  We prove two rigidity results for surfaces lying in the standard null
hypersurfaces of Schwarzschild spacetime satisfying certain mean curvature type
equations. The first is for the equation $\alpha_H = - d\log |H|$ studied in
\cite{WWZ}. The second is for the mean curvature vector of constant norm. The
latter is related to the Liouville and Obata Theorem in conformal geometry.
