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This manuscript investigates the relationship between higher-dimensional Hilbert Manifolds and Hilbert spaces using the diffeomorphic embedding theorem. The existence of diffeomorphisms between Hilbert spaces will be demonstrated, and we will provide exact constructions that demonstrate their mathematical flexibility in several scenarios. The more general case of compact complex Manifolds equipped with Hermitian metrics is investigated. Under specific conditions, the existence of diffeomorphic embeddings into a separable infinite-dimensional Hilbert space that preserve both the complex structure and the Hermitian geometry is demonstrated. These results pave the way for studying complex geometry within the framework of infinite-dimensional Hilbert spaces. [ABSTRACT FROM AUTHOR]

A novel family of bi-univalent holomorphic functions is introduced by the use of the Lindelöf principle. The upper bound of the second Hankel determinant, H 2 , 2 (χ) , is evaluated. Furthermore, specific results are obtained as special cases of the main conclusion. These cases coincide with certain recently obtained results and improve or enhance specific ones. [ABSTRACT FROM AUTHOR]

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk D. For these new classes, the authors establish initial coefficient bounds of | a 2 | and | a 3 | . Furthermore, the famous Fekete–Szegö coefficient inequality is also obtained for these new classes of functions. [ABSTRACT FROM AUTHOR]

In this study, we present a novel family of holomorphic and bi-univalent functions, denoted as E Ω (η , ε ; Ϝ ). We establish the coefficient bounds for this family by utilizing the generalized telephone numbers. Additionally, we solve the Fekete–Szegö functional for functions that belong to this family within the open unit disk. Moreover, our results have several consequences. [ABSTRACT FROM AUTHOR]

We discuss some results concerning continuous functions and small subsets or complex structure specific conditions. A characterization of holomorphic functions using potential theory methods is obtained. Moreover, we establish necessary and sufficient conditions for the product of two exponents of continuous functions having two special types to be plurisubharmonic. The obtained results are fundamental keys in the study of the complex structure of a product of several plurisubharmonic functions. [ABSTRACT FROM AUTHOR]

We investigate the study of convex, strictly plurisubharmonic and the special class consisted of convex and strictly plurisubharmonic functions in convex domains of Cn, n ≥ 1. Let h : Cn → C be pluriharmonic. We prove that {b ∈ C/|h + b| is a convex function of Cn} = ∅, or {α}, or C, where α ∈ C. Now let α1, α2, α3 : D → C be three holomorphic functions, D is a domain of Cn. Put ..., for (z, w) ∈ D x C. We prove that u is psh on D x C if and only if (α1 + α2 + α3) and (α1α2 + α1α3 + α2α3) are constant on D, or (α1 + α2 + α3) is non constant and α1 = α2 = α3 on D. [ABSTRACT FROM AUTHOR]

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space E over C. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of E like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an E-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete E to locally complete E. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space E we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the E-valued polydisc algebra. [ABSTRACT FROM AUTHOR]

We study a superminimal surface M immersed into a hyperquadric Q2 in several cases classified by two global defined functions τX and τY, which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f:M → Q2. In case both τX and τY are not identically zero, it is proved that f is superminimal if and only if f is totally real or i ○ f:M → ℂP3 is also minimal, where i:Q2 → ℂP3 is the standard inclusion map. In the rest case that τX ≡ 0 or τY ≡ 0, the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q2 are completely described. [ABSTRACT FROM AUTHOR]

Let A1;A2 2 Cnf0g and n;m 2 Nnf0g: Using alge- braic methods, we prove that there exist three analytic functions ' : Cm ! C and g1; g2 : Cn ! C such that v is convex and strictly plurisubharmonic on Cn Cm if and only if m = 1; n 2 f1; 2g; there exists c 2 C such that j ' + c j2 is convex and strictly subharmonic on C and the functions g1 and g2 have fundamental representations over Cn: v(z;w) =j A1'(w) g1(z) j2 + j A2'(w) g2(z) j2; for (z;w) 2 Cn Cm: At the end, we prove an additional theorem by analytic and algebraic methods. [ABSTRACT FROM AUTHOR]

We introduce a notion stability for subgroups of local complex analytic diffeomorphisms having a common fixed point, in several complex variables. This notion, called L-stability, is inspired in the notion of stability of Lyapunov for singular points and closed orbits of ordinary differential equations. It is also connected to the notion of stability for a proper leaf of a foliation in the classical sense of Reeb. We first classify in terms of the unitary group. Then we prove analytic linearization for a L-stable map and on the classification of L-stable linear groups. This is related to the study of subgroups of U(n), the unitary matrix group. [ABSTRACT FROM AUTHOR]

