Abstract. We study stable, two-sided, smooth, properly immersed solutions to the Gaussian Isoperimetric Problem. That is, we study hyper-surfaces Σ n ⊂ R n+1 that are second order stable critical points of minimizing Aµ(Σ) = Σ e −|x| 2 /4 dA for compact variations that preserve weighted volume. Suchshow that such Σ satisfy the curvature condition H = x, N /2 + C where C is a constant. We also derive the Jacobi operator L for the second variation of such Σ. Our first main result is that for non-planar Σ, bounds on the index of L, acting on volume preserving variations, gives us that Σ splits off a linear space. A corollary of this result is that hyper-planes are the only stable smooth, complete, properly immersed solutions to the Gaussian Isoperimetric Problem, and that there are no hypersurfaces of index one. Finally, we show that for the case of Σ 2 ⊂ R 3 , there is a gradient decay estimate for fixed bound |C| ≤ M (C is from the curvature condition) and Σ obeying an appropriate Aµ condition. This shows that for fixed C, in the limit as R → ∞, stable (Σ, ∂Σ) ⊂ (B 2R (0), ∂B 2R (0)) with good volume growth bounds approach hyper-planes.

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.

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