Prescribing Morse Scalar Curvatures: Blow-Up Analysis

Blow-up in the Parabolic Scalar Curvature Equation

The parabolic scalar curvature equation is a reaction-diffusion type equation on an (n − 1)-manifold Σ, the time variable of which shall be denoted by r. Given a function R on [r0, r1)×Σ and a family of metrics γ(r) on Σ, when the coefficients of this equation are appropriately defined in terms of γ and R, positive solutions give metrics of prescribed scalar curvature R on [r0, r1)× Σ in the fo...

Sequences of Embedded Minimal Disks Whose Curvatures Blow up on a Prescribed Subset of a Line

Blow { up Analysis on the Heat

By blow{up analysis on the descending ow for the Yamabe quotient of a compact Reimannian manifold (M n ; g); n 3 , we prove that the ow globally exists and subconverges to u1 such that the scalar curvature of u 4 n?2

Blow Up Analysis for Anomalous Granular Gases

We investigate in this article the long-time behaviour of the solutions to the energydependant, spatially-homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under statistical description, that dissipates energy during collisions. We assume that the gas is “anomalous”, in the sense that energy dissipation increases when tempera...

PRESCRIBING SCALAR CURVATURE ON Sn

on S for n ≥ 3. In the case R is rotationally symmetric, the well-known Kazdan-Warner condition implies that a necessary condition for (1) to have a solution is: R > 0 somewhere and R′(r) changes signs. Then, (a) is this a sufficient condition? (b) If not, what are the necessary and sufficient conditions? These have been open problems for decades. In Chen & Li, 1995, we gave question (a) a nega...