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We prove a sharp inequality for hypersurfaces in the ndimensional Anti-deSitter-Schwarzschild manifold for general $n \ge  3$. This inequality generalizes the classical Minkowski inequality [19] for surfaces in the three dimensional Euclidean space. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established in [4].

Let $M^n$ be a complete noncompact K\"ahler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $\mathbb{C}^n$. This confirms the uniformization conjecture of Yau under the assumption $M$ has maximal volume growth.

We give existence results for simple closed curves with prescribed geodesic curvature on S2, which correspond to periodic orbits of a charge in a magnetic field.

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations for critical points for the associated variational problems. Using the Euler-Lagrange equations, we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points. Moreover, we use the formalism to construct a horizontal second fundamental form, II0, for vertically rigid spaces and, as a first application, use II0 to show that minimal surfaces cannot have points of horizontal positive curvature and that minimal surfaces in Carnot groups cannot be locally strictly horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.