Area and Gauss–Bonnet inequalities with scalar curvature

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 = (f₁,...., 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”.