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arxiv: 2607.01142 · v1 · pith:UI36MVYInew · submitted 2026-07-01 · 🧮 math.CV

On the geometry of locally growing Loewner chains

Eveliina Peltola , Anne Schreuder This is my paper

Pith reviewed 2026-07-02 02:37 UTC · model grok-4.3

classification 🧮 math.CV
keywords Loewner chainsdriving functiongenerating functionhull growthpath-connectednesslocal connectednessconformal mapsboundary behavior
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The pith

Left-continuity of the generating function ensures path-connected and locally connected hulls with right limits in locally growing Loewner chains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies topological features of hulls arising from Loewner chains that obey a general local growth property. It first revisits Loewner's theorem to associate any such collection of hulls with a real-valued driving function W that may fail to be continuous. The authors then introduce the notion of a generating function η that records the points added chronologically to the hulls and can be continuous, càdlàg, càglàd, or neither. They prove that left-continuity of η forces the hulls to be path-connected and locally connected and guarantees the existence of right limits for η. By contrast, the absence of left-continuity produces pathological boundary behavior.

Core claim

For a collection of hulls satisfying the local growth property, a driving function W exists by the revisited Loewner theorem; a generating function η exists under further conditions on the chain; left-continuity of η implies that the growing hulls are path-connected and locally connected and that right limits of η exist, whereas failure of left-continuity produces pathological boundary behavior of the hulls.

What carries the argument

The generating function η, which encodes the chronological addition of points or compact boundary sets to the hulls and may possess any combination of left and right limits.

Load-bearing premise

The collection of hulls satisfies a general local growth property that permits association with a possibly discontinuous real-valued driving function via the revisited Loewner theorem.

What would settle it

Construct or exhibit a locally growing collection of hulls whose associated generating function is left-continuous yet whose hulls are neither path-connected nor locally connected.

Figures

Figures reproduced from arXiv: 2607.01142 by Anne Schreuder, Eveliina Peltola.

Figure 1.1
Figure 1.1. Figure 1.1: Illustration of the local growth property (Definition 1.1). 1.1. General theory of locally growing Loewner chains. By a variation of the arguments leading to Loewner’s classical theorem, we obtain a bijection between real-valued càdlàg functions and locally growing hulls in the following sense. This result was partly inspired by [PS25], where we study random Loewner chains whose driving functions are gen… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Illustration of locally growing hulls forming a bubble Bt. Theorem 1.4. Let K = (Kt)t≥0 be a family of left-locally growing hulls. Then, for all t ≥ 0, we have Kt ∪ R =  [ s<t Ks ∪ R  ∪ Bt ∪ Pt, where • Bt = T s<t Kt\Ks  \(∂Kt∪R) is a bubble, and it is either empty or open and simply connected; • Pt = (∂Kt ∩ H) \ S s<t Ks is compact and connected. Theorem 1.4 is an immediate consequence of Theorem 4.6… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Illustration of spiraling hulls (Example 1.10). 1.3. Examples and counterexamples. Let us illustrate some possible behaviors for the growing hulls. Example 1.10 (The logarithmic spiral: Continuous driver — no generating function). See also Fig￾ure 1.3. In [MR05], Marshall & Rohde constructed a logarithmic spiral spinning around a circle as an example of a Loewner chain which has a (1/2-Hölder) continuous… view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: Illustration of a path-connected but not locally connected comb (Example 1.11). Example 1.11 (The comb space: Càdlàg driver — non-locally connected frontier). See also [PITH_FULL_IMAGE:figures/full_fig_p008_1_4.png] view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: Illustration of Example 1.12 (inspired by [Bel19, [PITH_FULL_IMAGE:figures/full_fig_p009_1_5.png] view at source ↗
Figure 1.6
Figure 1.6. Figure 1.6: Illustration of a path-connected but not locally connected double-comb (Example 1.13). i.e., the infimum of all times witnessing that behavior. We apply it to investigate path-connectedness of the hulls (see Proposition 4.13), which will be needed in Section 5. In the final Section 5, we consider topological properties of Loewner hulls that are generated by a func￾tion. Firstly, we show that a generating… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Illustration for the proof of Proposition 3.15. Furthermore, C in δ := ∂B(W(t−), r) ∩ H is a crosscut separating gt−δ(Kt \ Kt−δ) from ∞ in H. Using this, the fact that g −1 t−δ is a conformal bijection, and by the finite length estimate (3.9), we see that S in δ := g −1 t−δ (Cin δ ∩ H) is a crosscut in H \ Kt−δ separating Kt \ Kt−δ from ∞ in H \ Kt−δ with diam(S in δ ) < ε. This proves that the hulls K a… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Illustration for the proof of Proposition 3.16. 3.2. Proof of Loewner’s theorem: Constructing the driving function. In this section, we prove that every family (Kt)t≥0 of locally growing hulls admits a càdlàg function W : [0,∞) → R such that the mapping-out functions gt : H \ Kt → H solve the Loewner equation (LE) with driving function W. This proves the direction (1) =⇒ (2) in Theorem 1.2 and shows that… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Illustration of locally growing hulls with a closing bubble (Proposition 4.4). In the course of the proof, we have in fact obtained a slightly stronger result: Corollary 4.5. Let K be a family of left-locally growing hulls. Then, for all t > 0, we have Bt =  \ s<t Kt \ Ks  \ (∂Kt ∪ R) and Bt ⊂ \ s<t Kt \ Ks ⊂ Bt ∪ ∂Kt ∪ R. Proof. Proposition 4.4 gives  \ s<t Kt \ Ks  \ (∂Kt ∪ R) ⊂ Bt :=  \ s<t Kt \ … view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Locally growing hulls generated by a simple curve. Remark 4.8. By [LMR10, Section 5], every open, simply connected, and bounded set A ⊂ H can be the bubble of a locally growing Loewner chain. In fact, [LMR10] establishes a stronger claim: For a compact, connected set A ⊂ H, there exists a sufficiently smooth spiral winding infinitely often around A with limit set ∂A. In the setting of Theorem 4.6, if we … view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Illustration for the proof of Lemma 4.10: We have z ∈ S s<t Ks ∪ R. Proof. Let (Sn)n∈Z≥0 be a null-chain representing z. Then, for every n ∈ Z≥0, there exists εn > 0 such that Sn ⊂ B(z, εn) and εn → 0 as n → ∞. Moreover, because (Sn)n∈Z≥0 represents the grown end at time t, for every n ∈ Z≥0, there exists δn > 0 such that Sn is a crosscut in H \ Kt−δn separating Kt \ Kt−δn from ∞, and δn → 0 as n → ∞. In… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Illustration for the proof of Lemma 5.9. The following result generalizes [Bel19, Theorem 5.22], which proves that a locally connected Loewner chain with a continuous driving function is generated by a right-continuous function. Theorem 5.10. Let K = (Kt)t≥0 be a family of locally growing hulls. Let W : [0,∞) → R be the associated càdlàg driving function. Assume that the radial limits (5.5) exist for all… view at source ↗
read the original abstract

