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We study stable smooth solutions to the isoperimetric type problem for aGaussian weight on Euclidean Space. That is, we study hypersurfaces $\Sigma^n\subset \mathbb R^{n+1}$ that are second order stable critical points ofcompact variations that minimize Gaussian weighted area and preserve Gaussianweighted volume. We show that such $\Sigma$ satisfy a curvature condition, andderive the Jacobi operator $L$ for the second variation of such $\Sigma$.  Our first main result is that for non-planar $\Sigma$, bounds on the index of$L$, acting on volume preserving variations, gives us that $\Sigma$ splits offa linear space. A corollary of this result is that hyperplanes are the onlystable smooth complete solutions to this Gaussian isoperimetric type problem,and that there are no hypersurfaces of index one. Finally, we show that for thecase of $\Sigma^2 \subset \mathbb R^3$, there is a gradient decay estimatedepending on bounds for the curvature condition and an appropriate area growthbound. This shows that, in the limit as $R \to \infty$, stable $(\Sigma,\partial\Sigma) \subset (B_{2R}(0), \partial B_{2R}(0))$ with good area growthbounds approach hyperplanes.

The Hyperplane is the Only Stable, Smooth Solution to the Isoperimetric Problem in Gaussian Space