Anisotropic version of a theorem of H. Hopf

Given a positive function F on S ² which satisfies a convexity condition, we define a function for surfaces in which is a generalization of the usual mean curvature function. We prove that an immersed topological sphere in with = constant is the Wulff shape, up to translations and homotheties.      

A space cubic and its one-parameter automorphism groups

We discuss all automorphisms of which have a space cubic (twisted cubic) as a fixed figure. These automorphisms build up a three-parameter subgroup of all collineations of . In this paper we study the one-parameter subgroups of , their paths and tangent complexes.      

Affine normal curvature of hypersurfaces from the point of view of singularity theory

We consider smoothly embedded hypersurfaces under the action of the special affine group . We construct a differential invariant, called affine normal curvature, which assigns to a point and a tangent direction a number. We prove some of its nice properties which connect it with affine principal directions, affine umbilics, and affine mean curvature.      

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

On the Structure of the C *-Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols

We study the C ^*-algebra generated by Toeplitz operators with piece-wise continuous symbols acting on the Bergman space on the unit disk in . We describe explicitly each operator from this algebra and characterize Toeplitz operators which belong to the algebra. To the memory of G. S. Litvinchuk     

Complete surfaces of constant curvature in H<Superscript>2</Superscript> × R and S<Superscript>2</Superscript> × R

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or . J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034. J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.     

Parameterized partition relations on the real numbers

We consider several kinds of partition relations on the set of real numbers and its powers, as well as their parameterizations with the set of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition of there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of there is and a sequence of perfect sets such that the product lies in one piece of the partition, where is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. This work was supported by the research projects MTM 2005-01025 of the Spanish Ministry of Science and Education and 2005SGR-00738 of the Generalitat de Catalunya. A substantial part of the work was carried out while the second-named author was ICREA Visiting Professor at the Centre de Recerca Matemàtica in Bellaterra (Barcelona), and also during the first-named author’s stays at the Instituto Venezolano de Investigaciones Científicas and the California Institute of Technology. The authors gratefully acknowledge the support provided by these institutions.     

Cyclicity of bicyclic operators and completeness of translates

We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if is the shift operator acting on the weighted space of sequences , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if . On the other hand one can see that S is not cyclic if the series diverges. We show that the question of Herrero whether either S or S* is cyclic on admits a positive answer when the series is convergent. We also prove completeness results for translates in certain Banach spaces of functions on .     

Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p ∈ S, let be the compositum of all extensions unramified outside S of the form , for . If , let be the intersection of the fixed fields by , for i = 1, . . , n. We provide a wide family of elliptic curves such that the rank of is infinite for all n ≥ 0 and all , subject to the parity conjecture. Similarly, let be a polarized abelian variety, let K be a quadratic number field fixed by , let S be an infinite set of primes of and let be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over . We show that, under certain hypotheses, the -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in . As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.     

Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

The integral monodromy of hyperelliptic and trielliptic curves

We compute the and monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group . We prove that the monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group . Rachel Pries was partially supported by NSF grant DMS-04-00461.     

Dy namic of Abelia n Subgroups of GL( n, $$\mathbb{C}$$): A Structure Theorem

In this paper, we characterize the dynamic of every Abelian subgroups of , or . We show that there exists a -invariant, dense open set U in saturated by minimal orbits with a union of at most n -invariant vector subspaces of of dimension n−1 or n−2 over . As a consequence, has height at most n and in particular it admits a minimal set in . This work is supported by the research unit: systèmes dynamiques et combinatoire: 99UR15-15     

Syzygies of grids of projective lines

In this paper we study the syzygy modules of a grid or a fat grid of . We compute the minimal free resolution for the ideal of a complete grid in , and we conjecture this resolution in . Moreover we compute the minimal free resolution for the ideal of an incomplete grid of . We also conjecture the minimal free resolution for the ideal of a fat complete grid in .      

Smooth structures,normalized Ricci flows,and finite cyclic groups

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that for any finite cyclic group , where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group , which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of and .     

Totally geodesic shadow boundary submanifolds and a characterization for {\mathbb{S}}^n

We prove that the sphere is the only compact immersed hypersurface in Euclidean space each of whose shadow boundaries is a totally geodesic submanifold. Furthermore, we give conditions for the shadow boundary of a submanifold of to be regular. Received: 25 January 2007     

Derived McKay correspondence via pure-sheaf transforms

In most cases where it has been shown to exist the derived McKay correspondence can be written as a Fourier–Mukai transform which sends point sheaves of the crepant resolution Y to pure sheaves in . We give a sufficient condition for to be the defining object of such a transform. We use it to construct the first example of the derived McKay correspondence for a non-projective crepant resolution of . Along the way we extract more geometrical meaning out of the Intersection Theorem and learn to compute θ-stable families of G-constellations and their direct transforms.     

Constant mean curvature graphs in a class of warped product spaces

We introduce a new existence result for compact normal geodesic graphs with constant mean curvature and boundary in a class of warped product spaces. In particular, our result includes that of normal geodesic graphs with constant mean curvature in hyperbolic space over a bounded domain in a totally geodesic .      

Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

We establish a new 3G-Theorem for the Green’s function for the half space We exploit this result to introduce a new class of potentials that we characterize by means of the Gauss semigroup on . Next, we define a subclass of and we study it. In particular, we prove that properly contains the classical Kato class . Finally, we study the existence of positive continuous solutions in of the following nonlinear elliptic problem

A Slow Transient Diffusion in a Drifted Stable Potential

We consider a diffusion process X in a random potential of the form , where is a positive drift and is a strictly stable process of index with positive jumps. Then the diffusion is transient and converges in law towards an exponential distribution. This behaviour contrasts with the case where is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as “slow” as in the recurrent setting.      

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.