We study a perturbation begin{equation} Delta u + P | abla u| = h | abla u|, end{equation} of spacetime Laplacian equation in an initial data set $(M, g, p)$ where $P$ is the trace of the symmetric 2-tensor $p$ and $h$ is a smooth function.
We study hypersurfaces with prescribed null expansion in an initial data set. We propose a notion of stability and prove a topology theorem. Eichmairs Perron approach toward the existence of marginally outer trapped surface adapts to the settings of
hypersurfaces with prescribed null expansion with only minor modifications.
We study harmonic maps from a 3-manifold with boundary to $mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $pi / 2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.
