Area and Gauss–Bonnet inequalities with scalar curvature

Abstract

The Gauss–Bonnet theorem states for any compact surface $(S,g)$ that the quantity

vanishes identically. Let $(X,g)$ be a compact Riemannian manifold of dimension $n\ge 3$ with smooth boundary, associated with a continuous map $f=(f_1,\dots ,f_{n-2}):X\to [0,1{]}^{n-2}$, where $\mathrm{Lip}{f}_i\le d_i^{-1}$ 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 $\Sigma$ homologous to the $f$-pullback of a generic point such that each component $S$ of $\Sigma$ satisfies $Q_{GB}^X(S)\le C_n(d_i)\cdot \mathrm{area}(S)$, where

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”.