Gap and rigidity theorems of $\lambda$-hypersurfaces
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Abstract:
We study $\lambda$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int _{\Sigma } e^{-\frac {|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete $\lambda$-hypersurfaces in terms of the norm of the second fundamental form $|A|$. Second, we show that in one dimension, the only smooth complete and embedded $\lambda$-hypersurfaces in $\mathbb {R}^2$ with $\lambda \geq 0$ are lines and round circles. Moreover, we establish a Bernstein-type theorem for $\lambda$-hypersurfaces which states that smooth $\lambda$-hypersurfaces that are entire graphs with polynomial volume growth are hyperplanes. All the results can be viewed as generalizations of results for self-shrinkers.References
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Additional Information
- Qiang Guang
- Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
- MR Author ID: 1232860
- ORCID: setImmediate$0.5634441217556445$2
- Email: guang@math.ucsb.edu
- Received by editor(s): October 11, 2017
- Received by editor(s) in revised form: January 29, 2018
- Published electronically: June 29, 2018
- Communicated by: Guofang Wei
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4459-4471
- MSC (2010): Primary 53C44, 53C42
- DOI: https://doi.org/10.1090/proc/14111
- MathSciNet review: 3834671