Loewner chains are ubiquitous in the theory of slit mappings, and hence in the study of bounded conformal maps. They have attracted new interest in the past decades through their applications to statistical physics and fractal geometry, particularly in contexts involving randomness. In this article, we delve into topological features of the growing hulls obtained from Loewner chains with a general local growth property, inspired by the classical works of Loewner and Pommerenke. We first revisit Loewner's theorem, associating to each locally growing collection of hulls a real-valued driving function W, possibly discontinuous. We then investigate the points chronologically added to the growing hulls, which may be part of a simply connected swallowed ``bubble'', or a compact connected boundary set. For continuous driving functions, the Loewner chain can often be associated with a continuous curve (dubbed ``generating curve''). Motivated by this, we introduce a more general notion of a ``generating function'' for the Loewner chain, and characterize when there exists such a function {\eta} (which can be continuous, c\`adl\`ag, c\`agl\`ad, or neither). We then investigate the necessity of left and right limits for {\eta} from the point of view of the topology of the growing hulls. We find in particular that left-continuity implies path-connectedness and local connectedness of the hulls, as well as the existence of right limits, whereas failure of left-continuity leads to pathological boundary behavior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper revisits Loewner's theorem to associate a (possibly discontinuous) real-valued driving function W to any collection of hulls satisfying a general local growth property. It introduces and characterizes a generating function η (which may be continuous, càdlàg, càglàd, or neither) for the Loewner chain, then proves that left-continuity of η implies path-connectedness and local connectedness of the hulls together with existence of right limits, while failure of left-continuity produces pathological boundary behavior.

Significance. If the derivations hold, the results supply a precise topological dictionary between regularity properties of the generating function and geometric features of the hulls in the discontinuous setting. This extends classical Loewner–Pommerenke theory in a manner directly relevant to random or irregular growth models in statistical physics and fractal geometry.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise formulation of the 'general local growth property' (including any measurability or capacity-normalization assumptions) rather than referring only to its inspiration from classical works.
  2. Notation for the generating function η and its left/right limits should be introduced with a dedicated paragraph or displayed definition early in the manuscript to avoid ambiguity when the function is neither càdlàg nor càglàd.
  3. The characterization of when a generating function exists (continuous or otherwise) would benefit from a concise summary table or flowchart relating regularity classes of W to those of η.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, accurate summary of our results on Loewner chains with local growth, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a self-contained theoretical development in complex analysis that revisits the classical Loewner association of hulls with a driving function under an explicit local growth hypothesis and then derives topological consequences for the hulls from continuity properties of the generating function η. All steps are presented as direct consequences of the stated assumptions and standard results in conformal mapping theory; no parameter fitting, self-referential definitions, load-bearing self-citations, or imported uniqueness theorems appear in the derivation chain. The central implication (left-continuity of η implies path-connectedness and existence of right limits) is framed as a logical consequence rather than a reduction to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on standard complex analysis background and introduces one new conceptual object.

axioms (1)
  • standard math Loewner's theorem associating each locally growing collection of hulls with a real-valued driving function W (possibly discontinuous)
    Revisited at the start of the paper as the foundation for the subsequent analysis.
invented entities (1)
  • generating function η no independent evidence
    purpose: To track chronologically added points to the growing hulls and characterize their regularity
    New object introduced to generalize the notion of a generating curve for continuous driving functions.

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