Abstract
In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate: any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11901594
Funding source: Natural Science Foundation of Fujian Province
Award Identifier / Grant number: 2022J05007
Funding source: Fundamental Research Funds for the Central Universities
Award Identifier / Grant number: 20720220042
Funding statement: The first author was supported in part by the National Natural Science Foundation of China, Grant No. 11901594, FRG Program of the Macau University of Science and Technology, No. FRG-22-076-MCMS, and the Science and Technology Development Fund, Macau SAR (No. 0022/2023/ITP1, No. 0133/2022/A). The second author was supported in part by the Fundamental Research Funds for the Central Universities (No. 20720220042) and Natural Science Foundation of Fujian Province (No. 2022J05007).
Acknowledgements
The second author thanks Professor Huai-Dong Cao and Professor Fei He for helpful conversations.
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