On the difference of functions with timelike graphs (Q2477446)

This note announces two results on time-like graphs \(\Gamma_f :=\{x_0=f(x_1,\dots,x_n)\}\) in \((n+1)\)-dimensional Minkowski space, with \((x_1,\dots,x_n)\in D\), where \(D\) is a piecewise smooth domain in \({\mathbb R}^n\): (i) a uniqueness result for \(C^2\) solutions of \(M[f]=H(x,f(x))\), where \(M[f]\) is the mean curvature of \(\Gamma_f\), \(H\) is a given function, assumed nonincreasing in its second argument, and \(f\) satisfies Neumann or Dirichlet conditions on different parts of \(\partial D\); (ii) a comparison principle for functions \(f\) and \(g\) such that the square of the gradient of \(f-g\), for the induced metric on \(\Gamma_f\), is nonpositive.