AI 中文总结
本研究针对四维乘积流形$\u211d S^2\times P$，在标量曲率下界、边界面平均曲率非负及二面角约束条件下，证明了其同伦与同调2-收缩均不超过$8\u03c0/\u03c3$的精确不等式，证实了Gromov的相关猜想。
AI 中文摘要
我们证明了四维乘积流形$\u211d S^2\times P$的一个精确2-收缩不等式，其中$P\ub282\u211d R^2$是任意凸多边形。设$h=g_{\u211d S^2}+g_{\u210e u}$为标准乘积度量。若$\u211d S^2\times P$上的黎曼度量$g$满足标量曲率$\u2265\u03c3>0$、所有余维为1的面上平均曲率非负，且二面角不大于$h$的对应二面角，则其同伦与同调2-收缩均至多为$\frac{8\u03c0}{\u03c3}$。该结果证实了Gromov的一个猜想。
英文摘要
We prove a sharp $2$-systolic inequality for four-dimensional products $\mathbb S^2\times P$, where $P\subset\mathbb R^2$ is an arbitrary convex polygon. Let $h=g_{\mathbb S^2}+g_{\mathrm{eu}}$ be the standard product metric. If a Riemannian metric $g$ on $\mathbb S^2\times P$ has scalar curvature $\geq σ>0$, nonnegative mean curvature on every codimension one face, and dihedral angles no larger than the corresponding dihedral angles of $h$, then both its homotopy and homology $2$-systoles are at most $ \frac{8π}σ.$ This confirms a conjecture of Gromov.
Comments18 pages. Comments are welcome
