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A summary of methods from functional analysis in application to the analysis of several complex variables, based upon Hormander's estimate for ∂ operators on pseudoconvex domains.

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Pars 1 and 2 are devoted to study of solutions of certain differential inequalities. Namely, in Part 1 we show that a germ of an analytic set (real or complex) admits a Gagliardo-Nirenberg type inequality with a certain exponent s>=1. At a regular point s=1, and the inequality becomes classical. As our examples show, s can be strictly greater than 1 even for an isolated singularity. In Part 2 we prove the property of unique continuation for solutions of differential inequality |\Delta u|<=|Vu| for a large class of potentials V. This result can be applied to the problem of absence of positive eigenvalues for self-adjoint Schroedinger operator -\Delta+V defined in the sense of the form sum. The results of Part 2 are joint with Leonid Shartser. In Parts 3 and 4 we derive the basic elements of complex function theory within some subalgebras of holomorphic functions (including extension from submanifolds, corona type theorem, properties of divisors, approximation property). Our key instruments and results are the analogues of Cartan theorems A and B for the `coherent sheaves' on the maximal ideal spaces of these subalgebras, and of Oka-Cartan theorem on coherence of the sheaves of ideals of the corresponding complex analytic subsets. More precisely, in Part 3 we consider the algebras of holomorphic functions on regular coverings of complex manifolds whose restrictions to each fiber belong to a translation-invariant Banach subalgebra of bounded functions endowed with sup-norm. The model examples of such subalgebras are Bohr's holomorphic almost periodic functions on tube domains, and all fibrewise bounded holomorphic functions on regular coverings of complex manifolds. In Part 4 the primary object of study is the subalgebra of bounded holomorphic functions on the unit disk whose moduli can have only boundary discontinuities of the first kind. The results of Parts 3 and 4 are joint with Alexander Brudnyi.

Pars 1 and 2 are devoted to study of solutions of certain differential inequalities. Namely, in Part 1 we show that a germ of an analytic set (real or complex) admits a Gagliardo-Nirenberg type inequality with a certain exponent s>=1. At a regular point s=1, and the inequality becomes classical. As our examples show, s can be strictly greater than 1 even for an isolated singularity. In Part 2 we prove the property of unique continuation for solutions of differential inequality |\Delta u|<=|Vu| for a large class of potentials V. This result can be applied to the problem of absence of positive eigenvalues for self-adjoint Schroedinger operator -\Delta+V defined in the sense of the form sum. The results of Part 2 are joint with Leonid Shartser. In Parts 3 and 4 we derive the basic elements of complex function theory within some subalgebras of holomorphic functions (including extension from submanifolds, corona type theorem, properties of divisors, approximation property). Our key instruments and results are the analogues of Cartan theorems A and B for the `coherent sheaves' on the maximal ideal spaces of these subalgebras, and of Oka-Cartan theorem on coherence of the sheaves of ideals of the corresponding complex analytic subsets. More precisely, in Part 3 we consider the algebras of holomorphic functions on regular coverings of complex manifolds whose restrictions to each fiber belong to a translation-invariant Banach subalgebra of bounded functions endowed with sup-norm. The model examples of such subalgebras are Bohr's holomorphic almost periodic functions on tube domains, and all fibrewise bounded holomorphic functions on regular coverings of complex manifolds. In Part 4 the primary object of study is the subalgebra of bounded holomorphic functions on the unit disk whose moduli can have only boundary discontinuities of the first kind. The results of Parts 3 and 4 are joint with Alexander Brudnyi.

The classical Lu Qi-keng Conjecture asks whether the Bergman kernel function for every domain is zero free. The answer is no, and several counterexamples exist in the literature. However, the more general Lu Qi-keng Problem, that of determining which domains in Cn have vanishing kernels, remains a difficult open problem in several complex variables. A challenge in studying the Lu Qi-keng Problem is that concrete formulas for kernels are generally difficult or impossible to compute. Our primary focus is on developing methods of computing concrete formulas in order to study the Lu Qi-keng Problem. The kernel for the annulus was historically the first counterexample to the Lu Qi-keng Conjecture. We locate the zeros of the kernel for the annulus more precisely than previous authors. We develop a theory giving a formula for the weighted kernel on a general planar domain with weight the modulus squared of a meromorphic function. A consequence of this theory is a technique for computing explicit, closed-form formulas for such kernels where the weight is associated to a meromorphic kernel with a finite number of zeros on the domain. For kernels associated to meromorphic functions with an arbitrary number of zeros on the domain, we obtain a weighted version of the classical Ramadanov's Theorem which says that for a sequence of nested bounded domains exhausting a limiting domain, the sequence of associated kernels converges to the kernel associated to the limiting domain. The relationship between the zeros of the weighted kernels and the zeros of the corresponding unweighted kernels is investigated, and since these weighted kernels are related to unweighted kernels in C^2, this investigation contributes to the study of the Lu Qi-keng Problem. This theory provides a much easier technique for computing certain weighted kernels than classical techniques and provides a unifying explanation of many previously known kernel formulas. We also present and explore a generalization of the Lu Qi-keng Problem.

In a series of works, Jun-Muk Hwang and Ngaiming Mok have developed a geometric theory of uniruled projective manifolds, especially those of Picard Number 1, relying on the study of Varieties of Minimal Rational Tangents (VMRT) from both the algebro-geometric and the G-structure perspectives. Based on this theory, Ngaiming Mok and Jaehyun Hong studied the standard embedding between two Rational Homogeneous Spaces (RHS) associated to long simple roots which are of different dimensions. In this thesis, I consider admissible pairs of RHS (X0, X) of Picard number 1 and locally closed complex submanifolds S ⊂ X inheriting VMRT sub-structures modeled on X0 = G0/P0 ⊂ X = G/P de_ned by taking intersections of VMRT of X with tangent space of S. Moreover, if any such S modeled on (X0, X) is necessarily the image of a standard embedding i : X0 → X, (X0, X) is said to be rigid. In this thesis, it is proved that an admissible pair (X0, X) is rigid whenever X is associated to a long simple root and X0 is non-linear and de_ned by a marked Dynkin sub-diagram. In the case of the pair (S0, S) of compact Hermitian Symmetric Spaces (cHSS), all the admissible pairs (S0, S) are completely classified. Based on this classification, a sufficient condition for the pair (S0, S) to be non-rigid is established through explicitly constructing a submanifold S ⊂ S such that S can never be obtained from the image of any standard embedding i : S0 → S. Besides, the term special pair is coined for those (S0; S) sorted out through classification, and the algebraicity of submanifolds modeled on special pairs is confirmed by checking a modified form of the non-degeneracy condition defined by Hong and Mok is satisfied. However, the question as to whether these special pairs are rigid, as pointed out in this thesis, remains to be investigated. Finally, pairs of hyperquadrics (Q^n, Q^m) are studied separately. Since non-rigidity is trivial, in these cases it is interesting to establish a characterization of the standard embedding i : Q^n→Q^m under some stronger condition. In this thesis, the latter problem is solved in terms of the partial vanishing of second fundamental forms. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thèse - Clermont-Ferrand. / Bibliography: l. [92].

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1989. / Includes bibliographical references (p. 103-105).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1980. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 94-95).