Gaussian Harmonic Forms and two-dimensional self-shrinking surfaces

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.