We give two new proofs of a classical result of Frank Forelli about a function onthat is holomorphic along complex lines. We also present a number of generalizations and variants and applications. [ABSTRACT FROM PUBLISHER]

We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, the non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov (Fatou theory in two dimensions, PhD thesis, University of Michigan, 2004). In 2014, it was shown in Astorg et al. (Ann Math, [math.DS], 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, , 2014), the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, , 2014). Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds. [ABSTRACT FROM AUTHOR]

An extension theorem for holomorphic functions with L growth condition is strengthened for the case of the extension from hypersurfaces with isolated singularities. [ABSTRACT FROM AUTHOR]

The main purpose of the present paper is to study almost complex structures conformalWeyl-Euler-Lagrangian equations on 4-dimensionalWalker manifolds for (conservative) dynamical systems. In this study, routes of objects moving in space will be modeled mathematically on 4-dimensional Walker manifolds that these are time-dependent partial differential equations. A Walker n-manifold is a semi-Riemannian n-manifold, which admits a field of parallel null r-planes, with r ≤ n/2. It is well-known that semi-Riemannian geometry has an important tool to describe spacetime events. Therefore, solutions of some structures about 4-Walker manifold can be used to explain spacetime singularities. Then, here we present complex analogues of Lagrangian mechanical systems on 4-Walker manifold. Also, the geometrical-physical results related to complex mechanical systems are also discussed for conformal Weyl-Euler-Lagrangian equations for (conservative) dynamical systems and solution of the motion equations using Maple Algebra software will be made. [ABSTRACT FROM AUTHOR]

We study 4-dimensional orientable Riemannian manifolds equipped with a minimal and conformal foliation $${\mathcal {F}}$$ of codimension 2. We prove that the two adapted almost Hermitian structures $$J_1$$ and $$J_2$$ are both cosymplectic if and only if $${\mathcal {F}}$$ is Riemannian and its horizontal distribution $${\mathcal {H}}$$ is integrable. [ABSTRACT FROM AUTHOR]

Let x be a nonzero vector in $${\mathbb{C}^{n}}$$ , and let $${U\subseteq \mathcal{M}_{n}}$$ be a domain containing the zero matrix. We prove that if φ is a holomorphic map from U into $${\mathcal{M}_{n}}$$ such that the local spectrum of T ∈ U at x and the local spectrum of φ( T) at x have always a common value, then T and φ( T) have always the same spectrum, and they have the same local spectrum at x a.e. with respect to the Lebesgue measure on U. If $${\varphi \colon U\rightarrow \mathcal{M}_{n}}$$ is holomorphic with φ(0) = 0 such that the local spectral radius of T at x equals the local spectral radius of φ( T) at x for all T ∈ U, there exists $${\xi \in \mathbb{C}}$$ of modulus one such that ξ T and φ( T) have the same spectrum for all T in U. We also prove that if for all T ∈ U the local spectral radius of φ( T) coincides with the local spectral radius of T at each vector x, there exists $${\xi \in \mathbb{C}}$$ of modulus one such that φ( T) = ξ T on U. [ABSTRACT FROM AUTHOR]

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian-Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein-Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z. 190 (1985) 39-43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom. 7 (2007) 207-214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property. [ABSTRACT FROM AUTHOR]

In this note, we consider locally invertible analytic mappings of a two-dimensional space over a non-archimedean field. Such a map is called semi-hyperbolic if its Jacobian has eigenvalues λ1 and λ2 so that λ1 = 1 and |λ2| ≠ 1. We prove that two analytic semi-hyperbolic maps are analytically equivalent if and only if they are formally equivalent, applying a generalized version of an estimation scheme from our earlier work [A. Jenkins and S. Spallone, A p-adic approach to local dynamics: Analytic flows and analytic maps tangent to the identity, Ann. Fac. Sci. Toulouse Math. (6) 18(3) (2009) 611-634]. [ABSTRACT FROM AUTHOR]

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure μ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Lie algebra” of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L2(ν)-closure of holomorphic polynomials by their values on the Cameron–Martin subgroup. [ABSTRACT FROM AUTHOR]

We study (φ;φ')-holomorphic maps between almost contact metric manifolds, in particular horizontally conformal (φ;φ')-holomorphic submersions, and obtain some criteria for the harmonicity of such maps. [ABSTRACT FROM AUTHOR]

