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Transactions of the American Mathematical Society

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Stabilities of homothetically shrinking Yang-Mills solitons

Authors: Zhengxiang Chen and Yongbing Zhang
Journal: Trans. Amer. Math. Soc. 367 (2015), 5015-5041
MSC (2010): Primary 53C44, 53C07
Published electronically: February 26, 2015
MathSciNet review: 3335408
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Abstract: In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and the second variation of the $ \mathcal {F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabilities have connections with the study of Type-I singularities of the Yang-Mills flow. Two byproducts are also included: We show that the Yang-Mills flow in dimension four cannot develop a Type-I singularity, and we obtain a gap theorem for homothetically shrinking solitons.

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Zhengxiang Chen
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany
Address at time of publication: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

Yongbing Zhang
Affiliation: School of Mathematical Sciences and Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui Province, People’s Republic of China

Keywords: Yang-Mills flow, stability, homothetically shrinking soliton
Received by editor(s): May 2, 2013
Published electronically: February 26, 2015
Additional Notes: This project was supported by NSFC No. 11201448
Article copyright: © Copyright 2015 American Mathematical Society

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