AI 中文总结
该研究针对完备黎曼流形，在Ric≥0及(m+1)-中间曲率Cₘ₊₁≥1的条件下，推导了通用体积增长界，还得到m=n−2时的特殊情况结论。
AI 中文摘要
设整数n≥2，0≤m≤n−2，(Mⁿ,g)为完备黎曼流形，Cₘ₊₁是Brendle–Hirsch–Johne引入的(m+1)-中间曲率。我们证明：若Ric≥0且Cₘ₊₁≥1，则对任意p∈M和R>0，体积Vol B_R(p)≤C(n,m)Rᵐ。特别地，当m=n−2时，若Ric≥0且Scal≥1，则Vol B_R(p)≤C(n)Rⁿ⁻²。
英文摘要
Let $n,m$ be integers such that $n\geq2$ and $0\leq m\leq n-2$. Let $C_{m+1}$ denote the $(m+1)$-intermediate curvature introduced by Brendle--Hirsch--Johne. We prove that there are constants $ν(n,m),C(n,m)>0$ such that the following holds. If $(M^n,g)$ is complete and connected and, for $δ\geq 0$, \[ \mathrm{Ric}\geq-δ^2, \qquad C_{m+1}\geq 1, \] then \[ δR\leqν(n,m) \quad\Longrightarrow\quad \mathrm{Vol} B_R(p) \leq C(n,m)R^m \quad \text{for every $p\in M$ and $R>0$.} \] In particular, taking $m=n-2$ and $δ=0$ gives Gromov's conjectured codimension-two volume growth estimate under $\mathrm{Ric} \geq0$ and $\mathrm{Scal} \geq1$.
Comments15 pages. Minor revision. Theorems 1.2 and 1.4 are generalized beyond the nonnegative-Ricci setting of v1, requiring only minor changes to their proofs. Added Theorem 1.6. Expanded the Acknowledgments, Disclosure of AI tools, and Addendum; other changes are expository
