§ 4.9 Mountain-pass self-expanders665EA484B8DC

Let $2\leq n\leq 6$, let $C\subset \mathbb R^{n+1}$ be a $C^3$-regular cone, and suppose two distinct strictly stable $C^2$-asymptotically conical self-expanders $\Gamma _-\preceq \Gamma _+$ approach $C$. The order means inclusion of their consistently oriented enclosed regions. Bernstein–Wang construct a further smooth properly embedded self-expander $\Gamma _0$ between them, meeting the open intervening region. The associated flow is $M_ t=\sqrt t\, \Gamma _0$, $t>0$, with $\mathbf H=X^\perp /2$.

If the cone is generic in the precise sense that every expander asymptotic to it has no nontrivial Jacobi field fixing infinity, $\Gamma _0$ can be chosen unstable for expander weighted area. The theorem does not prescribe its genus, connectedness, symmetry or uniqueness. Completeness follows from proper embeddedness without boundary. Cone asymptotics give Euclidean area growth and hence finite Gaussian area and entropy; no numerical value is supplied.

Corollary 1.3 gives an open set of cones admitting at least two connected and one disconnected smoothing, beyond the rotational setting. Sources: [BWMP20], Theorem 1.1 and Corollaries 1.2–1.3, pp. 1–2. In higher dimensions the general theorem allows a singular set of codimension seven; only its smooth $2\leq n\leq 6$ cases are catalogued here. The figure depicts ordering, not a computed profile or prescribed topology.