Abstract
In this paper, we study the shrinking gradient Ricci-harmonic soliton. Firstly using Chow–Lu–Yang’s argument, we give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci-harmonic solitons with \(S\ge \delta \) to have polynomial volume growth with order \(n-2\delta \). Secondly, we derive a Logarithmic Sobolev inequality, as an application, we prove that any noncompact shrinking gradient Ricci-harmonic soliton must have linear volume growth, generalizing previous result of Munteanu and Wang (Commun Anal Geom 20(1):55–94, 2012).
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Wu, G., Zhang, S. Volume Growth of Shrinking Gradient Ricci-Harmonic Soliton. Results Math 72, 205–223 (2017). https://doi.org/10.1007/s00025-017-0703-7
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DOI: https://doi.org/10.1007/s00025-017-0703-7