Area and Gauss-Bonnet inequalities with scalar curvature

DOI10.4171/CMH/570arXiv2112.07245OpenAlexW4392168398MaRDI QIDQ6385612

Jintian Zhu, Misha Gromov

Publication date: 14 December 2021

Abstract: Let $X$ be an $n$-dimensional Riemannian manifold with "large positive" scalar curvature. In this paper, we prove in a variety of cases that if $X$ "spreads" in $(n-2)$ directions {it "distance-wise"}, then it {it can't} much "spread" in the remaining 2-directions {it "area-wise".} Here is a geometrically transparent example of what we plan prove in this regard that illustrates the idea. Let $g$ be a Riemannin metric on $X=S^{2}imesmathbb{R}^{n-2}$, for which the submanifolds ${\displaystyle }$mbox {$mathbb{R}_s^{n-2}=simesmathbb{R}^{n-2}subsetX$ and $S_y^2=S^{2}imesysubsetX$}${\displaystyle }$ are {it mutually orthogonal} at all intersection points ${\displaystyle }$x=(s,y)in X=mathbb R_s^{n-2}cap S^2_y.${\displaystyle }$ (An instance of this is $g=g(s,y)=phi(s,y{)}^{2}d{s}^2+psi(s,y{)}^{2}d{y}^2$.) Let the Riemannian metric on $mathbb{R}_s^{n-2}$ induced from $(X,g)$, that is $g{|}_{mathbb{R}_s^{n-2}}$, be {it greater than the Euclidean} metric on $mathbb{R}_s^{n-2}=mathbb{R}^{n-2}$ for all $sin{S}^2$. (This is interpreted as "large spread" of $g$ in the $(n-2)$ Euclidean directions.) {sf If the {it scalar curvature of $g$ is strictly greater than that of the unit 2-sphere}, ${\displaystyle }$Sc(g) geq Sc(S^2)+varepsilon=2+varepsilon, mbox { }varepsilon>0,${\displaystyle }$ then, provided $nleq7$, } (this, most likely, is unnecessary) {sf there exists a smooth {it non-contractible} spherical surface $SsubsetX$, such that ${\displaystyle }$area(S)<area(S^2)=4pi.${\displaystyle }$} (This says, in a way, that $(X,g)$ "doesn't spread much area-wise" in the 2 directions complementary to the Euclidean ones.)

Full work available at URL: https://doi.org/10.4171/cmh/570

Mathematics Subject Classification ID

Global Riemannian geometry, including pinching (53C20) Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces (53C23)

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