Harmonic functions with polynomial growth
Harmonic functions with polynomial growth
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DOI：
10.4310/jdg/1214459897
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发表时间：
1997
影响因子：
2.5
通讯作者：
T. Colding;W. Minicozzi
中科院分区：
文献类型：
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作者：
T. Colding;W. Minicozzi
20年前，丘成桐将复分析的经典Liouville定理推广到具有非负Ricci曲率的开流形上，特别是证明了在这样的流形上的正调和函数必为常数。次线性增长的调和函数必须是常数为了进一步研究这些流形的分析性质，人们希望尽可能多地限制要考虑的函数类，同时使信息损失最小化cf从Cheng和Yau的结果可以看出，多项式增长的调和函数类是一个自然的候选者，注意它们必须至少是线性增长的。研究这些功能丘的动机，使以下猜想见，也见优秀的调查文章彼得李
Twenty years ago Yau generalized the classical Liouville theo rem of complex analysis to open manifolds with nonnegative Ricci curva ture Speci cally he proved that a positive harmonic function on such a manifold must be constant This theorem of Yau was considerably generalized by Cheng Yau see by means of a gradient estimate which implies the Harnack inequality As a consequence of this gradient estimate see one has that on such a manifold even a harmonic function of sublinear growth must be constant In order to study further the analytic properties of these manifolds one would like to restrict the class of functions to be considered as much as possible while minimizing loss of information cf From the results of Cheng and Yau it follows that a natural candidate is the class of harmonic functions of polynomial growth note that they must be of at least linear growth In fact in his study of these functions Yau was motivated to make the following conjecture see and see also the excellent survey article by Peter Li