Translating solitons to flows by powers of the Gaussian curvature in Riemannian products (Q2694011)

Let \(\mathbb{S}^n\) be the \(n\)-dimensional unit sphere, and let \(M\) be a Riemannian manifold in \(\mathbb{R}^n\). Assume that \(M=\mathbb{S}^n\), \(M=\mathbb{R}^n\) or that \(M\) is one of the hyperbolic spaces \(\mathbb{H}^m_\mathbb{R}\) , \(\mathbb{H}^m_\mathbb{C}\) , \(\mathbb{H}^m_\mathbb{K}\), \(\mathbb{H}^2_{\mathbb{O}}\), \((m\geq 1)\). Moreover, let \(\pi_{\mathbb{R}}\) be the projection of \(M\times \mathbb{R}\) on \(\mathbb{R}\) and let \(\alpha>0\), with \(\alpha\in (0,\frac{1}{2}]\) if \(M\neq \mathbb{S}^n\), and \(\alpha\in [\delta_n,\frac{1}{2}]\) if \(M=\mathbb{S}^n\), where \(\delta_n=\frac{1}{2}\), if \(n=2\), and \(\delta_n=\max\{\frac{1}{4},\frac{1}{n-1}\}\), if \(n>2\). Under the above conditions, the author proves that there exists a complete rotational strictly convex translating \(K^{\alpha}\)-soliton \(\Sigma\) in the closed half-space \(M\times[0,+\infty)\) with unbounded height function \({\pi_{\mathbb{R}}}_{\mid \Sigma}\). Here, a translating \(K^\alpha\)-soliton is an oriented strictly convex hypersurface \(\Sigma\subset M\times \mathbb{R}\) such that \(K^\alpha=\Theta\) on \(\Sigma\), where \(K,\Theta\) are the Gaussian curvature and the angle function of \(\Sigma\), respectively. In addiction, the author proves that: \begin{itemize} \item[(i)] \(\Sigma\) is an entire graph over \(M\), if \(M\neq \mathbb{S}^n\); \item[(ii)] \(\Sigma\) is a graph over an open hemisphere \(B\subset \mathbb{S}^n\), with \(\Sigma\) asymptotic to \(\partial B\times [0+\infty)\), if \(M=\mathbb{S}^n\). \end{itemize}