局部共形平坦流形上尖锐的 $\sigma_k$-曲率不等式的定量形式
The sharp $σ_k$-curvature inequality on locally conformally flat manifolds in quantitative form
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中文总结 AI 辅助
本文证明局部共形平坦流形上 $\sigma_k$-曲率不等式的定量稳定性：若等式近似成立，则度量接近极小元，并用最优 Sobolev 范数刻画接近程度，推广了先前结果。
中文摘要 AI 辅助
设 $2\leq k<n/2$，且 $(M^n,[g_0])$ 是一个闭的、连通的、局部共形平坦黎曼流形，其共形类 $[g_0]$ 中存在一个 $k$-容许度量。我们证明了 $M$ 上 $\sigma_k$-曲率不等式的一个稳定性结果，即若某个共形度量几乎满足等式，则该度量接近不等式的一个极小元。接近程度通过共形因子的 Sobolev 范数量化度量，即分别关于 $W^{1,2}$-范数和 $W^{1,2k}$-范数，且指数 $2$ 和 $2k$ 是最优的。这推广了 Frank 和作者之前的结果到 $2<k<n/2$ 的情形，并在额外的非退化假设下，推广到 Viaclovsky 最初考虑的全部流形类。
英文摘要
Let $2\leq k<n/2$ and let $(M^n,[g])$ be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g]$. We prove a stability result of the $σ_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied for some conformal metric, then its conformal factor is close to a minimizer of the inequality. Closeness is measured quantitatively in terms of Sobolev norms of the conformal factor, namely with respect to the $W^{1,2}$- and the $W^{1,2k}$-norm. In the non-degenerate case, these norms come with optimal exponents $2$ and $2k$, respectively, whereas in general the exponents are $2+γ$ and $\max\{2k,2+γ\}$ for some $γ\geq 0$ originating from a Łojasiewicz inequality. This extends a previous result by Frank and the author from $k=2$ and the sphere to $2\leq k<n/2$ and the full class of manifolds originally considered by Viaclovsky. It also extends a previous result by Engelstein--Neumayer--Spolaor from $k=1$ to the setting of fully non-linear scalar curvatures.
