§ 2.48 Complete bi-rotational shrinking continuationsFDFAD175AF52

The 2026 preprint of Silva Neto studies maximal $O(p+1)\times O(q+1)$-invariant shrinking profiles in $\mathbb R^{p+q+2}$, $p,q\ge 1$, without imposing properness or embeddedness. The profile equations are those of Example 4.16 with both $+x/2,+y/2$ replaced by $-x/2,-y/2$. A maximal profile is closed off smoothly when it reaches an axis. The stated completeness theorem excludes the Clifford minimal cone; in particular the constructed escaping ends admit complete continuations, which need not be proper. The source distinguishes conical escaping ends from recurrent ends meeting both distinguished cylinders infinitely often. A noncompact complete example is proper precisely when all its ends are conical, under the source's classification. The almost-everywhere alternative “closed or nonproper” does not by itself construct a specified nonproper topological type. We therefore record no general topology, entropy value, or embedding criterion for arbitrary initial data. The known round and cylindrical solutions are not new entries; the McGrath compact construction is not included here. The flow is $\sqrt{-t}\, \Sigma $, $t<0$. The figure integrates a finite segment of the profile ODE; its display endpoints are not a certification of the complete global continuation. All global results remain attributed preprint statements.

Sources: [Neto2026a].