This survey gives the concept, the historical background and the results which lay ground for the investigations in the field of M. M. Djrbashian factorization theory and related problems of analysis. It gives some applications and results which reveal the fields of further, contemporary investigations. [ABSTRACT FROM AUTHOR]

Existence and uniqueness of invariant quadratic forms on spaces of holomorphic functions is studied. Invariance is with respect to the action of the automorphism group of the underlying domain given by [image omitted], where ϕ is an automorphism and Jϕ is its Jacobian determinant. Quadratic forms invariant under this action are called q-invariant forms. For q > 0, it is shown, under mild assumptions on the Bergman kernal, that there is a unique q-invariant form, which is positive definite and determines a unique functional Hilbert space. In the case of the ball, there is a unique q-invariant form for all real q, which is positive definite for q > 0, and positive semi-definite for q = 0 and a discrete set of negative q. In those cases when the form is positive semi-definite, integral formulas are obtained for the associated (semi) norms. [ABSTRACT FROM AUTHOR]

Let ϕ(z) = (ϕ1(z),...,ϕn(z)) be a holomorphic self-map of B and ψ(z) a holomorphic function on B, where B is the unit ball of ℂn. Let 0 < p, s < +∞, -n - 1 < q < +∞, q+s > -1 and α ≥ 0, this paper characterizes boundedness and compactness of weighted composition operator Wψ,ϕ induced by ϕ and ψ between the space F(p, q, s) and α-Bloch space $\mathcal{B}^{\alpha}$. [ABSTRACT FROM AUTHOR]

We examine the classical Hartogs extension phenomenon from a number of different points of view. Several different proofs are offered, some of them are new. Connections with other parts of the subject are developed. [ABSTRACT FROM AUTHOR]

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain of D’Angelo finite type in ℂn (n1) is an automorphism. [ABSTRACT FROM AUTHOR]

A Fischer pair (FP) for a vector space E is a pair (u, v) of linear maps on E, not necessarily everywhere defined, such that E=ker u⊕Im v (Fischer decomposition). Thus, in particular, every densely defined closed operator u on a Hilbert space E forms a Fischer pair together with its adjoint u*, whenever Im u, or equivalently, Im u* is closed since then Im u*=ker u⊥. The question of when a given pair of maps (u, v) is a FP is related to the well-posedness of the (abstract) Cauchy–Goursat problem for u, v in E. We establish some Fischer pairs, for spaces that are built up by homogeneous Hilbert–Schmidt polynomials on a Hilbert space, consisting of differential and multiplication operators. In particular we study Fischer decompositions of the space of entire functions of Hilbert–Schmidt type. As a basis we generalize Fischers theorem for homogeneous polynomials in n variables to Hilbert–Schmidt polynomials. [ABSTRACT FROM AUTHOR]

The PDE-preserving operators O on the space of nuclearly entire functions of bounded type HNb(E) on a Banach space E are characterized. An operator is PDE-preserving when it preserves homogenous solutions to homogeneous convolution equations. We establish a one to one correspondence between O and a set O of sequences of entire functionals, i.e. exponential type functions. In this way, algebraic structures on O, such as ring structures, can be carried over to O and vice versa. In particular, it follows that O is a non-commutative ring (algebra) with unity with respect to composition and the convolution operators form a commutative subring (subalgebra). We discuss range and kernel properties, for the operators in O, and characterize the projectors (onto polynomial spaces) in O by determining the corresponding elements in O. [ABSTRACT FROM AUTHOR]

In this paper, we study volume-preserving great circle flows on the 3-sphere S3, and introduce a moduli space for these, consisting of distance-decreasing holomorphic maps from connected open subsets of S2 to S2; one consequence is that the only volume-preserving great circle flows defined on the entire 3-sphere are those tangent to Hopf fibrations. [ABSTRACT FROM AUTHOR]

We construct automorphisms of C n which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances, this can be done while also approximating a given automorphism on a compact set. [ABSTRACT FROM AUTHOR]

We describe a classification of star algebras on the cotangent bundle of a complex manifold, locally isomorphic to the algebra of pseudo-differential operators; this requires a slight extension of the usual definition of star algebras. We show that in dimension ≥ 3 these are essentially trivial and come from algebras of differential operators on X; in dimension 1 and 2 there are many more, which we describe. [ABSTRACT FROM AUTHOR]