Global Solutions of Semilinear Parabolic Equations on Negatively Curved Riemannian Manifolds

We are concerned with global existence for semilinear parabolic equations on Riemannian manifolds with negative sectional curvatures. A particular attention is paid to the class of initial conditions which ensure existence of global solutions. Indeed, we show that such a class is crucially related to the curvature bounds. A crucial point in our arguments is the construction of positive bounded supersolutions to the eigenvalue equation.