## Volume estimate about shrinkers

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- by Xu Cheng and Detang Zhou PDF
- Proc. Amer. Math. Soc.
**141**(2013), 687-696 Request permission

## Abstract:

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.## References

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## Additional Information

**Xu Cheng**- Affiliation: Instituto de Matematica, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
- Email: xcheng@impa.br
**Detang Zhou**- Affiliation: Instituto de Matematica, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
- Email: zhou@impa.br
- Received by editor(s): July 4, 2011
- Published electronically: September 27, 2012
- Additional Notes: Both authors were partially supported by CNPq and FAPERJ, Brazil.
- Communicated by: Chuu-Lian Terng
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**141**(2013), 687-696 - MSC (2010): Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-2012-11922-7
- MathSciNet review: 2996973