[Paper Review] Holomorphic evolution: metamorphosis of the Loewner equation

Filippo Bracci|arXiv (Cornell University)|Dec 13, 2011

Meromorphic and Entire Functions22 references3 citations

TL;DR

This paper presents a unified, geometric, and dynamical framework for the Loewner equation by generalizing classical Loewner theory through evolution families, Herglotz vector fields, and Loewner chains. It establishes that the Loewner range—defined as the abstract basin of attraction—can be determined via the asymptotic behavior of the Kobayashi pseudometric, with key results showing when the range is biholomorphic to C^n or a fiber bundle over a complex submanifold.

ABSTRACT

This is a survey on recent results on the Loewner theory in one and several complex manifolds

Motivation & Objective

• To unify and generalize classical Loewner theory by emphasizing the dynamical and geometric aspects of evolution families.
• To define Herglotz vector fields and evolution families in higher-dimensional complex manifolds using infinitesimal generators of semigroups.
• To characterize the Loewner range of an evolution family as an abstract basin of attraction via intertwining maps and geometric invariants.
• To establish criteria for the Loewner range to be biholomorphic to C^n or a fiber bundle over a complex submanifold using the Kobayashi pseudometric.
• To connect the Loewner theory to hyperbolic dynamics and random dynamical systems through the abstract basin of attraction.

Proposed method

• Formalizes evolution families as non-autonomous holomorphic semigroups on complex manifolds, generalizing the classical Loewner flow.
• Introduces $L^d$-Herglotz vector fields as time-dependent holomorphic vector fields generating evolution families via non-autonomous ODEs.
• Defines Loewner chains as intertwining maps conjugating evolution families to the dynamics on the Loewner range.
• Uses the Kobayashi pseudometric to analyze asymptotic behavior of tangent vectors under evolution, leading to the invariant $\beta^s_z(v)$.
• Applies the limit $\beta^s_z(v) = \lim_{t\to\infty} \kappa_M(\varphi_{s,t}(z); (d\varphi_{s,t})_z(v))$ to determine the biholomorphic type of the Loewner range.
• Applies results from hyperbolic geometry and automorphism groups (e.g., $\text{aut}(M)$) to classify Loewner ranges in complete hyperbolic manifolds.

Experimental results

Research questions

• RQ1 Under what conditions is the Loewner range of an evolution family biholomorphic to $\mathbb{C}^n$?
• RQ2 How can the Kobayashi pseudometric be used to characterize the geometry of the Loewner range in higher dimensions?
• RQ3 What is the relationship between the dynamics of an evolution family and the biholomorphic type of its Loewner range?
• RQ4 When does an algebraic evolution family on the unit ball $\mathbb{B}^n$ admit a Loewner range containing an open domain in $\mathbb{C}^n$?
• RQ5 How does the abstract basin of attraction in Loewner theory relate to random dynamical systems and hyperbolic dynamics?

Key findings

• The Loewner range of an evolution family is biholomorphic to $\mathbb{C}^n$ if and only if $\beta^s_z(v) = 0$ for all non-zero $v \in T_zM$ at some $z$ and $s$.
• If $\dim_{\mathbb{C}}\{v \in T_zM : \beta^s_z(v) = 0\} = 1$, then the Loewner range is a fiber bundle with fiber $\mathbb{C}$ over a closed complex submanifold.
• For a complete hyperbolic manifold $M$ with compact quotient $M/\text{aut}(M)$, if $\beta^s_z(v) \neq 0$ for all non-zero $v$, then the Loewner range is biholomorphic to $M$.
• In the unit ball $\mathbb{B}^n$, if $\dim_{\mathbb{C}}\{v : \beta^s_z(v) = 0\} \leq 1$, then the Loewner range contains an open domain in $\mathbb{C}^n$.
• The pullback of the Kobayashi metric on the Loewner range satisfies $f_s^*\kappa_N(z;v) = \beta^s_z(v)$, establishing a direct link between dynamics and geometry.
• The construction of Loewner chains is categorical and provides the PDE Loewner equation in full generality, unifying classical and abstract formulations.