AI 中文总结
该研究针对Wei和Wylie的问题，在维度8k+1和8k+2中构造出反例，证明同伦球虽存在满足Ric_g + Hess_g f > 0的加权核度量，但因α-不变量非零而无Ric≥0的黎曼度量。
AI 中文摘要
Wei和Wylie提出问题：具有非负Bakry-Émery里奇曲率且势函数有界的完备加权流形，是否一定容许具有非负里奇曲率的黎曼度量？我们在每个满足k≥1的维度8k+1和8k+2中给出该问题的否定答案。我们的主要几何结果是，每个维度至少为7的光滑同伦球都容许加权核度量(g,e^{-f})，满足Ric_g + Hess_g f > 0。另一方面，在满足k≥1的维度8k+1和8k+2中，我们证明具有非零α-不变量的同伦球不容许满足Ric≥0的黎曼度量。由于具有非零α-不变量的同伦球存在于每个维度8k+1和8k+2中，且紧致性使得势函数有界，因此这些流形提供了所需的反例。
英文摘要
Wei and Wylie asked whether a complete weighted manifold with nonnegative Bakry--Émery Ricci curvature and bounded potential must admit a Riemannian metric with nonnegative Ricci curvature. We answer this question negatively in every dimension \(8k+1\) and \(8k+2\), where \(k\geq1\). Our main geometric result is that every smooth homotopy sphere of dimension at least seven admits a weighted core metric \((g,e^{-f})\) with \(\Ric_g+\Hess_g f>0\). On the other hand, in dimensions \(8k+1\) and \(8k+2\), where \(k\geq1\), we prove that a homotopy sphere with nonzero \(α\)-invariant admits no Riemannian metric with \(\Ric\geq0\). Since homotopy spheres with nonzero \(α\)-invariant exist in every dimension \(8k+1\) and \(8k+2\), and since compactness makes the potential bounded, these manifolds provide the required counterexamples.
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