AI 中文总结
该研究针对完备梯度Kähler-Ricci孤立子上的实值多重调和函数，引入孤立子势诱导的全纯量，证明稳态和收缩情形下不同$p$范围的$L^p$刘维尔定理，并构造例子说明$0<p<1$的扩展依赖孤立子结构。
AI 中文摘要
我们在梯度可积性假设下，研究完备梯度Kähler-Ricci孤立子上实值多重调和函数的刘维尔型定理。对于完备梯度Kähler-Ricci孤立子$(M,g,J,f)$和实值多重调和函数$u$，探究$u$必须为常数的条件。通过引入由孤立子势诱导的全局定义全纯量，得到超越调和函数已有范围的新刘维尔型结果：稳态情形下，当$0<p<\nfty$且$\nint_M|\nabla u|^p\udf16<\nfty$时，证明$u$为常数；收缩情形下，对$0<p\neq2$证明同一结论。最后构造完备Kähler例子，表明扩展至$0<p<1$的范围本质依赖孤立子结构，在一般完备Kähler流形上不成立。
英文摘要
We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient Kähler-Ricci solitons under gradient integrability assumptions. For a complete gradient Kähler-Ricci soliton $(M,g,J,f)$ and a real-valued pluriharmonic function $u$, we investigate conditions under which $u$ must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that $u$ is constant whenever $$ \int_M|\nabla u|^p\mathrm{d}v<\infty $$ for some $0<p<\infty$. In the shrinking case, we prove the same conclusion for $0<p\leq 2$. Finally, we construct a complete Kähler example showing that the extension to the range $0<p<1$ relies essentially on the soliton structure and does not hold on general complete Kähler manifolds.
CommentsThis version corrects a minor error in Lemma 3.3 as well as some typographical errors
