Generalized inverse mean curvature flows in spacetime (Q2472464)

Let \(S\) be an oriented, space-like, closed, \(C^2\) embedded \(2\)-surface in a space-time \((V, g)\). Let \(\vec\xi\) be a \(C^2\) normal vector field on \(S\), \(\vec H\) the mean curvature vector of the surface, and \(M_H(S)\) its Hawking mass. Let \(S_\lambda\) be any flow of space-like two-surfaces starting at \(S\) with \(\vec\xi\) as normal component of the initial velocity. The authors examine necessary conditions on flows of \(2\)-surfaces in space-time under which the Hawking quasi-local mass is monotone. One of these conditions states that the mean curvature vector \(\vec H\) is space-like or null on \(S\), and \(\vec\xi\) points into the same causal quadrant as \(\vec H\). They consider a subclass of uniformly expanding flows which can be studied for null as well as for space-like directions. In the null case, the flow exists locally. In the space-like case, the uniformly expanding condition leaves a \(1\)-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general.