关于完备收缩的Ricci-harmonic孤子的研究
ON COMLETE SHRINKING RICCI-HARMONIC SOLITONS

本文研究了收缩的Ricci-harmonic孤子的几何性质的问题.利用文献[4]在Ricci孤子下的方法,获得了每个紧致Ricci-harmonic孤子是一个梯度孤子的结论,推广了Perelman等人在Ricci孤子下的结果.此外,利用文献[14]在Ricci孤子下的方法,获得了完备非紧梯度收缩的Ricci-harmonic孤子具有比至多欧氏增长更加精确的体积增长估计的结果,推广了文献[14]在Ricci孤子下的结果.
In this paper, we study the geometry of shrinking Ricci-harmonic solitons. By utilizing the method of Manola, Gabriele and Carlo [4] under the Ricci soliton, we prove the result that every compact shrinking Ricci-harmonic soliton is a gradient one, which extends the result in the case of Ricci solition. Moreover, by utilizing the method of Zhang [14], we prove a more precise volume growth estimate than that of at most Euclidean growth for the complete non-compact gradient shrinking Ricci-harmonic soliton, which extends the result of Zhang [14] in the case of Ricci solition