C ∞-isometric imbeddings of the hyperbolic plane and of cylinders with hyperbolic metric in spherical spaces

Summary C^∞ — isometric imbeddings of the hyperbolic plane and of the two types of orientable cylinders with hyperbolic metric in spherical8-space are constructed, furthermore C^∞ — isometric imbeddings of the non-orientable cylinder with hyperbolic metric in Euclidean8-space and in spherical10-space. To Enrico Bompiani on his scientific Jubilee.     

Hele–Shaw flow on hyperbolic surfaces

Consider a complete simply connected hyperbolic surface. The classical Hadamard theorem asserts that at each point of the surface, the exponential mapping from the tangent plane to the surface defines a global diffeomorphism. This can be interpreted as a statement relating the metric flow on the tangent plane with that of the surface. We find an analogue of Hadamard's theorem with metric flow replaced by Hele–Shaw flow, which models the injection of (two-dimensional) fluid into the surface. The Hele–Shaw flow domains are characterized implicitly by a mean value property on harmonic functions.     

Poincaré-Andronov-Hopf bifurcation with hyperbolic rotation symmetry

For differential equations in Banach spaces with degenerate Fredholm operator at the derivative the general theorem about group symmetry inheritance by the relevant Lyapounov–Schmidt branching equation allows to construct its general form on admitted group symmetry. Here the solving of this problem is illustrated on the examples of the presence of SH(2) and H(2) symmetries. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A 2 × 2 hyperbolic traffic flow model

A new traffic flow model is presented. It accounts for various qualitative features of the evolution of the density and speed of cars along a crowded road. The model consists of a 2 × 2 system of nonlinear hyperbolic conservation laws generating a Cauchy problem which is well posed for all reasonable initil data. A similar result can be proved for the initial boundary value problem. The presence of a speed limit is also considered.     

Hyperbolic subalgebras of hyperbolic Kac–Moody algebras

We investigate regular hyperbolic subalgebras of hyperbolic Kac–Moody algebras via their Weyl groups. We classify all subgroup relations between Weyl groups of hyperbolic Kac–Moody algebras, and show that for every pair of a group and subgroup there exists at least one corresponding pair of algebra and subalgebra. We find all types of regular hyperbolic subalgebras for a given hyperbolic Kac–Moody algebra, and present a finite algorithm classifying all embeddings.     

Poincaré–Sobolev equations in the hyperbolic space

We study the a priori estimates, existence/nonexistence of radial sign changing solution, and the Palais–Smale characterisation of the problem ${-\Delta_{{\mathbb B}^{N}}u - \lambda u = |u|^{p-1}u, u\in H^1({\mathbb B}^{N})}$ in the hyperbolic space ${{\mathbb B}^{N}}$ where ${1 < p\leq\frac{N+2}{N-2}}$ . We will also prove the existence of sign changing solution to the Hardy–Sobolev–Mazya equation and the critical Grushin problem.     

New examples of Kobayashi hyperbolic surfaces in ℂℙ3

We show that a generic small deformation of the union of two generic cones in ³ of degree 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in ³ of any given degree d 8.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 90–94, 2005Original Russian Text Copyright © by B. Shiffman and M. ZaidenbergThe second authors research was supported in part by NSF grant No. DMS-0100474.Translated by B. Shiffman and M. Zaidenberg     

Cannon–Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E_Γ of F, and the main theorem of Dowdall–Taylor [DT14] states that in this situation E_Γ is hyperbolic if and only if Γ is purely atoroidal.     

Solvability of the fractional hyperbolic Keller–Segel system

We study a new nonlocal approach to the mathematical modelling of the chemotaxis problem, which describes the random motion of a certain population due to a substance concentration. Considering the initial–boundary value problem for the fractional hyperbolic Keller–Segel model, we prove the solvability of the problem. The solvability result relies mostly on fractional calculus and kinetic formulation of scalar conservation laws.     

The hyperbolic meaning of the Milnor–Wood inequality

We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several well-known results, including the Milnor–Wood inequality, using purely hyperbolic-geometric methods. Our methods express inequalities in Milnor’s function as equalities, with the deficiency from equality given by an area in the hyperbolic plane. We find that the twist of certain products found in surface group presentations is equal to the area of certain hyperbolic polygons arising as their fundamental domains.     

On orthogonality-preserving plücker transformations of hyperbolic spaces

A complete overview of all orthogonality-preserving Plücker transformations in finite dimensional hyperbolic spaces with dimension other than three is given. In the Cayley-Klein model of such a hyperbolic space all Plücker transformations are induced by collineations of the ambient projective space.     

Ritt's theorem and the Heins map in hyperbolic complex manifolds

Let x be a Kobayashi hyperbolic complex manifold, and assume that X does not contain compact complex submanifolds of positive dimension (e.g., X Stein). We shall prove the following generalization of Ritt's theorem: every holomorphic self-map f:X→X such that f(X) is relatively compact in X has a unique fixed point τ(f)∈X, which is attracting. Furthermore, we shall prove that τ(f) depends holomorphically on f in a suitable sense, generalizing results by Heins, Joseph-Kwack and the second author.     

Minimisers of the Allen–Cahn equation on hyperbolic graphs

We investigate minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic graph. Under some natural conditions on the graph, we show the existence of non-constant uniformly-bounded minimal solutions with prescribed asymptotic behaviours. For a phase field model on a hyperbolic graph, such solutions describe energy-minimising steady-state phase transitions that converge towards prescribed phases given by the asymptotic directions on the graph.     

Boundary behavior of Gleason''''s problem in hyperbolic harmonic Bergman spaces

Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B, dVβ) in the real unit ball B in Rn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem.     

Local rigidity for complex hyperbolic lattices and Hodge theory

We prove general local rigidity results for cocompact complex hyperbolic lattices using Hodge theory.     

A parabolic–hyperbolic system modeling the tumor growth with angiogenesis

In this paper, we study the parabolic–hyperbolic system about the growth of a tumor. The model is a coupled system of PDEs with Robin boundary, which involves nutrient density, extracellular matrix and matrix degrading enzyme. By transforming the free boundary into a fixed boundary and using strict mathematical analysis, we can prove the existence and uniqueness of the radially symmetric stationary solution. By the fixed point theorem, we obtain the existence and uniqueness of the radially symmetric solution globally in time.