Geometric structure of complete Riemannian manifolds and the scalar curvature equation

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This project is a reserch on the scalar curvature equation that is an analytic formulation of the problem "Which kind of smooth function on a Riemannian manifold can be realized as the scalar curvature of a Riemannian metric which is pointwise conformal to the given metric ?" In this project, we deal with the case of noncompact complete Riemannian manifolds.To take a bload view of the problem, as a reserch of geometric structure of complete Riemannian manifolds, we held, in 1999-2002, a series of meetings on the variational problems, e.g. harmonic maps, spectral geometry and the collapse of Riemannian manifolds, the graph theory, the motion of elastic curves etc., each of which has something in common with the scalar curvature equation.In 2000, the head investigator wrote a survey on the scalar curvature equation on open Riemannian manifolds.In 2000-2002, he also investigated on the separating phenomenon which occurs with concentration of curvature, which we can regard as a kind of bubble, and got some estimates on the separation of a Riemannian manifold by a new invariant called relative weight of end-pairs, in the model case of minimal surfaces.