发表机构
University of Wollongong（伍伦贡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对满足正里奇曲率下界及改进数量曲率下界的闭黎曼流形，结合系数适配的雅可比比较与积分洗牌比较，得到改进体积界，其在一阶下与Bray猜想一致，还给出度量球的平均体积比较。
AI 中文摘要
我们研究满足正里奇曲率下界及改进数量曲率下界的闭黎曼流形的体积比较。利用Brown和Freedman提出的数量雅可比解的有限洗牌比较的积分扩展，我们证明：若闭黎曼流形$(N^n, g)$满足$\text{Ric}_g\boldsymbol{\text{保留原符号}}\boldsymbol{\text{≥}}(n-1)g$且数量曲率$R_g\boldsymbol{\text{保留原符号}}\boldsymbol{\text{≥}}n(n-1)(1+\boldsymbol{\text{ε}})$，则其体积满足$\boldsymbol{|N|_g}\boldsymbol{\text{保留原符号}}\boldsymbol{\text{≤}}\frac{1}{\boldsymbol{\text{√(1+nε)}}}\boldsymbol{|S^n|}$。实际上，仅假设$\text{Ric}_g\boldsymbol{\text{保留原符号}}\boldsymbol{\text{≥}}(n-1)g$，我们可证$\frac{|N|_g}{|\boldsymbol{S^n}|}\boldsymbol{\text{保留原符号}}\boldsymbol{\text{≤}}\frac{1}{|N|_g}\boldsymbol{\text{∫}}_N\boldsymbol{(\frac{R_g}{n-1}-(n-1))^{-1/2}}d\text{vol}_g$，等号当且仅当$N$与单位球面等距时成立。证明结合了系数适配的雅可比比较与积分洗牌比较，得到的估计保留了完整的里奇谱，所得体积界在$\boldsymbol{\text{ε}}$一阶下与Bray猜想预测的因子一致，该论证的另一结果是涉及数量曲率的度量球的平均体积比较。
英文摘要
We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. We prove that if a closed Riemannian manifold $(N^n, g)$ satisfies $\operatorname{Ric}_g\ge (n-1)g$ and the scalar curvature $R_g\ge n(n-1)(1+\varepsilon)$, then its volume satisfies $$\lvert N\rvert_g \le \frac{1}{\sqrt{1+n\varepsilon}}\lvert\mathbb S^n\rvert. $$ In fact, assuming only $\mathrm{Ric}_g\ge(n-1)g$, we can prove that $$\frac{|N|_g}{\left|\mathbb{S}^n\right|} \le \frac{1}{|N|_g} \int_N\left(\frac{R_g}{n-1}-(n-1)\right)^{-\frac{1}{2}} d \mathrm{vol}_g. $$ The equality holds if and only if $N$ is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with a new integral shuffling comparison for scalar Jacobi solutions, inspired by Brown and Freedman \cite{BrownFreedman2022}. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in $\varepsilon$ with the factor appearing in Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature. We also obtain a volume comparison theorem under a weighted integral lower bound on the Branson $Q$-curvature.
CommentsCorrected some typos and added an application to Q-curvature. 24 pages. Comments are welcome
