We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into IRn, n = 3, 4, if there is one conformal immersion with Willmore energy smaller than a certain bound Wn,p depending on codimension and genus p of the Riemann surface. For tori in codimension 1, we know W3,1 = 8 .

In this paper we first introduce a transform for convex functions and use it to prove a Bernstein theorem for a Monge-Amp`ere equation in half space.We then prove the optimal global regularity for a class of Monge-Amp`ere type equations arising in a number of geometric problems such as Poincar´e metrics, hyperbolic affine spheres, and Minkowski type problems.

If M is a hyperbolic 3-manifold with a quasigeodesic flow, then we show that 1(M) acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is M¨obius-like but not conjugate into PSL(2,R).We conjecture that the latter possibility cannot occur.

We study the symplectic structure of the holomorphic coadjoint orbits, generalizing a theorem of McDuff on the symplectic structure of Hermitian symmetric spaces of noncompact type.

Let X be a smooth affine surface, X ! G2 m be a finite morphism. We study the affine curves on X, with bounded genus and number of points at infinity, obtaining bounds for their degree in terms of Euler characteristic. A typical example where these bounds hold is represented by the complement of a three-component curve in the projective plane, of total degree at least 4. The corresponding results may be interpreted as bounding the height of integral points on X over a function field. In the language of Diophantine Equations, our results may be rephrased in terms of bounding the height of the solutions of f(u, v, y) = 0, with u, v, y over a function field, u, v S-units. It turns out that all of this contain some cases of a strong version of a conjecture of Vojta over function fields in the split case. Moreover, our method would apply also to the nonsplit case. We remark that special cases of our results in the holomorphic context were studied by M. Green already in the seventies, and recently in greater generality by Noguchi, Winkelmann, and Yamanoi [12]; however, the algebraic context was left open and seems not to fall in the existing techniques.