This paper studies domains of determination of linear strictly hyperbolic second order operators P. For an open set, a set Z is a domainof determination when the values of solutions of the differential equationPu=0 are determined on Z by their values in. Fritz John's global H\"olmgren theorem implies that points that canbe reached by deformations of noncharacteristic hypersufaces with initial surface and boundaries in belong to a domain of determination provided thatlocal uniqueness holdsat noncharacteristic surfaces. Using spacelike hypersurfacesyields sharp finite speed results whose domains of determinationare described in terms of influence curves that never exceedthe local speed of propagation. This paper studies deformations of noncharacteristicnonspacelike hypersurfaces. We prove that points reachable by (repeated) deformations by noncharacteristic nonspacelikehypersurfaces coincide exactly with the set of pointsreachable by (repeated) homotopies of timelike arcs whoseinitial curves and endpoints belong to. When the set is a small neighborhood of a forward timelikearc connecting a to b, a natural candidate for Z is the intesection of the future of a with the past of b. This candidate is exact for D'Alembert's equation. We prove that it is also exact when a, b are points close together on a fixed timelikearc. The timelike homotopy criterion fuels the construction of surprising examples for which the domain of determination is strictly larger (resp. strictly smaller) than the future-intersect-past candidate.
Rousseau et al. (Wed,) studied this question.