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Volume 52, Issue 2
On the Triviality of a Certain Kind of Shrinking Solitons

Zhuhong Zhang

DOI: 10.4208/jms.v52n2.19.04

J. Math. Study, 52 (2019), pp. 169-177.

Published online: 2019-05

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  • Abstract

In this paper, we study shrinking gradient Ricci solitons whose Ricci tensor has one eigenvalue of multiplicity at least $n−2.$ Firstly, we show that if the minimal eigenvalue of Ricci tensor has multiplicity at least $n−1$ at each point, then the soliton are Einstein. While on the shrinking gradient Ricci solitons whose maximal eigenvalue has multiplicity at least $n−1,$ the triviality are also true if we naturally require the positivity of Ricci tensor.
We further prove that if the maximal (or minimal) eigenvalue of Ricci tensor has multiplicity at least $n−2$ at each point , and in addition the sectional curvature is bounded from above, then the soliton are Einstein.

  • Keywords

Einstein manifold, shrinking gradient Ricci soliton, positive Ricci curvature, pinched sectional curvature.

  • AMS Subject Headings

53C24, 53C25

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

juhoncheung@sina.com (Zhuhong Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JMS-52-169, author = {Zhang , Zhuhong}, title = {On the Triviality of a Certain Kind of Shrinking Solitons}, journal = {Journal of Mathematical Study}, year = {2019}, volume = {52}, number = {2}, pages = {169--177}, abstract = {

In this paper, we study shrinking gradient Ricci solitons whose Ricci tensor has one eigenvalue of multiplicity at least $n−2.$ Firstly, we show that if the minimal eigenvalue of Ricci tensor has multiplicity at least $n−1$ at each point, then the soliton are Einstein. While on the shrinking gradient Ricci solitons whose maximal eigenvalue has multiplicity at least $n−1,$ the triviality are also true if we naturally require the positivity of Ricci tensor.
We further prove that if the maximal (or minimal) eigenvalue of Ricci tensor has multiplicity at least $n−2$ at each point , and in addition the sectional curvature is bounded from above, then the soliton are Einstein.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v52n2.19.04}, url = {http://global-sci.org/intro/article_detail/jms/13157.html} }
TY - JOUR T1 - On the Triviality of a Certain Kind of Shrinking Solitons AU - Zhang , Zhuhong JO - Journal of Mathematical Study VL - 2 SP - 169 EP - 177 PY - 2019 DA - 2019/05 SN - 52 DO - http://doi.org/10.4208/jms.v52n2.19.04 UR - https://global-sci.org/intro/article_detail/jms/13157.html KW - Einstein manifold, shrinking gradient Ricci soliton, positive Ricci curvature, pinched sectional curvature. AB -

In this paper, we study shrinking gradient Ricci solitons whose Ricci tensor has one eigenvalue of multiplicity at least $n−2.$ Firstly, we show that if the minimal eigenvalue of Ricci tensor has multiplicity at least $n−1$ at each point, then the soliton are Einstein. While on the shrinking gradient Ricci solitons whose maximal eigenvalue has multiplicity at least $n−1,$ the triviality are also true if we naturally require the positivity of Ricci tensor.
We further prove that if the maximal (or minimal) eigenvalue of Ricci tensor has multiplicity at least $n−2$ at each point , and in addition the sectional curvature is bounded from above, then the soliton are Einstein.

Zhang , Zhuhong. (2019). On the Triviality of a Certain Kind of Shrinking Solitons. Journal of Mathematical Study. 52 (2). 169-177. doi:10.4208/jms.v52n2.19.04
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