Sets of Minimal Capacity and Extremal Domains

Herbert R Stahl

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Sets of Minimal Capacity and Extremal Domains

Herbert R Stahl

Classical Analysis and ODEs · Submitted 16 May 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In the paper a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.79 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let f be a function meromorphic in a neighborhood of infinity. The central problem in the present investigation is to find the largest domain D \subset C to which the function f can be extended in a meromorphic and singlevalued manner. 'Large' means here that the complement C\D is minimal with respect to (logarithmic) capacity. Such extremal domains play an important role in Pad'e approximation. In the paper a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved. The topological structure of sets of minimal capacity is studied, and analytic tools for their characterization are presented; most notable are here quadratic differentials and a specific symmetry property of the Green function in the extremal domain. A local condition for the minimality of the capacity is formulated and studied. Geometric estimates for sets of minimal capacity are given. Basic ideas are illustrated by several concrete examples, which are also used in a discussion of the principal differences between the extremality problem under investigation and some classical problems from geometric function theory that possess many similarities, which for instance is the case for Chebotarev's Problem.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let f be a function meromorphic in a neighborhood of infinity.	background 1.00	0.06	0.01	asserted 0.92
s1	The central problem in the present investigation is to find the largest domain D \subset C to which the function f can be extended in a meromorphic and singlevalued manner.	background 0.78	0.28	0.05	asserted 0.99
s2	'Large' means here that the complement C\D is minimal with respect to (logarithmic) capacity.	background 0.86	0.21	0.04	asserted 0.99
s3	Such extremal domains play an important role in Pad'e approximation.	background 0.95	0.07	0.03	asserted 1.00
s4	In the paper a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved.	result 0.77	0.95	0.78	asserted 1.00
s5	The topological structure of sets of minimal capacity is studied, and analytic tools for their characterization are presented; most notable are here quadratic differentials and a specific symmetry property of the Green function in the extremal domain.	contribution 0.97	0.91	0.07	asserted 1.00
s6	A local condition for the minimality of the capacity is formulated and studied.	contribution 0.85	0.92	0.04	asserted 0.99
s7	Geometric estimates for sets of minimal capacity are given.	result 0.61	0.90	0.04	asserted 1.00
s8	Basic ideas are illustrated by several concrete examples, which are also used in a discussion of the principal differences between the extremality problem under investigation and some classical problems from geometric function theory that possess many similarities, which for instance is the case for Chebotarev's Problem.	contribution 0.54	0.83	0.04	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper

Original abstract

Let f be a function meromorphic in a neighborhood of infinity. The central problem in the present investigation is to find the largest domain D \subset C to which the function f can be extended in a meromorphic and singlevalued manner. 'Large' means here that the complement C\D is minimal with respect to (logarithmic) capacity. Such extremal domains play an important role in Pad'e approximation. In the paper a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved. The topological structure of sets of minimal capacity is studied, and analytic tools for their characterization are presented; most notable are here quadratic differentials and a specific symmetry property of the Green function in the extremal domain. A local condition for the minimality of the capacity is formulated and studied. Geometric estimates for sets of minimal capacity are given. Basic ideas are illustrated by several concrete examples, which are also used in a discussion of the principal differences between the extremality problem under investigation and some classical problems from geometric function theory that possess many similarities, which for instance is the case for Chebotarev's Problem.

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References

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Minimal degree rational open up mappings and related questions
math.CV 2019-08 conditional novelty 7.0 of 10

For any n disjoint smooth arcs, there exists a rational function of type (n+1,n) that maps the complement of a compact set onto the complement of the arcs conformally, with degree n+1 minimal.
Existence of a "maximal" domain of meromorphy for an analytic function outside a polar compact set
math.CV 2026-08 accept novelty 6.0 of 10

A correct proof is given for the existence of a compact set of minimal logarithmic capacity among all compact sets on whose complement a given analytic germ extends meromorphically.
Numerical computation of the Schwarz function
math.NA 2025-01 conditional novelty 6.0 of 10

The Schwarz function of a planar curve can be computed numerically by fitting a rational function with the AAA algorithm to the curve's boundary values.

How the literature uses it

9 arXiv papers cite this in 33 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 16 sentences without a confident label

Contests it 1 paper

This mistake was made in original paper [ref] , in [ref] Stahl referred to [ref] , and he repeated this mistake in [ref] .

Existence of a "maximal" domain of meromorphy for an analytic function outside a polar compact set · Introduction · contest 0.95

Uses its method or tool 1 paper

We show this using the $S$-property which was introduced and studied in detail by Stahl (see, e.g., [ref] and Definition [ref] below).

Extremal polynomials and polynomial preimages · Widom factors for $T_n^$ · 1.00

Compares with it 1 paper

Let us remark that Stahl uses closed trajectories (with endpoints included, see [ref] , p. 11) while Strebel uses open trajectories, i.e. endpoints are not part of trajectories, see [ref] , p. 25, Theorem 5.5.

Minimal degree rational open up mappings and related questions · Appropriate set of arcs for given rational function · 0.94

Cites it as support 2 papers

By [ref] , $$ is therefore the solution to the Chebotarev problem corresponding to the points $a$ and $b$.

Extremal polynomials and polynomial preimages · The $S$-property and Chebotarev sets · 0.97
Alternatively, see Theorem 4 from [ref] .

Minimal degree rational open up mappings and related questions · Appropriate set of arcs for given rational function · 0.97

Cites it as background 6 papers

Many results from the former part are contained in [ref] (sometimes without proofs) and many results from the latter part are contained in [ref] .

Existence of a "maximal" domain of meromorphy for an analytic function outside a polar compact set · Preliminary results · 1.00
Stahl in the 1985–1986s (see [ref] and [ref] and there references given therein) when considering the problem of the limit distribution of the zeros and poles of Padé approximants in the class of multivalued analytic functions $^∘(Σ)$.

On a new approach to the problem of the zero distribution of Hermite-Pad\'e polynomials for a Nikishin system · Section · 1.00
The set $S=S(f_*)$ is the so-called Stahl's compact set of minimal capacity (see [ref] and [ref] ).

Rational Hermite-Pad\'e Approximants vs Pad\'e Approximants. Numerical Results · Definition of diagonal Padé approximant. · 0.85

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