§ 4.8 One-sided expanders from mean-convex cones5C7DD27422AD

(Ding's smooth minimizing expanders) Let $C\subset \mathbb R^{n+1}$ be an $n$-dimensional $C^{3,\beta }$-regular cone, $0<\beta <1$, regular away from its vertex, with a $C^{3,\beta }$ embedded link in $S^ n$. Assume that $C$ is mean convex but not area minimizing, that $\partial \Omega =C$, and that its mean curvature vector points into the chosen domain $\Omega $ wherever it is nonzero. Then there is a unique smooth complete embedded $E$-minimizing self-expanding hypersurface $M\subset \Omega $ with tangent cone $C$ at infinity. Its mean curvature vector is everywhere nonzero and points into $\Omega $, and

is a foliation of $\Omega $ moving by mean curvature flow. The normalization is $\vec H=X^\perp /2$.

This is an existence theorem for a family determined by the cone and the chosen side, not an explicit parametrization. The uniqueness just stated includes the chosen-side and minimizing hypotheses. Its topology, symmetry, and number of ends depend on the chosen cone. Source: [Ding20], Theorem 1.1, printed p. 361, supplied PDF p. 3, and Theorem 6.12, printed pp. 402–403, PDF pp. 44–45. The broader minimizing-current existence theorem in that paper can have singularities and is not being substituted for this smooth theorem. The figure is a schematic of the cone, selected domain, and expanding leaves.