We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil3 and the Lie group Sol3 endowed with their standard left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil3 that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol3 that lies on one side of a special plane Et (see the discussion just before Theorem 1.5 for the definition of a special plane in Sol3), then S is the special plane Eu for some u 2 R.

We show that the scaling limit exists and is invariant under dilations and rotations. We give some tools that might be useful to show universality.

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

The present paper is devoted to the theory of discontinuous Markoff processes, that is processes where the transitions between states take place either by “jumps” of some specified kind, or by other means. States are taken as point$x$in an abstract space;$phases$are points ($x, t$) in the product state×time space; sets of states are denoted by$X$, sets of phases by$S$.