Area and Gauss–Bonnet inequalities with scalar curvature

Misha Gromov, Jintian Zhu

Research output: Contribution to journalArticlepeer-review


The Gauss–Bonnet theorem states for any compact surface (S, g) that the quantity QGBS (S) = ∫S Sc(S, s) ds + ∫S mean:curv.(S, b) db - 4π χ(S) vanishes identically. Let (X, g) be a compact Riemannian manifold of dimension n ≥ 3 with smooth boundary, associated with a continuous map f = (f1,...., fn-2): X → [0, 1]n-2, where Lip fi ≤ di-1 for positive constants di . For a universal constant Cn(di) depending only on di and n, we show that there is a compact surface ∑ homologous to the f -pullback of a generic point such that each component S of ∑ satisfies QGBX (S) ≤ Cn(di) · area(S), where QGBX (S) = ∫S Sc(X, s) ds ∫S mean.curv.(X, b) db -4π χ(S). As corollaries, if X has “large positive” scalar curvature, we prove in a variety of cases that if X “spreads” in (n - 2) directions “distance-wise”, then it cannot much “spread” in the remaining 2-directions “area-wise”.

Original languageEnglish (US)
Pages (from-to)355-395
Number of pages41
JournalCommentarii Mathematici Helvetici
Issue number2
StatePublished - 2024


  • Gauss–Bonnet inequality
  • area inequality
  • scalar curvature
  • μ-bubble method

ASJC Scopus subject areas

  • General Mathematics


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