Solving one extreme problem in a class of locally single-leaf functions

Центральное место в теории конформных отображений занимает решение экстремальных задач на классах однолистных отображений. В известных классах нормированных голоморфных функций $S$ и $C$ решение «проблемы коэффициентов» связано с получением точных оценок модулей тейлоровских коэффициентов элементов классов. Аналогичные задачи ставятся для классов локально однолистных отображений. В.Г.Шеретов ввел в рассмотрение классы локально конформных отображений, генерируемых с помощью интегральных структурных формул из элементов классов $S$ и $C$. В статье решена задача о точной оценке модуля тейлоровского коэффициента в этом классе. The central place in the theory of conformal maps is occupied by the solution of extreme problems on classes of single-leaf maps. In the known classes of normalized holomorphic functions S and C, the solution of the "coefficient problem" is associated with obtaining accurate estimates of the modules of the Taylor coefficients of class elements. Similar problems are posed for classes of locally single-leaf mappings. V.G.Sheretov introduced classes of locally conformal mappings generated using integral structural formulas from elements of classes S and C. The article solves the problem of an accurate estimation of the modulus of the Taylor coefficient in this class.

Some geometric obstructions to warped product immersions in locally conformal Kaehler space forms

Being motivated by a well-known Nash’s embedding theorem, Chen introduced a method to discover the relationship for the extrinsic invariants controlled by the intrinsic one. In this paper, we extend Chen’s inequality for the intrinsic and extrinsic invariants for pointwise bi-slant warped products in locally conformal Kaehler space forms with quarter-symmetric and semi-symmetric connections. The equality case of the inequality is also investigated. Several applications of the inequality are given. Furthermore, we provide two non-trivial examples of such immersions.

An isogeometric formulation of locally-conformal perfectly matched layer for acoustic scattering problems

This paper presents an isogeometric formulation of the locally-conformal perfectly matched layer (PML) for time-harmonic acoustic scattering problems. The new formulation is a generalization of the conventional locally-conformal PML, in which the NURBS patch supporting the PML domain is transformed from real space to complex space. This is achieved by complex coordinate stretching, based on a stretching vector field indicating the directions in which incident sound waves are absorbed. The performance of the isogeometric PML formulation is discussed through several acoustic scattering problems, spanning from one to three dimensions. It is found that the proposed method presents superior computational accuracy, high geometric adaptivity, and good robustness against challenging geometric features. The geometry-preserving ability inherent in the isogeometric framework could bring extra benefits by eliminating geometric errors that are unavoidable in the conventional PML. Meanwhile, these properties are not sensitive to the location of the sound source or the depth of the PML domain.

Topological and Dynamical Aspects of Jacobi Sigma Models

The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories—recently introduced by the authors—which share and generalise relevant features of Poisson sigma models, such as gauge invariance under diffeomorphisms and finite dimension of the reduced phase space. After reviewing the main novelties and peculiarities of these models, we perform a detailed analysis of constraints and ensuing gauge symmetries in the Hamiltonian approach. Contact manifolds as well as locally conformal symplectic manifolds are discussed, as main instances of Jacobi manifolds.

Some Properties of Curvature Tensors and Foliations of Locally Conformal Almost Kähler Manifolds

We investigate a class of locally conformal almost Kähler structures and prove that, under some conditions, this class is a subclass of almost Kähler structures. We show that a locally conformal almost Kähler manifold admits a canonical foliation whose leaves are hypersurfaces with the mean curvature vector field proportional to the Lee vector field. The geodesibility of the leaves is also characterized, and their minimality coincides with the incompressibility of the Lee vector field along the leaves.

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/4, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.