AI 中文总结
针对满足$\text{Ric}\geqslant g$的闭凯勒流形，通过融入$\bar\partial$-拉普拉斯算子的第一个正特征值，建立改进刘维尔定理，得到改进的索伯列夫与Beckner不等式，还推导了该流形直径的新上界。
AI 中文摘要
设$(M,g,J)$为满足$\text{Ric}\geqslant g$的闭凯勒流形，我们将$\bar\partial$-拉普拉斯算子的第一个正特征值融入微分恒等式论证，建立了与Beckner-Sobolev不等式相关的欧拉-拉格朗日方程的改进刘维尔定理。由此，当第一个特征值足够大时，我们得到了改进的索伯列夫不等式和Beckner不等式，它们对已知的黎曼和凯勒估计进行了精细化，还推导了$(M,g)$直径的新上界。
英文摘要
Let $(M,g,J)$ be a closed Kähler manifold satisfying $\operatorname{Ric}\geqslant g$. We establish improved Liouville theorems for the Euler--Lagrange equations associated with the Beckner--Sobolev inequalities by incorporating the first positive eigenvalue of the $\bar\partial$-Laplacian into a differential-identity argument. As a consequence, we obtain improved Sobolev and Beckner inequalities that refine the known Riemannian and Kähler estimates when the first eigenvalue is sufficiently large. We also derive new upper bounds for the diameter of $(M,g)$.
