AI 中文总结
该研究针对n≥3的无边界光滑流形，证明满足数量曲率下界的黎曼度量可由数量曲率等于该下界的度量局部一致逼近，所得结果强化了黎曼逆Burnett猜想。
AI 中文摘要
设M为连通光滑无边界n维流形，n≥3，κ为实数，若M为开流形则κ≤0。我们证明每个满足Scal_{g0}≥κ的光滑黎曼度量g0，均为满足Scal_{gi}=κ且在W^{1,∞}中局部一致有界的光滑黎曼度量序列gi的局部一致极限。结合Gromov的C^0稳定性定理，作为推论得到对任意α∈(0,1)，有等式overline{{g:Scal_g=κ}}^{C^{0,α}_{loc}}={g:Scal_g≥κ}，且α<1的限制是最优的。当κ=0时，该结果证明并强化了Huneau与Luk提出的黎曼逆Burnett猜想。
英文摘要
Let $M$ be a connected smooth $n$-manifold without boundary, where $n\geq3$, and let $κ\in\mathbb{R}$, with $κ\leq0$ if $M$ is open. We prove that every smooth Riemannian metric $g_0$ with $\mathrm{Scal}_{g_0}\geqκ$ is a locally uniform limit of smooth Riemannian metrics $g_i$ with $\mathrm{Scal}_{g_i}=κ$ that are locally uniformly bounded in $W^{1,\infty}$. As a corollary, combining this with Gromov's $C^0$-stability theorem, we obtain the perhaps surprising identity \[ \overline{\{g:\mathrm{Scal}_g=κ\}}^{\,C^{0,α}_{\mathrm{loc}}}=\{g:\mathrm{Scal}_g\geqκ\}, \quad \forall α\in(0,1). \] The restriction $α<1$ is sharp. At $κ=0$, this proves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk.
Comments26 pages, all comments welcome!
