Gaussian Harmonic Forms and two-dimensional self-shrinking surfaces
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- by Matthew McGonagle PDF
- Proc. Amer. Math. Soc. 143 (2015), 3603-3611 Request permission
Abstract:
We consider two-dimensional self-shrinkers $\Sigma ^2$ for the Mean Curvature Flow of polynomial volume growth immersed in $\mathbb R^n$. We look at closed one forms $\omega$ satisfying the Euler-Lagrange equation associated with minimizing the norm $\int _\Sigma dV e^{-|x|^2/4} |\omega |^2$ in their cohomology class. We call these forms Gaussian Harmonic one Forms (GHF).
Our main application of GHF’s is to show that if $\Sigma$ has genus $\geq 1$, then we have a lower bound on the supremum norm of $|A|^2$. We also may give applications to the index of $L$ acting on scalar functions of $\Sigma$ and to estimates of the lowest eigenvalue $\eta _0$ of $L$ if $\Sigma$ satisfies certain curvature conditions.
References
- Huai-Dong Cao and Haizhong Li, A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension, Calc. Var. Partial Differential Equations 46 (2013), no. 3-4, 879–889. MR 3018176, DOI 10.1007/s00526-012-0508-1
- 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
- H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765, DOI 10.1007/978-1-4612-2034-3
- 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
- Jürgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext, Springer, Heidelberg, 2011. MR 2829653, DOI 10.1007/978-3-642-21298-7
- Antonio Ros, One-sided complete stable minimal surfaces, J. Differential Geom. 74 (2006), no. 1, 69–92. MR 2260928
- Francisco Urbano, Second variation of one-sided complete minimal surfaces, Rev. Mat. Iberoam. 29 (2013), no. 2, 479–494. MR 3047425, DOI 10.4171/RMI/727
Additional Information
- Matthew McGonagle
- Affiliation: Department of Mathematics, Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218-2686
- Address at time of publication: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 9815-4350
- Email: mmcgona1@math.washington.edu
- Received by editor(s): January 2, 2013
- Published electronically: April 20, 2015
- Communicated by: Michael Wolf
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3603-3611
- MSC (2010): Primary 53A10, 53C42; Secondary 53C44
- DOI: https://doi.org/10.1090/proc12750
- MathSciNet review: 3348801