Mean curvature flow in a Riemannian manifold endowed with a Killing vector field
Let \((N^{n+1},g)\) be an \((n+1)\)-dimensional Riemannian manifold endowed with a Killing vector field \(\nu\). Suppose that the distribution orthogonal to \(\nu\) is of constant rank and integrable. Let \(\mathbb{L}^n\) be a complete integral leaf of that distribution and let \(M\subset \mathbb{L}^n\) be a bounded domain with boundary \(\partial M \in C^3\). The Killing graph of a differentiable function \(u:\bar{M}\to \mathbb{R}\) is the hypersurface \(\Sigma_u \subset N\) parametrized by the map \(X(x)=\zeta(x,u(x))\), \(x\in \bar{M}\), where \[ \zeta :\bar{M}\times \mathbb{R} \to N \] is the flow generated by the Killing vector field \(\nu\) with inital values in \(N\). In this paper, the author studies the Killing graphs over a bounded regular domain \(M\) in an integral distribution orthogonal to a Killing vector field with prescribed variable contact angle. Under some condition between the geometry of the domain and the contact angle, the author shows that the solutions of the mean curvature flow of Killing graphs with capillarity type boundary condition converge to a translating solution.
scientific article; zbMATH DE number 7291160
scientific article