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51. CMB 2010 (vol 53 pp. 516)

Maurmann, Quinn; Engelstein, Max; Marcuccio, Anthony; Pritchard, Taryn
 Asymptotics of Perimeter-Minimizing Partitions We prove that the least perimeter $P(n)$ of a partition of a smooth, compact Riemannian surface into $n$ regions of equal area $A$ is asymptotic to $n/2$ times the perimeter of a planar regular hexagon of area $A$. Along the way, we derive tighter estimates for flat tori, Klein bottles, truncated cylinders, and MÃ¶bius bands. Category:53C42

52. CMB 2010 (vol 53 pp. 412)

Calvaruso, G.
 Einstein-Like Lorentz Metrics and Three-Dimensional Curvature Homogeneity of Order One We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field. Keywords:Lorentz manifolds, curvature homogeneity, Einstein-like metricsCategories:53C50, 53C20, 53C30

53. CMB 2009 (vol 53 pp. 206)

Atçeken, Mehmet
 Semi-Slant Submanifolds of an Almost Paracontact Metric Manifold In this paper, we define and study the geometry of semi-slant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semi-slant submanifold to be semi-slant product and obtain integrability conditions for the distributions involved in the definition of a semi-slant submanifold. Keywords:paracontact metric manifold, slant distribution, semi-slant submanifold, semi-slant productCategories:53C15, 53C25, 53C40

54. CMB 2009 (vol 40 pp. 257)

Baikoussis, Christos
 A characterization of real hypersurfaces in complex space forms in terms of the Ricci tensor We study real hypersurfaces of a complex space form $M_n(c)$, $c\ne 0$ under certain conditions of the Ricci tensor on the orthogonal distribution $T_o$. Categories:53C40, 53C15

55. CMB 2009 (vol 40 pp. 108)

Schaer, J.
 Continuous Self-maps of the Circle Given a continuous map $\delta$ from the circle $S$ to itself we want to find all self-maps $\sigma\colon S\to S$ for which $\delta\circ\sigma = \delta$. If the degree $r$ of $\delta$ is not zero, the transformations $\sigma$ form a subgroup of the cyclic group $C_r$. If $r=0$, all such invertible transformations form a group isomorphic either to a cyclic group $C_n$ or to a dihedral group $D_n$ depending on whether all such transformations are orientation preserving or not. Applied to the tangent image of planar closed curves, this generalizes a result of Bisztriczky and Rival [1]. The proof rests on the theorem: {\it Let $\Delta\colon\bbd R\to\bbd R$ be continuous, nowhere constant, and $\lim_{x\to -\infty}\Delta(x)=-\infty$, $\lim_{x\to+\infty}\Delta (x)=+\infty$; then the only continuous map $\Sigma\colon\bbd R\to\bbd R$ such that $\Delta\circ\Sigma=\Delta$ is the identity $\Sigma=\id_{\bbd R}$. Categories:53A04, 55M25, 55M35

56. 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

57. 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 morphismCategories:53C15, 58E20

58. CMB 2009 (vol 52 pp. 132)

Shen, Zhongmin
 On Projectively Flat $(\alpha,\beta)$-metrics The solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called $(\alpha,\beta)$-metrics defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension $n \geq 3$. Categories:53B40, 53C60

59. CMB 2009 (vol 52 pp. 87)

Lee, Junho
 Holomorphic 2-Forms and Vanishing Theorems for Gromov--Witten Invariants On a compact K\"{a}hler manifold $X$ with a holomorphic 2-form $\a$, there is an almost complex structure associated with $\a$. We show how this implies vanishing theorems for the Gromov--Witten invariants of $X$. This extends the approach used by Parker and the author for K\"{a}hler surfaces to higher dimensions. Category:53D45

60. CMB 2008 (vol 51 pp. 359)

Cho, Jong Taek; Ki, U-Hang
 Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Structure Jacobi Operator Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $(A)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator. Keywords:complex space form, real hypersurface, structure Jacobi operatorCategories:53B20, 53C15, 53C25

