| The topic of unique continuation is a topic of interest in many areas of mathematics. Perhaps the most well-known case is the one encountered in complex analysis. In general, it can be stated as follows: if Q is a differential operator in R n, u is a function verifying Qu = 0, and u vanishes to infinite order at a point x0, then u is identically 0 near x. There is a long history of strong unique continuation problems in the elliptic case, starting with the work of Aronszajn and Cordes. In this thesis, we study the hyperbolic case, namely the wave operator, with the Laplacian defined with respect to a certain metric in a Riemannian manifold. We write the Laplacian in the normal form, then we conjugate Q with the aid of a Gaussian, using a method of Tataru. Then we take advantage of an a priori estimate for the conjugate operator (which is the same as a Carleman estimate for Q) to get u = 0 near x, by using a technique of Hormander. |
