# Volume Growth of Shrinking Gradient Ricci-Harmonic Soliton

Volume Growth of Shrinking Gradient Ricci-Harmonic Soliton 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 $$S\ge \delta$$ S ≥ δ to have polynomial volume growth with order $$n-2\delta$$ n - 2 δ . 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# Volume Growth of Shrinking Gradient Ricci-Harmonic Soliton

, Volume 72 (2) – Jun 13, 2017
19 pages

Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0703-7
Publisher site
See Article on Publisher Site

### 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 $$S\ge \delta$$ S ≥ δ to have polynomial volume growth with order $$n-2\delta$$ n - 2 δ . 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).

### Journal

Results in MathematicsSpringer Journals

Published: Jun 13, 2017

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