Classification of surfaces in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map

In this article, we study submanifolds in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map. We give a characterization theorem for Lorentzian surfaces in the pseudo‐sphere with zero mean curvature vector in and 2‐type pseudo‐spherical Gauss map. We also prove that non‐totally umbilical proper pseudo‐Riemannian hypersurfaces in a pseudo‐sphere with non‐zero constant mean curvature has 2‐type pseudo‐spherical Gauss map if and only if it has constant scalar curvature. Then, for we obtain the classification of surfaces in with 2‐type pseudo‐spherical Gauss map. Finally, we give an example of surface with null 2‐type pseudo‐spherical Gauss map which does not appear in Riemannian case, and we give a characterization theorem for Lorentzian surfaces in with null 2‐type pseudo‐spherical Gauss map.     

Wiener Tauberian theorem for rank one semisimple Lie groups and for hypergeometric transforms

We prove a genuine analogue of the Wiener Tauberian theorem for , where G is a real rank one noncompact, connected, semisimple Lie group with finite centre. This generalizes the corresponding result on the automorphism group of the unit disk by Y. Ben Natan, Y. Benyamini, H. Hedenmalm, and Y. Weit. We extend this result for hypergeometric transforms and as an application we prove an analogue of Furstenberg theorem on harmonic functions for hypergeometric transforms.     

Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.     

Maximal regularity for non‐autonomous Robin boundary conditions

We consider a non‐autonomous Cauchy problem where is associated with the form , where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in (if and ) under a new regularity condition on the form with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.     

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

This paper gives an insight into making a mathematical bridge between the parabolic‐parabolic signal‐dependent chemotaxis system and its parabolic‐elliptic version. To be more precise, this paper deals with convergence of a solution for the parabolic‐parabolic chemotaxis system with strong signal sensitivity to that for the parabolic‐elliptic chemotaxis system where Ω is a bounded domain in () with smooth boundary, is a constant and χ is a function generalizing In chemotaxis systems parabolic‐elliptic systems often gave some guide to methods and results for parabolic‐parabolic systems. However, the relation between parabolic‐elliptic systems and parabolic‐parabolic systems has not been studied except for the case that . Namely, in the case that Ω is a bounded domain, it still remains to analyze on the following question: Does a solution of the parabolic‐parabolic system converge to that of the parabolic‐elliptic system as ? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.     

The Radon‐Nikodym property,generalized bounded variation and the average range

A generalized bounded variation characterization of Banach spaces possessing the Radon‐Nikodym property is given in terms of the average range. We prove that a Banach space X has the Radon‐Nikodym property if and only if for each function of generalized bounded variation on [0, 1], the average range is a nonempty set at almost all .     

Critical metrics of the total scalar curvature functional on 4‐manifolds

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980's that every CPE metric must be Einstein. We prove that a 4‐dimensional CPE metric with harmonic tensor must be isometric to a round sphere      

Cohomological characterisation of monads

Let X be an n‐dimensional smooth projective variety with an n‐block collection , with , of coherent sheaves on X that generate the bounded derived category . We give a cohomological characterisation of torsion‐free sheaves on X that are the cohomology of monads of the form where . We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadric or the Fano threefold V₅. We construct a family of monads on a Segre variety and apply our main result to this family.     

Note on Nikodym maximal in the variable Lebesgue spaces

In this paper we consider the k‐plane Nikodym maximal estimates in the variable Lebesgue spaces . We first formulate the problem about the boundedness of the k‐plane Nikodym maximal and show that the maximal estimate in is equivalent to that in for . So, the optimal Nikodym maximal estimate in follows from Cordoba's estimate.     

Recovery of periodicities hidden in heavy‐tailed noise

We address a parametric joint detection‐estimation problem for discrete signals of the form , , with an additive noise represented by independent centered complex random variables . The distributions of are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy‐tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., frequencies , their number N, and complex coefficients . For example, one of considered classes of noise is the following: are independent identically distributed random variables with and . The construction of estimators is based on detection of singularities of anti‐derivatives for Z‐transforms and on a two‐level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series.     

