We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge- Amp`ere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R3. We prove a general local existence result for a large class of Monge-Amp`ere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth curve.

Let X CPN be a smooth subvariety. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X which attempts to deform the given embedding into a balanced one. If L X is an ample line bundle, considering embeddings via H0(Lk) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldsons theorem relating balanced embeddings to metrics with constant scalar curvature [12]. In our proof we combine Donaldson乫s techniques with an asymptotic result of Liu and Ma [17] which, as we explain, describes the asymptotic behavior of the derivative of the map FS . Hilb whose fixed points are balanced metrics.

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.

We consider the inverse curvature flow of strictly convex hypersurfaces in the space form N of constant sectional curvature K_N with speed given by F^{−α}, where α∈(0,1] for K_N=0,−1 and α=1 for K_N=1, F is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual F_∗ approaching zero on the boundary of the positive cone Γ_+. We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flows.

We provide an exhaustive description of all simply connected positively curved cohomogeneity one manifolds. The description yields a complete classification in all dimensions but 7, where in addition to known examples, our list contains one exceptional space and two infinite families not yet known to carry metrics of positive curvature. The infinite families also carry a 3-Sasakian metric of cohomogeneity one, which is associated to a family of selfdual Einstein orbifold metrics on the 4-sphere constructed by Hitchin.