We calculate the dimension of the locus of Jacobian elliptic surfaces over P1 with a given Picard number, in the corresponding moduli space.

We study the Plateau Problem of finding an area minimizing disk bounding a given Jordan curve in Alexandrov spaces with curvature ≥ κ. These are complete metric spaces with a lower curvature bound given in terms of triangle comparison. Imposing an additional condition that is satisfied by all Alexandrov spaces according to a conjecture of Perel’man, we develop a harmonic map theory from two dimensional domains into these spaces. In particular, we show that the solution to the Dirichlet problem from a disk is H¨older continuous in the interior and continuous up to the boundary. Using this theory, we solve the Plateau Problem in this setting generalizing classical results in Euclidean space (due to J. Douglas and T. Rado) and in Riemannian manifolds (due to C.B. Morrey).

We show that expanding Kahler-Ricci solitons which have positive holomorphic bisectional curvature and are C^2-asymptotic to a conical Kahler manifold at infinity must be the U(n)-rotationally symmetric expand solitons constructed by Cao.

In this paper, we classify n-dimensional (n ≥ 3) complete non-compact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact nonflat locally conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.

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