Let X be a compact connected strongly pseudoconvex CR manifold of real dimension 2n . 1 in CN. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For n  3 and N = n+1, Yau found a necessary and sufficient condition for the interior regularity of the HarveyLawson solution to the complex Plateau problem by means of Kohn–Rossi cohomology groups on X in 1981. For n = 2 and N  n + 1, the problem has been open for over 30 years. In this paper we introduce a new CR invariant g(1,1)(X) of X. The vanishing of this invariant will give the interior regularity of the Harvey–Lawson solution up to normalization. In the case n = 2 and N = 3, the vanishing of this invariant is enough to give the interior regularity.

We study moduli spaces of O’Grady’s ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the K3[n] case, which are of dimension 19 and 20 respectively.

We study the question of local and global uniqueness of completions, based on null geodesics, of Lorentzian manifolds. We show local uniqueness of such boundary extensions. We give a necessary and sufficient condition for existence of unique maximal completions. The condition is verified in several situations of interest. This leads to existence and uniqueness of maximal spacelike conformal boundaries, of maximal strongly causal boundaries, as well as uniqueness of conformal boundary extensions for asymptotically simple space-times. Examples of applications include the definition of mass, or the classification of inequivalent extensions across a Cauchy horizon of the Taub space-time.

In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Khler Calabi-Yau surfaces or Hopf surfaces.

In this article, we study local holomorphic isometric embeddings from Bn into BN1 × · · · × BNm with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok, and Mok and Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.