Boundary and Lens Rigidity of Lorentzian Surfaces

Abstract:

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