Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder (Q6562504)

A one-parameter family \(\Sigma _{t}=F(M^{n},t)\) of complete convex embedded hypersurfaces defined by \(F:M^{n}\times \lbrack 0,T)\rightarrow \mathbb{R} ^{n+1}\) is a solution to the Gauss curvature flow if \(F(p,t)\) satisfies \N\[ \N\frac{\partial }{\partial t}F(p,t)=-K(p,t)\nu (p,t),\N\] \Nwhere \(K(p,t)\) is the Gauss curvature of \(\Sigma _{t}\) at \(F(p,t)\), and \(K(p,t)\) is the unit normal vector of \(\Sigma _{t}\) at \(F(p,t)\) pointing outward of the convex hull of \(\Sigma _{t}\). \N\NThe first main result proves that given a convex bounded domain \(\Omega \subset \mathbb{R}^{n}\), the translating soliton asymptotic to \(\Omega \times \mathbb{R}\) is the unique non-compact ancient solution asymptotic to \(\Omega \times \mathbb{R}\), this uniqueness holding up to translations along the \(e_{n+1}\) direction and reflection about \( \{x_{n+1}=0\}\). \N\NThe authors first recall the definition of translating soliton asymptotic to a cylinder and the existence and uniqueness result proved by \textit{J. Urbas} in [Invent. Math. 91, No. 1, 1--29 (1988; Zbl 0674.35026); Math. Ann. 311, No. 2, 251--274 (1998; Zbl 0910.53043)]. They also define the notions of weak subsolution, weak supersolution, weak solution, and weak ancient solution. They recall the existence and uniqueness of a weak solution starting at any convex hypersurface and properties of these solutions. The proof is finally obtained translating the weak ancient complete solution and observing this solution around the tip region, then proving that it converges to the unique translating soliton when \(t\rightarrow -\infty \). They derive the uniqueness of non-compact ancient solutions. \N\NThe second main result proves that given a convex bounded domain \(\Omega \subset \mathbb{R}^{n}\) with \(C^{1,1}\) boundary, there exists a compact ancient solution \(\Gamma _{t}\subset \mathbb{R}^{n+1}\) to the Gauss curvature flow which is defined for all \(t\in (-\infty ,T)\), it becomes extinct at \(T=-\frac{2V_{\Omega }}{\omega _{n}}\), and has asymptotic cylinder \(\Omega \times \mathbb{R}\). Here \(V_{\Omega }\) is the volume under the graph of the translating soliton asymptotic to \(\Omega \times \mathbb{R}\) . The proof is based on properties of a compact weak ancient solution \( \Gamma _{t}\). The third main result proves that given a convex bounded \( \Omega \subset \mathbb{R}^{n}\) with \(C^{1,1}\) boundary, let \(\Sigma _{t}\), \( t\in (-\infty ,T)\), be a compact ancient solution to the Gauss curvature flow in \(\mathbb{R}^{n+1}\) which is asymptotic to \(\Omega \times \mathbb{R}\) . Assuming that the solution becomes extinct at time \(T=-\frac{2V_{\Omega }}{ \omega _{n}}\), there is \(\nu \in \mathbb{R}\) such that \(\Sigma_{t}+ve_{n+1}= \Gamma_{t}\), for all \(t\in (-\infty ,T)\), where \(\Gamma _{t}\) is the solution constructed in the second main result. The authors here use tools found by \textit{T. Bourni} et al. in [J. Differ. Geom. 119, No. 2, 187--219 (2021; Zbl 1489.53122); Calc. Var. Partial Differ. Equ. 59, No. 4, Paper No. 133, 15 p. (2020; Zbl 1446.53071)].