AI 中文总结
本文通过构造欧氏单位球的共形变形，结合扰动判据与显式多项式共形因子，对所有n≥3否定了埃斯科瓦尔关于紧致黎曼流形边界主曲率与σ₁关系的猜想。
AI 中文摘要
埃斯科瓦尔（《函数分析杂志》165卷1期，101-116页，1999）提出猜想：对所有n≥3，n维紧致黎曼流形若满足非负里奇曲率，且所有边界主曲率下界为κ>0，则其σ₁≥κ。我们通过构造欧氏单位球的共形变形，对所有n≥3否定该猜想。首先建立扰动判据，再构造满足该判据的显式多项式共形因子。对所有足够小的t>0，所得度量gₜ=e^(2tΦ)g_(ℝⁿ)具有正里奇曲率，所有边界主曲率严格大于1，且σ₁(𝔹ⁿ,gₜ)<1。证明需多个计算，部分在Mathematica中完成，Mathematica代码附于本投稿。
英文摘要
Escobar (J Funct Anal 165(1):101-116, 1999) conjectured that for every $n\ge 3$, an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and all boundary principal curvatures bounded below by $κ>0$ must satisfy $σ_1\geq κ$. We disprove this conjecture for every $n\geq 3$ by constructing conformal deformations of the Euclidean unit ball. We first establish a perturbative criterion, then construct explicit polynomial conformal factors satisfying this criterion. For every sufficiently small $t>0$, the resulting metrics $g_t=e^{2tΦ}g_{\mathbb{R}^n}$ have positive Ricci curvature, every boundary principal curvature is strictly larger than $1$, and $σ_1(\mathbb{B}^n,g_t)<1$. The proof requires several computations, some of which were carried out in Mathematica. The Mathematica code is attached to this submission.
