We prove Li-Yau-Hamilton inequalties that extend Hamilton's matrix inequality for solutions of the Ricci flow with nonnegative curvature operators. To obtain our extensions, we apply the space-time formalism of S.-C. Chu and one of the authors to solutions of the Ricci flow modified by a cosmological constant. Then we adjoin to the Ricci flow the evolution of a 1-form and a 2-form flowing by a system of heat-type equations. By a rescaling argument, the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of Hamilton and one of the authors.

We prove that a finite topology properly embedded Bryant surface in a complete hyperbolic 3-manifold has finite total curvature. This permits us to describe the geometry of the ends of such a Bryant surface. Our theory applies to a larger class of Bryant surfaces, which we call quasi-embedded. We give many examples of these surfaces and we show their end structure is modelled on the quotient of a ruled Bryant catenoid end by a parabolic isometry. When the ambient hyperbolic 3-manifold is hyperbolic 3-space, the theorems we prove here were established by Collin, Hauswirth and Rosenberg, 2001.

We construct embedded minimal surfaces of finite total curvature in euclidean space by gluing catenoids and planes. We use Weierstrass Representation and we solve the Period Problem using the Implicit Function Theorem. As a corollary, we obtain the existence of minimal surfaces with no symmetries.

We construct complete bounded minimal surfaces in ℝ³ with arbitrary topological genus.