Metric Flips with Calabi Ansatz

Jian Song; Yuan Yuan

doi:10.1007/s00039-012-0151-1

Metric Flips with Calabi Ansatz

Jian Song, Yuan Yuan

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

Abstract

We study the limiting behavior of the Kähler-Ricci flow on ℙ(O _ℙn ⊕ O _ℙn(-1) ^⊕(m+1)) for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to ℙ ⁿ or contracts a subvariety of codimension m + 1 in the Gromov-Hausdorff sense. We also show that the Kähler-Ricci flow resolves a certain type of cone singularities in the Gromov-Hausdorff sense.

Original language	English (US)
Pages (from-to)	240-265
Number of pages	26
Journal	Geometric and Functional Analysis
Volume	22
Issue number	1
DOIs	https://doi.org/10.1007/s00039-012-0151-1
State	Published - Feb 2012
Externally published	Yes

Keywords

Gromov-Hausdorff convergence
Kähler-Ricci flow
flip
small contraction

ASJC Scopus subject areas

Analysis
Geometry and Topology

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10.1007/s00039-012-0151-1

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title = "Metric Flips with Calabi Ansatz",

abstract = "We study the limiting behavior of the K{\"a}hler-Ricci flow on ℙ(O ℙn ⊕ O ℙn(-1) ⊕(m+1)) for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to ℙ n or contracts a subvariety of codimension m + 1 in the Gromov-Hausdorff sense. We also show that the K{\"a}hler-Ricci flow resolves a certain type of cone singularities in the Gromov-Hausdorff sense.",

keywords = "Gromov-Hausdorff convergence, K{\"a}hler-Ricci flow, flip, small contraction",

author = "Jian Song and Yuan Yuan",

note = "Funding Information: The first named author is supported in part by National Science Foundation grant DMS-0847524 and a Sloan Foundation Fellowship.",

year = "2012",

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doi = "10.1007/s00039-012-0151-1",

language = "English (US)",

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T1 - Metric Flips with Calabi Ansatz

AU - Song, Jian

AU - Yuan, Yuan

N1 - Funding Information: The first named author is supported in part by National Science Foundation grant DMS-0847524 and a Sloan Foundation Fellowship.

PY - 2012/2

Y1 - 2012/2

N2 - We study the limiting behavior of the Kähler-Ricci flow on ℙ(O ℙn ⊕ O ℙn(-1) ⊕(m+1)) for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to ℙ n or contracts a subvariety of codimension m + 1 in the Gromov-Hausdorff sense. We also show that the Kähler-Ricci flow resolves a certain type of cone singularities in the Gromov-Hausdorff sense.

AB - We study the limiting behavior of the Kähler-Ricci flow on ℙ(O ℙn ⊕ O ℙn(-1) ⊕(m+1)) for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to ℙ n or contracts a subvariety of codimension m + 1 in the Gromov-Hausdorff sense. We also show that the Kähler-Ricci flow resolves a certain type of cone singularities in the Gromov-Hausdorff sense.

KW - Gromov-Hausdorff convergence

KW - Kähler-Ricci flow

KW - flip

KW - small contraction

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SP - 240

EP - 265

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

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Metric Flips with Calabi Ansatz

Abstract

Keywords

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