Mid Sweden University

We first investigate the holomorphic extension of some classical and accessible classes of domains in $\mathbb{C}^n$ to know to what extent their envelopes are constructive. We give alternative proofs to some of the classical theorems using only first-principle arguments and without involving higher Stein geometry. Next, we address an open problem asked by M. Jarnicki/P. Pflug and construct a counter-example to provide a negative answer to a related open question asked by J. Noguchi, which asks whether the envelopes of holomorphy of truncated tube domains are always schlicht. We also provide a sufficient condition for schlichtness of a tube domain $X+iY$ in $\mathbb{C}^2$, for which $X\subset \mathbb{R}^2$ is a convex domain consisting of finitely many holes with strictly convex $\mathcal{C}^2$-boundary, and $Y\subset \mathbb{R}^2$ is a convex domain.

• We investigate the holomorphic extension of some classical and accessible classes of domains in ℂⁿ to determine to what extent their envelopes are constructive. We give alternative proofs of some of the classical theorems using only first-principle arguments and without involving higher Stein geometry.
• We address an open problem raised by M. Jarnicki and P. Pflug and construct a counterexample to provide a negative answer to their question. We prove that the envelopes of holomorphy of truncated tube domains need not always be schlicht, and in fact can be infinite-sheeted. We also provide a sufficient condition for the schlichtness of a tube domain X+iY in ℂ², where X ⊂ ℝ² is a domain obtained by removing, from a convex domain, finitely many strictly convex holes with C² boundaries, and Y ⊂ ℝ² is a convex domain.
• We prove that if X = X_σ is the affine toric variety corresponding to an affine simplicial strictly convex cone σ, then for every Reinhardt domain D ⊂ X, there is a Stein Reinhardt domain D̂ ⊂ X that is a holomorphic extension of D. In particular, D̂ is the schlicht envelope of holomorphy of D. Moreover, we prove that there is a finite subgroup Γ of GL_n(ℂ), a morphism π: ℂⁿ → X, and an isomorphism φ: ℂⁿ/Γ → X such that π = φ ∘ π_Γ, where π_Γ: ℂⁿ → ℂⁿ/Γ is the quotient mapping. With this, we conclude that for every Reinhardt domain D ⊂ X, the domain π⁻¹(D) is Reinhardt in ℂⁿ.
• We introduce special domains and discuss the geometry of these domains. We show that every pseudoconvex truncated tube domain is a special domain. One of our main theorems establishes the schlichtness of the envelope of special domains in ℂⁿ (n ≥ 2) and also generalizes Jarnicki-Pflug's theorem. We provide two additional higher-dimensional generalizations of this result by ensuring the schlichtness of the envelope of tube domains.
• Finally, we introduce a collection of new open problems in the theory of the envelope of holomorphy and the schlichtness phenomenon. In particular, some of our problems focus on the class of truncated tube domains. By listing these well-posed open questions, we highlight a strategic gap for future research.

Some of the results presented in this thesis were previously published in the author's licentiate thesis.