§ 3.23 Translators with general closed-shrinker endsC44A1D57ECC3

For a closed smooth embedded self-shrinker $S\subset \mathbb R^ n$ of Gaussian Morse index $I$, Sun–Wang construct an $I$-parameter family of complete embedded translators in $\mathbb R^{n+1}$ whose unique tangent flow at backward infinity is $S\times \mathbb R$ with multiplicity one. The construction may have a singular set of dimension at most $n-7$; it is smooth for $n\le 6$, namely ambient dimension at most seven. In higher dimensions it is a weak geometric construction, not an unconditional smooth hypersurface theorem. The Angenent-torus and spherical-doubling input cases have their own existing pages. Additional inputs include other closed embedded high-entropy shrinkers and Riedler's $g=3,4,6$ isoparametric examples. The excluded compact product-of-spheres construction is not used as an input here. The translating flow is $\Sigma +te_ z$, $t\in \mathbb R$. The topology of a cross-sectional shrinker must not be recorded as the genus of the higher-dimensional translator. The theorem specifies the tangent flow, not a literal product end at finite height. The figure shows increasing asymptotic cross-sections and labels the uncomputed core; no arbitrary tube is asserted to solve the translator equation.

Sources: [SunWang2025].