AI 中文总结
该研究确定了闭连通光滑流形上，满足广义共形拉普拉斯算子正定的黎曼度量空间的同伦性质，推广了相关结果并解决了Gromov同伦论强化猜想，还处理了等变与带边界情形。
AI 中文摘要
设n≥2，Mⁿ为闭连通光滑流形，R^γ(M)为M上光滑黎曼度量g构成的空间，满足广义共形拉普拉斯算子-γΔ_g+R_g严格正定。我们证明：当n=2且γ≥0，或n≥3且0≤γ≤4(n-1)/(n-2)时，包含映射R⁰(M)↪R^γ(M)是同伦等价，将Botvinnik-Rosenberg与Li-Mantoulidis的结果推广到所有维数及最大系数范围；还证明当n≥3且γ>4(n-1)/(n-2)时，R^γ(M)可缩且非空，在最大可能系数范围内解决了Gromov猜想（《标量曲率四讲》6.1.2节猜想3）的同伦论强化版本，还处理了等变情形与带边界流形的情况。
英文摘要
Let $n\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^γ(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-γΔ_g+\mathrm{R}_g$ is strictly positive. We prove that if $n=2$ and $γ>0$, or if $n\ge3$ and $0< γ\leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\hookrightarrow R^γ(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik--Rosenberg and Li--Mantoulidis. Then, we prove that if $n\ge3$ and $γ>4(n-1)/(n-2)$, the space $R^γ(M)$ is contractible, and hence nonempty. This solves a homotopy-theoretic strengthening of a conjecture of Gromov (Conjecture 3, Section 6.1.2, "Four Lectures on Scalar Curvature") in the maximal possible coefficient range. Concerning Gromov's conjecture we also treat the equivariant case and the case of manifolds with boundary.
Comments21 pages. Comments are welcome! v2: Title changed and minor edits
