  We study stable smooth solutions to the isoperimetric type problem for a
Gaussian weight on Euclidean Space. That is, we study hypersurfaces $\Sigma^n
\subset \mathbb R^{n+1}$ that are second order stable critical points of
compact variations that minimize Gaussian weighted area and preserve Gaussian
weighted volume. We show that such $\Sigma$ satisfy a curvature condition, and
derive 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 off
a linear space. A corollary of this result is that hyperplanes are the only
stable smooth complete solutions to this Gaussian isoperimetric type problem,
and that there are no hypersurfaces of index one. Finally, we show that for the
case of $\Sigma^2 \subset \mathbb R^3$, there is a gradient decay estimate
depending on bounds for the curvature condition and an appropriate area growth
bound. 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 growth
bounds approach hyperplanes.
