[@HamiltonSingularities], [@Ivey3DSoliton], [@Naber4D], [@NiWallach], [@Perelman2], [@PetersenWylie], and even a limited extension of this classification to dimensions four and higher would help advance the understanding of the long-time behavior of solutions to the equation.

There are indications that the asymptotic behavior of complete noncompact shrinking solitons may be rigid enough to support a classification into general structural types. At present, all known examples of complete noncompact shrinkers which are not locally reducible as products are connected at infinity and smoothly asymptotic to regular cones (see [@FeldmanIlmanenKnopf], [@DancerWang], [@Yang]). Recent results of Munteanu-Wang [@MunteanuWangKaehler; @MunteanuWangConical; @MunteanuWangGRSStructure] suggest that these two types may actually exhaust the possibilities for complete noncompact four-dimensional shrinkers.

In this paper, we consider shrinkers of the second type, whose geometries are asymptotically conical in a sense which we now make precise. Given a closed Riemannian manifold $(\Sigma, g_{\Sigma})$ of (real) dimension $n-1$ and a fixed $a \in {\mathbb{R}}$, let $E_a = E_a(\Sigma)$ denote the cylinder $(a, \infty) \times \Sigma$ and $g_c = dr^2 + r^2g_{\Sigma}$ the regular conical metric on $E_0$. Further, for any fixed $\lambda > 0$, let $\rho_{\lambda}:E_0\to E_0$ denote the dilation map $\rho_{\lambda}(r,\sigma){\doteqdot}({\lambda}r, \sigma)$. By an *end* $V\subset (M, g)$, we will mean an unbounded connected 