Rigidity of asymptotically conical shrinking gradient Ricci solitons

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

We show that if two gradient shrinking Ricci solitons are asymptotic along some end of each to the same regular cone ((0, ∞) x ∑, dr<sup>2</sup> + r<sup>2</sup> g<inf>∑</inf>), then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on ℝ<sup>n</sup>.

Original languageEnglish (US)
Pages (from-to)55-108
Number of pages54
JournalJournal of Differential Geometry
Volume100
Issue number1
StatePublished - May 1 2015

Fingerprint

Ricci Soliton
Shrinking
Rigidity
Solitons
Gradient
Symmetric Cone
Metric
Isometric
Geodesic
Completeness
Cone
Infinity
Restriction
Theorem

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Analysis
  • Geometry and Topology

Cite this

Rigidity of asymptotically conical shrinking gradient Ricci solitons. / Kotschwar, Brett; Wang, Lu.

In: Journal of Differential Geometry, Vol. 100, No. 1, 01.05.2015, p. 55-108.

Research output: Contribution to journalArticle

@article{2ecd7a67db9f45dab811aa2480ae1881,
title = "Rigidity of asymptotically conical shrinking gradient Ricci solitons",
abstract = "We show that if two gradient shrinking Ricci solitons are asymptotic along some end of each to the same regular cone ((0, ∞) x ∑, dr2 + r2 g∑), then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on ℝn.",
author = "Brett Kotschwar and Lu Wang",
year = "2015",
month = "5",
day = "1",
language = "English (US)",
volume = "100",
pages = "55--108",
journal = "Journal of Differential Geometry",
issn = "0022-040X",
publisher = "International Press of Boston, Inc.",
number = "1",

}

TY - JOUR

T1 - Rigidity of asymptotically conical shrinking gradient Ricci solitons

AU - Kotschwar, Brett

AU - Wang, Lu

PY - 2015/5/1

Y1 - 2015/5/1

N2 - We show that if two gradient shrinking Ricci solitons are asymptotic along some end of each to the same regular cone ((0, ∞) x ∑, dr2 + r2 g∑), then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on ℝn.

AB - We show that if two gradient shrinking Ricci solitons are asymptotic along some end of each to the same regular cone ((0, ∞) x ∑, dr2 + r2 g∑), then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on ℝn.

UR - http://www.scopus.com/inward/record.url?scp=84928957644&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928957644&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84928957644

VL - 100

SP - 55

EP - 108

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 1

ER -