Gaussian Harmonic Forms and two-dimensional self-shrinking surfaces

Abstract

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

Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

“…Ros proved the bound in (a) for usual complete minimal surfaces in R 3 (see [17]); this was later improved to Ind(Σ) ≥ 2 3 (g + r) − 1 by O. Chodosh and D. Maximo ( [5]), where r is the number of ends. The inequality in b) improves the lower bound Ind f (Σ) ≥ g 3 proved in [15] under the additional condition that sup Σ |k 2 1 − k 2 2 | ≤ δ < 1 (here k 1 and k 2 are the principal curvatures of Σ).…”

Index estimate of self-shrinkers in $\mathbb {R}^3$ with asymptotically conical ends

“…We also define ⋆ µ = e x 2 4 ⋆, where ⋆ is the Hodge operator with respect to the Riemannian metric on Σ (see [1, §3]). The following existence of GHF of degree 1 is a slight generalization of the result in [11] by Mcgonagle, in which we also include the 1-forms constructed above and its weighted Hodge dual. The proof is similar but we include it here for completeness.…”

“…We will now show that each of these is cohomologous to a smooth Gaussian Harmonic 1-form. Let us follow [11] (see also [1]) and define…”

“…In this section we use the calculations by Mcgonagle in [11] together with our previous results to improve the index estimates of self-shrinkers when the ends are asymptotically conical. Given a 1-form ω in Σ and a vector v ∈ R 3 we will denote by ω # the unique vector field defined by ω(X) = g Σ (ω # , X) and v ♭ the 1-form on R 3 defined by v ♭ (X) = ⟨v, X⟩.…”

“…

We construct Gaussian Harmonic forms of finite Gaussian weighted L 2 -norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an extension of the index estimates of self-shrinkers in [11] under the existence of such ends. We show that the Morse index of a self-shrinker is greater or equal to 2g+r−1 3 , where r is the number of asymptotically conical ends.

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