A characterization of domains in <Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript> with noncompact automorphism group

Let D be a bounded domain in C ² with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.     

Infinite-dimensionality of the automorphism groups of homogeneous Stein manifolds

We show that the group of holomorphic automorphisms of a Stein manifold X with dim X ≥ 2 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group.     

On the quantum homology algebra of toric Fano manifolds

We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.      

Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance

We show that if a hyperbolic 3-manifold M has two toroidal Dehn fillings with distance at least 3, then ∂M consists of at most three tori. As a result, we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3-manifolds with boundary a union of at least 4 tori. S. Lee was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314-C00024). M. Teragaito was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 19540089.     

Discontinuity of the Lempert Function and the Kobayashi-Royden Metric of the Spectral Ball

Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given. This work was initialized during the stay of the first and second named authors at the Jagiellonian University, Kraków in October, 2006, supported by the EGIDE program. They wish to thank both institutions. The third author was supported by the KBN research grant No. 1 PO3A 005 28.     

Local solvability in top degree of the semilinear Kohn equation

We study the local solvability problem for a class of semilinear equations whose linear part is the Kohn Laplacian, acting on top degree forms. We also study the validity of the Poincaré lemma, in top degree, for semilinear perturbations of the tangential Cauchy–Riemann complex.     

Inégalité d’Ahlfors en dimension supérieure

We prove an Ahlfors’ like inequality for the holomorphic curves with boundary of a complex compact Kobayashi-hyperbolic manifold.     

Almost complex and complex structures on real projective Stiefel manifolds

We give a complete list of real projective Stiefel manifolds which admit almost complex structures and show that many of them are in fact complex manifolds. The first named author was supported in part by Grants 1/1486/94 and 2/1225/96 of VEGA (Slovakia) during the preparation of this work.     

A gradient estimate in the Calabi–Yau theorem

We prove a C ¹-estimate for the complex Monge–Ampère equation on a compact Kähler manifold directly from the C ⁰-estimate, without using a C ²-estimate. This was earlier done only under additional assumption of non-negative bisectional curvature.     

Strong Morita equivalence of higher-dimensional noncommutative tori. II

We show that two C*-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K ₀-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C*-algebras of arbitrary finitely generated abelian groups. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada, held by George A. Elliott.     

Notes on a class of one-dimensional Landau-Brazovsky models

In the paper, a class of one-dimensional Landau–Brazovsky models is investigated. We present a sufficient condition under which the corresponding functional achieves its minimum. Moreover, a nonexistence result for nontrivial critical points is given. Project supported by the Doctoral Programme Foundation of the Ministry of Education of China.     

On the distribution of imaginary parts of zeros of the Riemann zeta function,II

We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Montgomery’s pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. The first author is supported by National Science Foundation Grant DMS-0555367. The second author is partially supported by the National Science Foundation and the American Institute of Mathematics (AIM). The third author is supported by National Science Foundation Grant DMS-0456615.     

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation on a closed manifold M, it is known that is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form (relatively to a suitable riemannian metric μ) is zero (cf. álvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group , where (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincaré Duality (cf. Kamber et and Tondeur in Astérisque 18:458–471, 1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group , when M is oriented. When M is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF). J. I. Royo Prieto was partially supported by EHU06/05, by a PostGrant from the Basque Government and by the MCyT of the Spanish Government. R. Wolak was partially supported by the KBN grant 2PO3A 021 25.     

Nondicritical $${\mathbb {C}^*}$$-actions on two-dimensional Stein manifolds

We study and classify actions of the complex multiplicative group on a nonsingular Stein surface with an isolated nondicritical singularity. We prove that the corresponding foliation exhibits a holomorphic first integral of a type F = f ⁿ g ^m where f and g are global holomorphic functions and . Under some additional conditions on the functions f and g we prove analytic linearization for the action. Our results can be viewed as extension of the original work of Masakazu Suzuki.     

An irregular complex valued solution to a scalar uniformly elliptic equation

We present an irregular weak solution of a uniformly elliptic scalar equation in divergence form with measurable coefficients. The solution has a square integrable gradient. Such examples have been known for dimension n ≥ 5 only. The author was partially supported by SFB 611.     

The influence of the boundary behavior on isometric immersions in the hyperbolic space

This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infinity. Our first result states that a proper isometric minimal immersion into the hyperbolic space with the asymptotic boundary contained in a sphere reduces codimension. This result is a corollary of a more general one that establishes a sharp lower bound for the sup-norm of the mean curvature vector of a Proper isometric immersion into the Hyperbolic space whose Asymptotic boundary is contained in a sphere. We also prove that a properly immersed hypersurface with mean curvature satisfying sup _{p∈Σ ||H(p)|| < 1} has no isolated points in its asymptotic boundary. Our main tool is a Tangency principle for isometric immersions of arbitrary codimension. This work is partially supported by CAPES, Brazil.     

On the Wu metric in unbounded domains

We discuss the properties of the Wu pseudometric and present counterexamples for its upper semicontinuity that answers the question posed by Jarnicki and Pflug. We also give formulae for the Wu pseudometric in elementary Reinhardt domains. Received: 12 September 2007     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.