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26. CMB 2011 (vol 56 pp. 44)

Biswas, Indranil; Dey, Arijit
Polystable Parabolic Principal $G$-Bundles and Hermitian-Einstein Connections
We show that there is a bijective correspondence between the polystable parabolic principal $G$-bundles and solutions of the Hermitian-Einstein equation.

Keywords:ramified principal bundle, parabolic principal bundle, Hitchin-Kobayashi correspondence, polystability
Categories:32L04, 53C07

27. CMB 2011 (vol 55 pp. 663)

Zhou, Chunqin
An Onofri-type Inequality on the Sphere with Two Conical Singularities
In this paper, we give a new proof of the Onofri-type inequality \begin{equation*} \int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{ \frac{1}{4\pi(\beta+1)} \int_S |\nabla u|^2 \,ds^2 + \frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\} \end{equation*} on the sphere $S$ with Gaussian curvature $1$ and with conical singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for $\beta\in (-1,0)$; here $p_1$ and $p_2$ are antipodal.

Categories:53C21, 35J61, 53A30

28. CMB 2011 (vol 55 pp. 723)

Gigli, Nicola; Ohta, Shin-Ichi
First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces
We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces $X$ with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.

Keywords:Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow
Categories:53C23, 28A35, 49Q20, 58A35

29. CMB 2011 (vol 55 pp. 611)

Özgür, Cihan; Mihai, Adela
Chen Inequalities for Submanifolds of Real Space Forms with a Semi-Symmetric Non-Metric Connection
In this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric non-metric connection, i.e., relations between the mean curvature associated with a semi-symmetric non-metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered.

Keywords:real space form, semi-symmetric non-metric connection, Ricci curvature
Categories:53C40, 53B05, 53B15

30. CMB 2011 (vol 55 pp. 108)

Guler, Dincer
On Segre Forms of Positive Vector Bundles
The goal of this note is to prove that the signed Segre forms of Griffiths' positive vector bundles are positive.

Categories:53C55, 32L05

31. CMB 2011 (vol 55 pp. 329)

Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.
Non-Discrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$
A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,n,\infty;k)$.

Keywords:complex hyperbolic triangle group
Categories:51M10, 32M15, 53C55, 53C35

32. CMB 2011 (vol 55 pp. 474)

Chen, Bin; Zhao, Lili
A Note on Randers Metrics of Scalar Flag Curvature
Some families of Randers metrics of scalar flag curvature are studied in this paper. Explicit examples that are neither locally projectively flat nor of isotropic $S$-curvature are given. Certain Randers metrics with Einstein $\alpha$ are considered and proved to be complex. Three dimensional Randers manifolds, with $\alpha$ having constant scalar curvature, are studied.

Keywords:Randers metrics, scalar flag curvature
Categories:53B40, 53C60

33. CMB 2011 (vol 55 pp. 138)

Li, Benling; Shen, Zhongmin
Projectively Flat Fourth Root Finsler Metrics
In this paper, we study locally projectively flat fourth root Finsler metrics and their generalized metrics. We prove that if they are irreducible, then they must be locally Minkowskian.

Keywords:projectively flat, Finsler metric, fourth root Finsler metric

34. CMB 2011 (vol 55 pp. 114)

Kon, S. H.; Loo, Tee-How
On Characterizations of Real Hypersurfaces in a Complex Space Form with $\eta$-Parallel Shape Operator
In this paper we study real hypersurfaces in a non-flat complex space form with $\eta$-parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.

Keywords:complex space form, Hopf hypersurfaces, ruled real hypersurfaces, $\eta$-parallel shape operator
Categories:53C40, 53C15

35. CMB 2011 (vol 54 pp. 716)

Okassa, Eugène
Symplectic Lie-Rinehart-Jacobi Algebras and Contact Manifolds
We give a characterization of contact manifolds in terms of symplectic Lie-Rinehart-Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold.

Keywords:Lie-Rinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifolds
Categories:13N05, 53D05, 53D10

36. CMB 2011 (vol 54 pp. 422)

Pérez, Juan de Dios; Suh, Young Jin
Two Conditions on the Structure Jacobi Operator for Real Hypersurfaces in Complex Projective Space
We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

Keywords:complex projective space, real hypersurface, structure Jacobi operator, two conditions
Categories:53C15, 53B25

37. CMB 2010 (vol 53 pp. 684)

Proctor, Emily; Stanhope, Elizabeth
An Isospectral Deformation on an Infranil-Orbifold
We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's theorem due to DeTurck and Gordon.

Keywords:spectral geometry, global Riemannian geometry, orbifold, nilmanifold
Categories:58J53, 53C20

38. CMB 2010 (vol 53 pp. 564)

Watanabe, Yoshiyuki; Suh, Young Jin
On $6$-Dimensional Nearly Kähler Manifolds
In this paper we give a sufficient condition for a complete, simply connected, and strict nearly Kähler manifold of dimension 6 to be a homogeneous nearly Kähler manifold. This result was announced in a previous paper by the first author.

Keywords:Nearly Kähler manifold, 6-dimension, Homogeneous, The 1st Chern Class, Einstein manifolds
Categories:53C40, 53C15

39. 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 metrics
Categories:53C50, 53C20, 53C30

40. 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 Möbius bands.


41. 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 product
Categories:53C15, 53C25, 53C40

42. 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.


43. 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

44. 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

45. 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 operator
Categories:53B20, 53C15, 53C25

46. 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

47. 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 submanifolds
Categories:53C42, 53C40

48. 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

49. 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 curvature
Categories:52A22, 53C65, 51C16

50. 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
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