发表机构
Capital Normal University（首都师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究格林函数的极点正则性与刚性，证明三维的相关结论无高维类比，对不同维数给出曲率条件或反例，揭示极点正则性与流形等距性质的关联。
AI 中文摘要
设\ntitle_cn\n为完备非抛物黎曼流形上的归一化极小正格林函数（即极小正格林函数存在），令\nb = G^{1/(2-n)}\n。Colding证明，在三维情形下，\n|\n∇\nb\n|²\n在极点处的C²正则性结合非负里奇曲率，可迫使该流形为欧氏空间。本文证明，仅极点正则性不存在高维类比：对所有\nn ≥ 4\n，存在显式完备非平坦旋转对称流形，其具有非负截面曲率与欧氏体积增长，且\n|\n∇\nb\n|²\n可延拓至极点处为光滑函数。对所有奇数\nn ≥ 5\n，本文给出极点处的有限阶曲率条件，该条件可推出\nG(p,x) = r^{2-n} + H_p + O₁(r)\n；在这些条件下，\n|\n∇\nb\n|²\n的\nC^{n-1}\n正则性可迫使该流形等距同构于\nℝⁿ\n。在五维情形下，仅需假设\nScal(p) = 0\n且\n|\n∇\nb\n|² ∈ C⁴\n即可。相比之下，对所有偶数\nn ≥ 4\n，存在完备非平坦例子，其在极点附近为欧氏空间且\n|\n∇\nb\n|²\n在极点处光滑。
英文摘要
Let G(p, .) be the normalized minimal positive Green function on a complete nonparabolic Riemannian manifold with nonnegative Ricci curvature, and set b = G^(1/(2-n)). Colding established the following rigidity result in dimension three: if |nabla b|^2 admits a C^2 regularity across the pole, then the manifold must be isometric to Euclidean space. We show that rigidity conclusions corresponding to the pole regularity condition alone do not hold in higher dimensions: for any n >= 4, we can construct explicit complete nonflat rotationally symmetric manifolds with nonnegative sectional curvature and Euclidean volume growth, such that |nabla b|^2 extends smoothly across the pole. For any odd integer n >= 3, we impose finite-order pointwise curvature conditions at the pole (where this condition is vacuous when n = 3), so that the Green function satisfies the asymptotic expansion G(p,x) = r^(2-n) + H_p + O_1(r). Under this curvature assumption, if |nabla b|^2 extends to a C^(n-2) function near the pole, then the manifold must be isometric to R^n, and this regularity threshold is also sharp. As special cases: in dimension three, if |nabla b|^2 is in C^1, then the manifold must be isometric to Euclidean space; in dimension five, it suffices to assume that the scalar curvature at the pole satisfies Scal(p) = 0 and |nabla b|^2 is in C^3. In contrast, for any even dimension n >= 4, there exist complete nonflat examples that are Euclidean in a neighborhood of the pole and admit a smooth extension of |nabla b|^2 across the pole.
CommentsVersion 2: Revised version. The sharp regularity exponent is corrected, n=3 is included in the odd-dimensional framework, sharpness and counterexamples are discussed, one proof is completed, and the exposition is improved. 20 pages, LaTeX
