Boundary and Lens Rigidity of Lorentzian Surfaces

Source abstract

Let g g be a Lorentzian metric on the plane R 2 \mathbb {R}^2 that agrees with the standard metric g 0 = − d x 2 + d y 2 g_0=-dx^2+dy^2 outside a compact set and so that there are no conjugate points along any time-like geodesic of ( R 2 , g ) (\mathbb {R}^2,g) . Then ( R 2 , g ) (\mathbb {R}^2,g) and ( R 2 , g 0 ) (\mathbb {R}^2,g_0) are isometric. Further, if ( M , g ) (M,g) and ( M ∗ , g ∗ ) (M^*,g^*) are two dimensional compact time oriented Lorentzian manifolds with space–like boundaries and so that all time-like geodesics of ( M , g ) (M,g) maximize the distances between their points and ( M , g ) (M,g) and ( M ∗ , g ∗ ) (M^*,g^*) are “boundary isometric”, then there is a conformal diffeomorphism between ( M , g ) (M,g) and ( M ∗ , g ∗ ) (M^*,g^*) and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.