非负曲率三维流形上的数量曲率增长
Scalar curvature growth on nonnegatively curved three-manifolds
- Shantou University（汕头大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对完备非紧非负截面曲率三维流形，通过水平集论证给出数量曲率积分的渐近上界，单端情形上极限不超过8π(1-渐近体积比)，双端情形极限不超过16π，且常数均可达到。
AI中文摘要：
设 $(M^3,g)$ 为完备、连通、非紧且无边界，并具有非负截面曲率的三维流形。我们给出一个水平集论证，用于数量曲率积分的渐近上界。在单端情形下，该论证得出 \\[ \limsup_{r\to\infty}\frac1r\int_{B_p(r)}\operatorname{Scal}\\,d V \le 8\pi\bigl(1-{V_M}\bigr). \\] 不假设极点、非塌缩条件或数量曲率界。证明使用一个几乎凸的光滑穷竭函数以及距离函数的独立光滑逼近。水平集的负曲率误差是可积的。在双端情形下，分裂定理给出一个实际极限，其上界为 $16\pi$。对于 $[0,1]$ 中每个给定的渐近体积比，单端上界均可达到；双端常数也可达到。单端结论是一个上极限界；不断言相应极限的存在性。
英文摘要：
Let $(M^3,g)$ be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, \[ \lim_{r\to\infty}\frac1r\int_{B_p(r)}\operatorname{Scal}\,d V =8π\bigl(χ(M)-V_M\bigr)\le8π(1-V_M), \] where $V_M$ is the asymptotic volume ratio. In the one-ended case, we show that $χ(M)\in\{0,1\}$. Thus positive asymptotic volume ratio gives the value $8π(1-V_M)$, while in the collapsed case the value is determined by the topology of the end. No pole or scalar-curvature bound is assumed. The proof combines separate smooth approximations of the Busemann and distance functions, integrable negative curvature errors, and a determinant estimate on the level surfaces. An averaged boundary estimate then gives convergence of the integrated extrinsic curvature. In the two-ended case, the splitting theorem gives the exact limit $8πχ(N)$ for the compact surface factor $N$. The one-ended upper bound is attained for every prescribed asymptotic volume ratio in $[0,1]$, and the two-ended bound is also sharp.
