## Abstract

The Gauss–Bonnet theorem states for any compact surface (S, g) that the quantity Q_{GB}^{S} (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 = (f_{1},...., f_{n-2}): X → [0, 1]^{n-2}, where Lip f_{i} ≤ d_{i}^{-1} for positive constants d_{i} . For a universal constant C_{n}(d_{i}) depending only on d_{i} 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 Q_{GB}^{X} (S) ≤ C_{n}(d_{i}) · area(S), where Q_{GB}^{X} (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 language | English (US) |
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Pages (from-to) | 355-395 |

Number of pages | 41 |

Journal | Commentarii Mathematici Helvetici |

Volume | 99 |

Issue number | 2 |

DOIs | |

State | Published - 2024 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics