The Kähler-Ricci flow on surfaces of positive Kodaira dimension
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If B, s, F are defined as in [50] and g denotes the horizontal component of the incomplete semi-flat metric on B, then F|ds|^2 = e^ψ g with ψ∈C∞(B), so that F∼|s|^{-α} for some α>0.
“Indeed, if we define B, s, F as in [50] and let g denote the horizontal component of the incomplete semi-flat metric on B, then F |ds| 2 = e ψ g with ψ ∈ C ∞ (B), so that F ∼ |s| -α for some α > 0.”
From Gravitational instantons from rational elliptic surfaces (2011) · cites this paper as [50]
Section 5.3 defines F on a disk B and studies its behaviour near singular fibres. But Lemma 5.4 (pp. 24–25) gives different behaviours by fibre type: F is bounded for some types and grows logarithmically for type Ib. Those cases contradict the unqualified conclusion that F is comparable to |s|^{-α} for some α>0; the cited passages also do not state the asserted identity with the horizontal metric. (LLM, conf=0.86)
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Experimental — accuracy not yet measuredAccording to Table 1, the estimate (3.17) in Song-Tian [50] is incorrect, but the error is irrelevant for the rest of that paper.
“According to Table 1, the estimate (3.17) in Song-Tian [50] is incorrect, but the error is irrelevant for the rest of that paper.”
From Gravitational instantons from rational elliptic surfaces (2011) · cites this paper as [50]
The copy’s Section 3 (pp. 7–8) contains estimates numbered (3.1)–(3.5), but searches for (3.17) did not locate the cited estimate. The copy also does not address whether an error in that estimate would be irrelevant to the rest of the paper. The citing authors attribute their assessment to their own Table 1, which is not evidence available from this source. (LLM, conf=0.76)
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Cites
Barth, W. P., Hulek, K., Peters, C. A. M., & Van de Ven, A. (2004) · Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics · doi:10.1007/978-3-642-57739-0 · Cited by 978 · see citing works
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Research works printed but not linked: 2
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Singular Kähler-Einstein metrics
Eyssidieux, P., Guedj, V., & Zeriahi, A. (2009) · Journal of the American Mathematical Society · doi:10.1090/s0894-0347-09-00629-8 · Cited by 305 · see citing works
Gravitational instantons from rational elliptic surfaces
Hein, H.-J. (2011) · Journal of the American Mathematical Society · doi:10.1090/s0894-0347-2011-00723-6 · Cited by 87 · see citing works
- Authors
- Song, J., & Tian, G.
- Affiliations
- Johns Hopkins University (US)
- Princeton University (US)
- Corresponding author
- Tian, Gang
- ORCID
- Venue
- Inventiones mathematicae
- Published
- 2007-09-19
- Volume
- 170
- Issue
- 3
- Pages
- 609-653
- ISSN
- 0020-9910 (ISSN-L)
- 1432-1297
- OpenAlex
- W2147197923
- Cited by
- 162
- References
- 1
- Citations by year
- 2026: 2 · 2025: 8 · 2024: 6 · 2023: 5 · 2022: 7 · 2021: 9 · 2020: 14 · 2019: 8 · 2018: 13 · 2017: 13 · 2016: 10 · 2015: 14 · 2014: 8 · 2013: 11 · 2012: 5
- FWCI
- 10.49
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Inventiones mathematicae (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Johns Hopkins University
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometry and complex manifolds
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Geometry and Topology › Algebraic Geometry and Number Theory
- Keywords
- Kähler-Ricci flow, minimal surfaces, Kähler-Einstein metrics, canonical models, canonical line bundle, metric classification, projective varieties
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