Two Problems On Non-positive Curved Manifolds

In this thesis we will discuss two problems related to non-positive curvature met-rics. The first problem comes from conformal geometry. Let (M, g) be a Poincare-Einstein manifold with a smooth denning function. We prove that there are infinitely many asymptotically hyperbolic metrics with constant Q-curvature in the conformal class of an asymptotically hyperbolic metric close enough to g. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.The second problem is the local and global inverse mapping problem on Hadamard manifolds. Let F:Mâ†'N be a C1map between Riemannian manifolds of the same dimension, M-complete, N-Cartan-Hadamard. We show that F is a C1diffeomorphism if infx∈M|d(BζoF)(x)|>0for all ζ∈N(∞) and Busemann functions Bζ. This generalizes the Cartan-Hadamard theorem and the Hadamard invertibility criterion, which requires infx∈M||DF(x)-1||-1=infζ∈N(∞) infx∈M|d(BζoF)(x)|>0. Our proofs use a version of the shooting method for two-point boundary value problems. These ideas lead to new results about the size of the critical set of a function f∈C2(Rn,R):a) If infx∈Rn|Hess f(x)v|>0for all v≠0then the function f has precisely one critical point. b) If g∈C2(Rn,R) is the C1local uniform limit of functions as in a), and Hess g(x) is nowhere singular, then g has at most one critical point. The totality of functions described in b) contains properly the class consisting of all C2strictly convex functions defined on Rn.