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Gaussian Harmonic Forms and two-dimensional self-shrinking surfaces

Author: Matthew McGonagle
Journal: Proc. Amer. Math. Soc. 143 (2015), 3603-3611
MSC (2010): Primary 53A10, 53C42; Secondary 53C44
Published electronically: April 20, 2015
MathSciNet review: 3348801
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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^{-\vert x\vert^2/4} \vert\omega \vert^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 $ \vert A\vert^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.

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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

Keywords: Mean Curvature Flow, self-shrinkers, harmonic one forms, genus, Gaussian harmonic
Received by editor(s): January 2, 2013
Published electronically: April 20, 2015
Communicated by: Michael Wolf
Article copyright: © Copyright 2015 American Mathematical Society

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