Approximation of Biholomorphisms on the Spirallike Domains in Certain Complex Manifolds and its Applications

Chatterjee, Sanjoy (2024) Approximation of Biholomorphisms on the Spirallike Domains in Certain Complex Manifolds and its Applications. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Sanjoy Chatterjee (16IP002)) 16IP002.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

A domain Ω ⊆ Cn is said to be a Runge domain if every holomorphic function f on Ω can be approximated by holomorphic polynomials uniformly on every compact subset of Ω. Runge domains can also be defined for complex manifolds. A domain Ω on a complex manifold M is said to be Runge in M if every holomorphic function defined on Ω can be approximated by a holomorphic function defined on M uniformly on every compact subset of Ω. A domain Ω in a complex manifold M is said to be a spirallike domain with respect to a complete holomorphic vector field V on M if the integral curve of the vector field with the initial point inside the domain Ω remains inside for all nonnegative realtime. In this thesis, we have shown that if Ω ⊂ M is a domain that is spirallike with respect to a complete globally asymptotic stable vector field V , and contains the equilibrium point of the vector field V , then Ω is Runge in M. As a consequence, we obtain that if Ω ⊂ Cn contains the origin, and spirallike with respect to a complete holomorphic globally asymptotic stable vector field V with V (0) = 0 (i.e., the origin is the globally asymptotic stable equilibrium point of the vector field V ) then Ω is a Runge domain in Cn. The next problem that we deal with here is: whether a biholomorphism Φ : Ω → Φ(Ω) ⊆ Cn, Ω ⊆ Cn is a domain, can be approximated by automorphisms of Cn. The Aut(Cn) is studied by Rosay and Rudin, Andersén and Lempert. Andersén- Lempert proved that every biholomorphism from starshaped domain onto a Runge domain can be approximated by elements of automorphisms of Cn. It raises the natural question of finding the class of domains where every biholomorphism with Runge image can be approximated by elements of Aut(Cn). Forstneriˇc and Rosay gave the following sufficient condition: Let Ω ⊂ Cn be domain and Φ: Ω → Φ(Ω) be a biholomorphism onto a Runge domain. If the given map Φ can be connected via a C1- smooth isotopy to another biholomorphism Φ0, such that Φ0 can be approximated by elements of Aut(Cn) uniformly over every compact subset of Ω then the given map can be approximated by Aut(Cn) locally uniformly on Ω. Although it establishes a sufficient condition for the provided biholomorphism to be approximated by Aut(Cn), yet it raises the question of the existence of such an isotopy within a given domain, which is, in general, quite difficult. In 2015, Hamada proved that if a domain Ω ⊂ Cn contains the origin and spirallike with respect to certain matrix A ∈ GL(n,C) then every biholomorphism from the domain Ω onto a Runge domain can be approximated by elements of Aut(Cn). In this thesis, we substantially extended Hamada’s result when the vector field is not necessarily linear. Our proof demonstrates that if a domain is spirallike with respect to a certain complete holomorphic vector field, it leads to the same conclusion as Hamada’s result. Our result, when restricted to the linear vector field already generalizes Hamada’s. In 1997, Varolin introduced the notion of the density property of a complex manifold, which is a precise way of stating that the manifold has a large automorphism group. He demonstrated that analogous approximation results to those of Forstneriˇc and Rosay also apply in Stein manifolds exhibiting the density property. In this thesis, we are able to prove that if M is a Stein manifold with the density property and Ω ⊂ M is a spirallike domain with a certain complete holomorphic vector field on M then every biholomorphism from the domain Ω onto a Runge domain can be approximated by elements of Aut(M) uniformly on every compact subset of Ω. Next, we provide an application of our approximation result in Cn in the context of the Loewner PDE. We have proved that any Loewner PDE associated to a Herglotz vector field defined on a complete hyperbolic domain which is spirallike domain with respect to certain holomorphic vector fields admits a unique univalent solution with values in Cn. We will now focus on exploring the polynomial convexity of the closure of a bounded pseudoconvex domain. Recall that for a compact subset K ⊂ Cn the Polynomially convex hull of K, denoted by bK , is defined by bK := {z ∈ Cn : |p(z)| ≤ supw∈K |p(w)|, ∀p ∈ C[z1, z2, · · · , zn]} and K is polynomially convex if bK = K. The notion of polynomial convexity has an important role in the context of uniform approximation of continuous functions by holomorphic polynomials. In C a compact set K is polynomially convex if and only if C\K is path connected. But in higher dimensions, no such characterization exists. In general, it is difficult to determine whether a given compact subset is polynomially convex. Here, we prove that the closure of a bounded pseudoconvex domain, which is strictly spirallike with respect to a complete globally asymptotically stable holomorphic vector field, is polynomially convex. We also provide a necessary and sufficient condition, in terms of polynomial convexity, on a univalent function defined on a strongly convex domain for embedding it into a filtering Loewner chain. Afterward, we applied this result within the context of finding dense holomorphic curves and hypercyclic operators. We show that for any bounded pseudoconvex strictly spirallike domain Ω in Cn and given any connected complex manifold Y , there exists a holomorphic map from the unit disc to the space of all holomorphic maps from Ω to Y whose image is dense in O(Ω, Y ). This also yields us the existence of a O(Ω, Y )-universal map for any generalized translation on Ω, which implies the hypercyclicity of certain composition operators on O(Ω, Y ).

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