where r = ∫ Rdμ/ ∫ dμ is the average scalar curvature (R is the scalar curvature) and Ric is the Ricci curvature tensor of h. Hamilton then spectacularly illustrated the success of this method by proving, when n = 3, that if the initial Riemannian metric has strictly positive Ricci curvature it evolves through time to a positively curved Einstein metric h∞ on M . And, because n = 3, such a Riem...

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kähler metric of constant scalar curvature on the blow-up according to [18]. We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus we can produce numerous examples of Kähle...

On modifying the gravitational action by addition of higher-derivative terms of curvature, it is obtained that the Ricci scalar behaves as a physical field as well as a geometrical field. Riccion is a particle giving the physical aspect of the Ricci scalar curvature. Here, it is probed about the possibility of riccion being a candidate for cosmic cold dark matter. PACS nos.: 98.80-k; 95.30 C.

In this paper we continue to study Riemannian manifolds (M, g) equipped with a smooth measure m. In particular, we show that the construction of conformally covariant operators due to Graham-Jenne-Mason-Sparling can be adapted to this setting. As a by-product, we define a family of scalar curvatures, one of which corresponds to Perelman’s scalar curvature function. We also study the variational...

Problem 1.1. Given a compact constant scalar curvature Kähler manifold (M,J, g, ω), of complex dimension m := dimC M , and having defined △ := {(p1, . . . , pn) ∈ M n : ∃ a 6= b pa = pb}, characterize the set PW = {(p1, . . . , pn, α1, . . . , αn)} ⊂ (M n \ △) × (0,+∞) for which M̃ = Blp1,...,pnM , the blow up of M at p1, . . . , pn has a constant scalar curvature Kähler metric (cscK from now on...