We then prove that a domain of dependence D contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function φ is in the Zygmund class. Moreover in this case the surface of constant curvature K contained in D has bounded principal curvatures, for every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302577&_mathId=si4.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=b4fca8723f12c207366643b9f77f38e0" title="Click to view the MathML source">K<0class="mathContainer hidden">class="mathCode">. In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302577&_mathId=si5.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=90dac9de45eec8af3c672e0dd0f221a4" title="Click to view the MathML source">∂Dclass="mathContainer hidden">class="mathCode">.
Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature K, as K varies in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302577&_mathId=si6.gif&_user=111111111&_pii=S0001870815302577&_rdoc=1&_issn=00018708&md5=e3b9a5a8544410f1045889e50e0913ee" title="Click to view the MathML source">(−∞,0)class="mathContainer hidden">class="mathCode">.