Index estimate of self-shrinkers in ℝ³ with asymptotically conical ends

Sarquis Aiex, Nicolau

doi:10.1090/proc/14306

Index estimate of self-shrinkers in $\mathbb {R}^3$ with asymptotically conical ends
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Proc. Amer. Math. Soc. 147 (2019), 799-809 Request permission

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

Edward L. Bueler, The heat kernel weighted Hodge Laplacian on noncompact manifolds, Trans. Amer. Math. Soc. 351 (1999), no. 2, 683–713. MR 1443866, DOI 10.1090/S0002-9947-99-02021-8
Otis Chodosh and Davi Maximo, On the topology and index of minimal surfaces, J. Differential Geom. 104 (2016), no. 3, 399–418. MR 3568626
Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755–833. MR 2993752, DOI 10.4007/annals.2012.175.2.7
Baptiste Devyver, On the finiteness of the Morse index for Schrödinger operators, Manuscripta Math. 139 (2012), no. 1-2, 249–271. MR 2959680, DOI 10.1007/s00229-011-0522-1
Qi Ding and Y. L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, Asian J. Math. 17 (2013), no. 3, 443–456. MR 3119795, DOI 10.4310/AJM.2013.v17.n3.a3
Avron Douglis and Louis Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8 (1955), 503–538. MR 75417, DOI 10.1002/cpa.3160080406
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 (1985), no. 1, 121–132. MR 808112, DOI 10.1007/BF01394782
Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
Tom Ilmanen, Lectures on mean curvature flow and related equations, unpublished notes. http://www.math.ethz.ch/ ilmanen/papers/pub.html, 1995.
Debora Impera and Michele Rimoldi, Stability properties and topology at infinity of $f$-minimal hypersurfaces, Geom. Dedicata 178 (2015), 21–47. MR 3397480, DOI 10.1007/s10711-014-9999-6
Debora Impera, Michele Rimoldi, and Alessandro Savo, Index and First Betti Number of $f$-Minimal Hypersurfaces and Self-Shrinkers. arXiv:1803.08268 [math.DG], 2018.
Chao Li, Index and topology of minimal hypersurfaces in $\Bbb {R}^n$, Calc. Var. Partial Differential Equations 56 (2017), no. 6, Paper No. 180, 18. MR 3722074, DOI 10.1007/s00526-017-1272-z
Matthew McGonagle, Gaussian harmonic forms and two-dimensional self-shrinking surfaces, Proc. Amer. Math. Soc. 143 (2015), no. 8, 3603–3611. MR 3348801, DOI 10.1090/proc12750
Antonio Ros, One-sided complete stable minimal surfaces, J. Differential Geom. 74 (2006), no. 1, 69–92. MR 2260928
Alessandro Savo, Index bounds for minimal hypersurfaces of the sphere, Indiana Univ. Math. J. 59 (2010), no. 3, 823–837. MR 2779062, DOI 10.1512/iumj.2010.59.4013
Lu Wang, Uniqueness of self-similar shrinkers with asymptotically conical ends, J. Amer. Math. Soc. 27 (2014), no. 3, 613–638. MR 3194490, DOI 10.1090/S0894-0347-2014-00792-X
Lu Wang, Asymptotic structure of self-shrinkers. arXiv:1610.04904 [math.DG], 2016.