Modifying the Weil-Petersson Geometry of Certain Moduli Spaces to Establish their Kobayashi

n the study of geometry, mathematicians are interested in how certain geometrical objects curve or osculate, and the “curvature ” of a geometrical object is a measure of how the geometrical object is curving at its various points. In the familiar case of two-dimensional surfaces in the three-dimensional Euclidean space, the curvature at a point of the surface is a number which may be positive, zero or negative. When the surface is a plane (which, of course, does not curve at all), the curvature is zero everywhere on the plane. By contrast, an ellipsoid (see Figure 1) and a horse saddle (or more formally, a hyperbolic paraboloid; see Figure 2) do curve a lot. Furthermore, I the ellipsoid and the horse saddle actually curve in different ways. To visualize this, one may take any point on the ellipsoid, and consider the tangent plane touching the ellipsoid at that point. Then one easily sees that the whole ellipsoid lies on one side of the tangent plane. On the other hand, when one considers the tangent plane touching the horse saddle at a given point, one observes that there are points of the horse saddle lying on each of the two sides of the tangent plane. The standard convention is that the ellipsoid is of positive curvature at each of its points, while the horse saddle is everywhere of negative curvature. An important class of higher dimensional geometrical objects www. innova t i onmagaz ine. com C o v e r S t o r y 2