61. CMB 2008 (vol 51 pp. 467)

Wang, Yue
 Coupled Vortex Equations on Complete KÃ¤hler 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

62. CMB 2008 (vol 51 pp. 448)

Sasahara, Toru
 Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed. Keywords:biharmonic maps, Sasakian manifolds, Legendrian submanifoldsCategories:53C42, 53C40

63. CMB 2007 (vol 50 pp. 347)

Pérez, Juan de Dios; Santos, Florentino G.; Suh, Young Jin
 Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Is of Codazzi Type We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type. Categories:53C15, 53B25

64. CMB 2007 (vol 50 pp. 321)

Blair, David E.
 On Lagrangian Catenoids Recently I. Castro and F. Urbano introduced the Lagrangian catenoid. Topologically, it is $\mathbb R\times S^{n-1}$ and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ${\mathbb C}^n$ is foliated by round $(n-1)$-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ${\mathbb C}^n$. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid. Categories:53C42, 53D12

65. CMB 2007 (vol 50 pp. 474)

Zhou, Jiazu
 On Willmore's Inequality for Submanifolds Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total square mean curvature $\int_M H^2 d\sigma$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$. Keywords:submanifold, mean curvature, kinematic formul, scalar curvatureCategories:52A22, 53C65, 51C16

66. CMB 2007 (vol 50 pp. 365)

Godinho, Leonor
 Equivariant Cohomology of $S^{1}$-Actions on $4$-Manifolds Let $M$ be a symplectic $4$-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations. Categories:53D20, 55N91, 57S15

67. CMB 2007 (vol 50 pp. 24)

Brown, Nathan; Finck, Rachel; Spencer, Matthew; Tapp, Kristopher; Wu, Zhongtao
 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups We classify the left-invariant metrics with nonnegative sectional curvature on $\SO(3)$ and $U(2)$. Category:53C20

68. 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 ballCategories:58E15, 53C07

69. CMB 2007 (vol 50 pp. 97)

Kim, In-Bae; Kim, Ki Hyun; Sohn, Woon Ha
 Characterizations of Real Hypersurfaces in a Complex Space Form We study a real hypersurface $M$ in a complex space form $\mn$, $c \neq 0$, whose shape operator and structure tensor commute each other on the holomorphic distribution of $M$. Categories:53C40, 53C15

70. CMB 2006 (vol 49 pp. 321)

Balser, Andreas
 Polygons with Prescribed Gauss Map in Hadamard Spaces and Euclidean Buildings We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration. Moreover, the same result holds for semistable configurations on arbitrary Euclidean buildings. Keywords:Euclidean buildings, Hadamard spaces, polygonsCategory:53C20

71. 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

72. CMB 2006 (vol 49 pp. 152)

Yun, Jong-Gug
 Comparison Geometry With\\$L^1$-Norms of Ricci Curvature We investigate the geometry of manifolds with bounded Ricci curvature in $L^1$-sense. In particular, we generalize the classical volume comparison theorem to our situation and obtain a generalized sphere theorem. Keywords:Mean curvature, Ricci curvatureCategory:53C20

73. CMB 2006 (vol 49 pp. 134)

Suh, Young Jin
 Real Hypersurfaces in Complex Two-Plane Grassmannians with Vanishing Lie Derivative In this paper we give a characterization of real hypersurfaces of type $A$ in a complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ which are tubes over totally geodesic $G_2(\mathbb{C}^{m+1})$ in $G_2(\mathbb{C}^{m+2})$ in terms of the {\it vanishing Lie derivative\/} of the shape operator $A$ along the direction of the Reeb vector field $\xi$. Categories:53C40, 53C15

74. 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

75. CMB 2005 (vol 48 pp. 561)

Foth, Philip
 A Note on Lagrangian Loci of Quotients We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group. Keywords:Quotients, involutions, real forms, Lagrangian lociCategory:53D20
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