A conformally invariant gap theorem characterizing CP2 via the Ricci flow

Research output: Contribution to journalArticle

Abstract

We extend the sphere theorem of Chang et al. (Publ Math Inst Ht Études Sci 98:105–434, 2003) to give a conformally invariant characterization of (CP2, gFS). In particular, we introduce a conformal invariant β(M4, [g]) ≥ 0 defined on conformal four-manifolds satisfying a ‘positivity’ condition; it follows from Chang et al.(2003) that if 0 ≤ β(M4, [g]) < 4 , then M4 is diffeomorphic to S4. Our main result of this paper is a ‘gap’ result showing that if b2+(M4)>0 and 4 ≤ β(M4, [g]) < 4 (1 + ϵ) for ϵ> 0 small enough, then M4 is diffeomorphic to CP2. The Ricci flow is used in a crucial way to pass from the bounds on β to pointwise curvature information.

Original languageEnglish (US)
Pages (from-to)721-746
Number of pages26
JournalMathematische Zeitschrift
Volume294
Issue number1-2
DOIs
StatePublished - Feb 1 2020

Fingerprint

Sphere Theorem
Conformal Invariants
Four-manifolds
Ricci Flow
Positivity
Curvature
Invariant
Theorem

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Bach flat
  • Ricci flow
  • Sphere theorem

Cite this

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abstract = "We extend the sphere theorem of Chang et al. (Publ Math Inst Ht {\'E}tudes Sci 98:105–434, 2003) to give a conformally invariant characterization of (CP2, gFS). In particular, we introduce a conformal invariant β(M4, [g]) ≥ 0 defined on conformal four-manifolds satisfying a ‘positivity’ condition; it follows from Chang et al.(2003) that if 0 ≤ β(M4, [g]) < 4 , then M4 is diffeomorphic to S4. Our main result of this paper is a ‘gap’ result showing that if b2+(M4)>0 and 4 ≤ β(M4, [g]) < 4 (1 + ϵ) for ϵ> 0 small enough, then M4 is diffeomorphic to CP2. The Ricci flow is used in a crucial way to pass from the bounds on β to pointwise curvature information.",
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A conformally invariant gap theorem characterizing CP2 via the Ricci flow. / Chang, Sun Yung A.

In: Mathematische Zeitschrift, Vol. 294, No. 1-2, 01.02.2020, p. 721-746.

Research output: Contribution to journalArticle

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AU - Chang, Sun Yung A.

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