50, 58J53, 53C20, 35P15

41. CMB 2006 (vol 49 pp. 36)

Daskalopoulos, Georgios D.; Wentworth, Richard A.
Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles
Using a modification of Webster's proof of the Newlander--Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class.

Categories:57M50, 58E20, 53C24

42. CMB 2005 (vol 48 pp. 340)

Andruchow, Esteban
Short Geodesics of Unitaries in the $L^2$ Metric
Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$

Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.
Categories:46L51, 58B10, 58B25

43. CMB 2005 (vol 48 pp. 283)

Thibault, Lionel; Zagrodny, Dariusz
Enlarged Inclusion of Subdifferentials
This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma$-enlargement of the subdifferential of $g$, then the difference of those functions is $ \gamma$-Lipschitz over their effective domain.

Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable function
Categories:49J52, 46N10, 58C20

44. CMB 2004 (vol 47 pp. 624)

Zhang, Xi
A Compactness Theorem for Yang-Mills Connections
In this paper, we consider Yang-Mills connections on a vector bundle $E$ over a compact Riemannian manifold $M$ of dimension $m> 4$, and we show that any set of Yang-Mills connections with the uniformly bounded $L^{\frac{m}{2}}$-norm of curvature is compact in $C^{\infty}$ topology.

Keywords:Yang-Mills connection, vector bundle, gauge transformation
Categories:58E20, 53C21

45. CMB 2004 (vol 47 pp. 607)

Plamenevskaya, Olga
A Residue Formula for $\SU(2)$-Valued Moment Maps
Jeffrey and Kirwan gave expressions for intersection pairings on the reduced space $M_0=\mu^{-1}(0)/G$ of a Hamiltonian $G$-space $M$ in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian $\SU(2)$-space. The definition of quasi-Hamiltonian $G$-spaces was recently introduced in .

Category:58F05

46. CMB 2004 (vol 47 pp. 515)

Frigon, M.
Remarques sur l'enlacement en théorie des points critiques pour des fonctionnelles continues
Dans cet article, \`a partir de la notion d'enlacement introduite dans ~\cite{F} entre des paires d'ensembles $(B,A)$ et $(Q,P)$, nous \'etablissons l'existence d'un point critique d'une fonctionnelle continue sur un espace m\'etrique lorsqu'une de ces paires enlace l'autre. Des renseignements sur la localisation du point critique sont aussi obtenus. Ces r\'esultats conduisent \`a une g\'en\'eralisation du th\'eor\`eme des trois points critiques. Finalement, des applications \`a des probl\`emes aux limites pour une \'equation quasi-lin\'eaire elliptique sont pr\'esent\'ees.

Categories:58E05, 35J20

47. CMB 2003 (vol 46 pp. 481)

Bachir, M.; Lancien, G.
On the Composition of Differentiable Functions
We prove that a Banach space $X$ has the Schur property if and only if every $X$-valued weakly differentiable function is Fr\'echet differentiable. We give a general result on the Fr\'echet differentiability of $f\circ T$, where $f$ is a Lipschitz function and $T$ is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm $\Vert \ \Vert_{\lip}$ on various spaces of Lipschitz functions.

Categories:58C20, 46B20

48. CMB 2002 (vol 45 pp. 378)

Fernández-López, Manuel; García-Río, Eduardo; Kupeli, Demir N.
The Local Möbius Equation and Decomposition Theorems in Riemannian Geometry
A partial differential equation, the local M\"obius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local M\"obius equation and an additional partial differential equation.

Keywords:submersion, Möbius equation, twisted product, warped product, product Riemannian manifolds
Categories:53C12, 58J99

49. CMB 2002 (vol 45 pp. 97)

Haas, Andrew
Invariant Measures and Natural Extensions
We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real M\"obius transformations. Included are the maps that are exactly $n$-to-$1$, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures.

Keywords:Continued fractions, interval maps, invariant measures
Categories:11J70, 58F11, 58F03

50. CMB 2002 (vol 45 pp. 3)

Azagra, D.; Dobrowolski, T.
Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications
We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces.

Categories:46B20, 58B99
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