收缩梯度李奇-调和孤立子的体积增长

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 $Sgeq delta$ to have polynomial volume growth with order $n-2delta$. 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 ofMunteanu-Wang.