A stability result for translating space-like graphs in Lorentz manifolds

In this paper, we investigate space-like graphs defined over a domain $Ω\subset M^{n}$ in the Lorentz manifold $M^{n}\times\mathbb{R}$ with the metric $-ds^{2}+σ$, where $M^{n}$ is a complete Riemannian $n$-manifold with the metric $σ$, $Ω$ has piecewise smooth boundary, and $\mathbb{R}$ denotes the Euclidean $1$-space. We can prove an interesting stability result for translating space-like graphs in $M^{n}\times\mathbb{R}$ under a conformal transformation.

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