We consider two-dimensional self-shrinkers Σ 2 for the Mean Curvature Flow of polynomial volume growth immersed in R n . We look at closed one forms ω satisfying the Euler-Lagrange equation associated with minimizing the norm Σ dV e −|x| 2 /4 |ω| 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 Σ has genus ≥ 1, then we have a lower bound on the supremum norm of |A| 2 . We also may give applications to the index

more »

... f L acting on scalar functions of Σ and to estimates of the lowest eigenvalue η 0 of L if Σ satisfies certain curvature conditions.