Subgroups and homology of extensions of centralizers of pro‐p groups

We study the growth of , where U is an open subgroup of and is a special class of pro‐p groups defined in 7 . Furthermore for non‐abelian we prove the core property: for pro‐p subgroups such that H is finitely generated and N is non‐trivial normal in G the index is always finite.     

New gap results on the 4‐dimensional sphere

A result showed by M. Gursky in 4 ensures that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. In this note, we prove that there exists a universal number i ₀ such that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. Moreover, there exists a universal such that any metric g on the 4‐dimensional sphere with nonnegative sectional curvature, and is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in 4 .     

Solvability near the characteristic set for a class of first‐order linear partial differential operators

In this work we deal with solvability of first‐order differential equations in the form , where L is a planar complex vector field, elliptic everywhere except along a simple closed curve Σ on which it is tangent and vanishes of order . In contrast with the local solvability, it is shown that the zero order term p has influence in the solvability in a full neighborhood of Σ.     

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings of the unit disk , where h and g are analytic with , and determine the Bohr radius if any one of the following conditions holds:

• 1. h is bounded in .
• 2. h satisfies the condition in with .
• 3. both h and g are bounded in .
• 4. h is bounded and .

We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in is strictly less than 1. In addition, we determine the Bohr radius for the space of analytic Bloch functions and the space of harmonic Bloch functions. The paper concludes with two conjectures.     

On semi‐isogenous mixed surfaces

Let C be a smooth projective curve and G be a finite subgroup of whose action is mixed, i.e. there are elements in G exchanging the two isotrivial fibrations of . Let be the index two subgroup . If G⁰ acts freely, then is smooth and we call it semi‐isogenous mixed surface. In this paper we give an algorithm to determine semi‐isogenous mixed surfaces with given geometric genus, irregularity and self‐intersection of the canonical class. As an application we classify irregular semi‐isogenous mixed surfaces with and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with . We provide an example of a minimal surface of general type with and .     

Well‐posedness of non‐autonomous linear evolution equations in uniformly convex spaces

This paper addresses the problem of well‐posedness of non‐autonomous linear evolution equations in uniformly convex Banach spaces. We assume that for each t is the generator of a quasi‐contractive, strongly continuous group, where the domain D and the growth exponent are independent of t . Well‐posedness holds provided that is Lipschitz for all . Hölder continuity of degree is not sufficient and the assumption of uniform convexity cannot be dropped.     

On admissible topologies in JB*‐triples

Given a complex JB*‐triple X, we define and study admissible topologies on X, i.e., locally convex topologies τ on X coarser than the norm topology, invariant under the group of surjective linear isometries of X, and such that the triple product is jointly ‐continuous on bounded subsets of X. As a consequence of the joint ‐continuity of the triple product, all holomorphic automorphisms of the open unit ball are homeomorphisms of and the natural action is jointly ‐continuous on .     

On the automorphism group of the asymptotic pants complex of an infinite surface of genus zero

The braided Thompson group is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a Hatcher–Thurston cell complex whose vertices are asymptotically trivial pants decompositions. We prove that the automorphism group of this complex is also an asymptotic mapping class group in a weaker sense. Moreover is obtained by by first adding new elements called half‐twists and further completing it.     

Flat connections and cohomology invariants

The main goal of this article is to construct some geometric invariants for the topology of the set of flat connections on a principal G‐bundle . Although the characteristic classes of principal bundles are trivial when , their classical Chern–Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps to the cohomology group , where S is null‐cobordant ‐manifold, once a G‐invariant polynomial p of degree r on is fixed. For , this gives a homomorphism . The map is shown to be globally gauge invariant and furthermore it descends to the moduli space of flat connections , modulo cohomology with integer coefficients. The construction is also adapted to complex manifolds. In this case, one works with the set of connections with vanishing (0, 2)‐part of the curvature, and the Dolbeault cohomology. Some examples and applications are presented.