Steklov eigenvalues on annulus

Chengjie Yu; Luen-fai Tam; Xu-Qian Fan

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Steklov eigenvalues on annulus

Xu-Qian Fan, Luen-Fai Tam, Chengjie Yu

Differential Geometry · Submitted 29 Oct 2013

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

The authors' main claim We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,210 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.82 · human participants 0.01 · controlled comparison 0.23 · randomized assignment 0.01 · states quantitative result 0.27 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We also obtain some partial results on the comparison of the normalized Stekov eigenvalues of rotationally symmetric metrics and general conformal metrics on the cylinder. A counter example is constructed to show that for that the first normalized Steklov eigenvalue of rotationally symmetric metric may not be larger.

background contribution method result interpretation main claim

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	We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1.	result 0.73	0.96	0.87	asserted 1.00
s1	The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen.	background 0.99	0.03	0.03	asserted 1.00
s2	We give geometric description in terms of minimal surfaces for metrics attaining the supremum.	result 0.57	0.95	0.13	asserted 1.00
s3	We also obtain some partial results on the comparison of the normalized Stekov eigenvalues of rotationally symmetric metrics and general conformal metrics on the cylinder.	result 0.54	0.93	0.06	asserted 0.91
s4	A counter example is constructed to show that for that the first normalized Steklov eigenvalue of rotationally symmetric metric may not be larger.	result 0.90	0.93	0.06	hedged possibility 0.50

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

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arXiv listing abstract v1

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We also obtain some partial results on the comparison of the normalized Stekov eigenvalues of rotationally symmetric metrics and general conformal metrics on the cylinder. A counter example is constructed to show that for that the first normalized Steklov eigenvalue of rotationally symmetric metric may not be larger.

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1 arXiv paper cite this in 1 sentence

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

Cites it as background 1 paper

Higher eigenvalues on annuli have been studied in [ref] .

Spectral geometry of the Steklov problem · Surfaces with large Steklov eigenvalues · 0.88

What it cites · 3 sentences without a confident label

Uses the method or tool of 1 paper

In [ref] , by using minimal surfaces as in [ref] , Fraser and Schoen showed that the maximum of the first normalized Steklov eigenvalues for the annulus is achieved by the critical catenoid.

Minimal surfaces and eigenvalue problems · Introduction · 0.80

Cites as background 2 papers

There have been many works on estimating the Steklov eigenvalues, see [ref] and references therein.

Sharp eigenvalue bounds and minimal surfaces in the ball · Introduction · 1.00
There have been many works on estimating the Steklov eigenvalues, see [ref] and references therein.

Minimal surfaces and eigenvalue problems · Introduction · 1.00

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