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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.

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Consider a compact K¨ahler manifold Mm with Ricci curvature lower bound RicM ≥ −2 (m + 1) . Assume that its universal cover fM has maximal bottom of spectrum 1 fM = m2. Then we prove that fM is isometric to the complex hyperbolic space CHm.

We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].