Topology of Kähler Ricci solitons

We study the issue of connectedness at infinity of gradient Kähler Ricci solitons. For shrinking Kähler Ricci solitons, we show they must be connected at infinity. We also show the same holds true for expanding Kähler Ricci solitons with proper potential functions. As a separate issue, we obtain a sharp pointwise lower bound for the weight function of any smooth metric measure space, in terms of a lower bound of the associated Bakry - Émery curvature.