On space-like surfaces with parallel mean curvature vector of an indefinite space form (Q1904465)

The authors consider spacelike surfaces in indefinite space forms with negative definite normal spaces. That is to say, they consider a spacelike surface isometrically immersed in \(N^{p+2}_p(c)\), a pseudo-Riemannian space form with signature \((p,2)\) and constant curvature \(c\). They prove several theorems. The first is a pinching theorem for the square norm of the second fundamental form of a complete spacelike surface with parallel mean curvature vector in \(N^{p+2}_p(c)\). Then they estimate the Gaussian curvature of a conformal metric on a spacelike surface \(M\) with parallel mean curvature vector in \(N^{p+2}_p(c)\). One of their results is the following Theorem: Let \(M\) be a spacelike surface with parallel mean curvature vector in \(N^{p+2}_p(c)\), \(H\) the mean curvature and \(K\) the Gaussian curvature of \(M\) with the induced metric \(ds^2_M\). At non-umbilic points in \(M\), \(H^2-c+K>0\). So we can define the conformal metric \[ \overline{ds}^2=(H^2-c+K)^b ds^2_M \] for any real number \(b\). Then the Gaussian curvature \(\overline K\) of the metric \(\overline{ds}^2\) satisfies \[ \overline K\leq-{(2b-1)K\over (H^2-c+K)^b}, \] and equality holds if there exists a complete three-dimensional totally geodesic submanifold \(N^3_1(c)\) in \(N^{p+2}_p(c)\) such that \(M\subset N^3_1(c)\).