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The associated families of semi-homogeneous complete hyperbolic affine spheres

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Zhicheng Lin, Erxiao Wang

Differential Geometry · Submitted 10 Apr 2015

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

The authors' main claim After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand's examples.

how it is stated · asserted 1.00 read the record →

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arXiv ID1504.02551 (version 1)
Pith IDRNNRIPJO, Pith's permanent record of this paper and its review
CitationErxiao Wang, Zhicheng Lin. "The associated families of semi-homogeneous complete hyperbolic affine spheres." arXiv:1504.02551 (2015). https://pith.science/paper/RNNRIPJO
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,246 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.48 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Hildebrand classified all semi-homogeneous cones in $\mathbb{R}^3$ and computed their corresponding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand's examples. The generic member of these affine spheres is given by Weierstrass $\mathscr{P}, ~\zeta ~\text{and} ~\sigma$ functions. In general any regular convex cone in $\mathbb{R}^3$ has a natural associated $S^1$-family of such cones, which deserve further studies.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Hildebrand classified all semi-homogeneous cones in $\mathbb{R}^3$ and computed their corresponding complete hyperbolic affine spheres. background 1.00 0.03 0.06 asserted 1.00
s1 We compute isothermal parametrizations for Hildebrand's new examples. contribution 0.88 0.96 0.09 asserted 1.00
s2 After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand's examples. contribution 0.81 0.95 0.22 asserted 1.00
s3 The generic member of these affine spheres is given by Weierstrass $\mathscr{P}, ~\zeta ~\text{and} ~\sigma$ functions. result 0.95 0.80 0.27 asserted 1.00
s4 In general any regular convex cone in $\mathbb{R}^3$ has a natural associated $S^1$-family of such cones, which deserve further studies. future work 0.89 0.15 0.23 asserted 0.91

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

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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

Hildebrand classified all semi-homogeneous cones in $\mathbb{R}^3$ and computed their corresponding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand's examples. The generic member of these affine spheres is given by Weierstrass $\mathscr{P}, ~\zeta ~\text{and} ~\sigma$ functions. In general any regular convex cone in $\mathbb{R}^3$ has a natural associated $S^1$-family of such cones, which deserve further studies.

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