Index estimate of self-shrinkers in $\mathbb {R}^3$ with asymptotically conical ends
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Abstract:
We construct Gaussian harmonic forms of a finite weighted $L^2$-norm on noncompact surfaces that detect each asymptotically conical end. As an application we prove an extension of the index estimates of self-shrinkers (see Matthew McGonagle [Proc. Amer. Math. Soc. 143 (2015), no. 8, 3603–3611]) under the existence of such ends. We show that the Morse index of a self-shrinker is greater than or equal to $\frac {2g+r-1}{3}+1$, where $r$ is the number of asymptotically conical ends.References
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Additional Information
- Nicolau Sarquis Aiex
- Affiliation: Earth Sciences Building Room 4118, Department of Mathematics, 2207 Main Mall, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
- Email: nsarquis@math.ubc.ca
- Received by editor(s): April 29, 2018
- Received by editor(s) in revised form: May 25, 2018
- Published electronically: October 31, 2018
- Communicated by: Jia-Ping Wang
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 799-809
- MSC (2010): Primary 53C21, 53C42; Secondary 53C44
- DOI: https://doi.org/10.1090/proc/14306
- MathSciNet review: 3894918