The paper develops an existence theory for solutions of the Abreu equation, which include extremal metrics on toric surfaces. The technique employed is a continuity method, combined with “blow-up” arguments. General existence results are obtained, assuming a hypothesis (the “M-condition”) on the solutions, which is shown to be related to the injectivity radius.

Let .. be the fundamental group of a compact riemannian manifold X of sectional curvature K ≤ .1 and dimension n ≥ 3. We suppose that .. = A .C B is the free product of its subgroups A and B amalgamated over the subgroup C. We prove that the critical exponent δ(C) of C satisfies δ(C) ≥ n.2. The equality occurs if and only if there exist an embedded compact hypersurface Y . X, totally geodesic, of constant sectional curvature .1, whose fundamental group is C and which separates X in two connected components whose fundamental groups are A and B respectively. Similar results hold if .. is an HNN extension, or more generally if .. acts on a simplicial tree without fixed point.

Let T be the Teichm¨uller space of marked genus g, n punctured Riemann surfaces with its bordification T the augmented Teichm ¨uller space of marked Riemann surfaces with nodes, [Abi77,Ber74]. Provided with the WP metric, T is a complete CAT(0) metric space, [DW03,Wol03, Yam04]. An invariant of a marked hyperbolic structure is the length ℓ of the geodesic α in a free homotopy class. A basic feature of Teichm¨uller theory is the interplay of two-dimensional hyperbolic geometry, Weil-Petersson (WP) geometry and the behavior of geodesic-length functions. Our goal is to develop an understanding of the intrinsic local WP geometry through a study of the gradient and Hessian of geodesic-length functions. Considerations include expansions for the WP pairing of gradients, expansions for the Hessian and covariant derivative, comparability models for the WP metric, as well as the behavior of WP geodesics, including a description of the Alexandrov tangent cone at the augmentation. Approximations and applications for geodesics close to the augmentation are developed. An application for fixed points of group actions is described. Bounding configurations and functions on the hyperbolic plane is basic to our approach. Considerations include analyzing the orbit of a discrete group of isometries and bounding sums of the inverse square exponential-distance.

We prove the existence of embedded minimal surfaces of arbitrary genus g ≥ 3 in any flat 3-torus. In fact, we construct a sequence of such surfaces converging to a planar foliation of the 3-torus. In particular, the area of the surface can be chosen arbitrarily large.

Evolving smooth, compact hypersurfaces in Rn+1 with normal speed equal to a positive power k of the mean curvature improves a certain ‘isoperimetric difference’ for k > n.1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n 6 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.