AI 中文总结
该研究针对维数$n\geq3$的黎曼流形，在里奇曲率与标量曲率满足特定条件时，证明其体积不超过标准$n$维球面体积，从而证实了1997年提出的Bray猜想。
AI 中文摘要
设$(M^n,g)$是维数$n\geq3$的连通、闭、光滑黎曼流形，存在正常数$\varepsilon_n<1$，若其里奇曲率$\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$且标量曲率$R_g\geq n(n-1)$，则其体积$V_g(M^n)$不超过标准$n$维球面的体积，这证实了Bray在1997年提出的猜想。
英文摘要
Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.
CommentsThis is our submitted version to a journal. This version includes the rigidity in the equality case and an application section
