  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 $\frac{2g+r-1}{3}$, where $r$ is the
number of asymptotically conical ends.
