Bilinéarité et géométrie affine attachées aux espaces de spineurs complexes, minkowskiens ou autres
The notions of left Bol and Bol-Bruck actions are introduced. A purely algebraic analogue of a Nono family (Lie triple family), the so called Sabinin-Nono family, is given. It is shown that any Sabinin-Nono family is a left Bol-Bruck action. Finally it is proved that any local Nono family is a local left Bol-Bruck action. On general matters see [L.V. Sabinin 91, 99].
The existence of a singular curve in is proven, whose curvature can be extended to an function. The curve is the boundary of a two dimensional set, minimizing the length plus the integral over the set of the extension of the curvature. The existence of such a curve was conjectured by E. De Giorgi, during a conference held in Trento in July 1992.
Let be a bounded domain in with smooth boundary . We consider the equation , under zero Neumann boundary conditions, where is open, smooth and bounded and is a small positive parameter. We assume that there is a -dimensional closed, embedded minimal submanifold of , which is non-degenerate, and certain weighted average of sectional curvatures of is positive along . Then we prove the existence of a sequence and a positive solution such that in the sense of measures, where ...