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 form g = udr + rγ. If the area element of rγ is expanding for increasing r, then the equation is, in fact, uniformly parabolic, and the basic existence problem is to take positive initial data at some r = r0 and solve for u on the maximal interval of existence, which above was implicitly assumed to be I = [r0, r1); one often hopes that r1 = ∞. However, the case of greatest physical interest, R > 0, often leads to blow up in finite time so that r1 < ∞. It is the purpose of the present work to investigate the situation in which the blow-up nonetheless occurs in such a way that g is continuously extendible to M̄ = [r0, r1]× Σ as a manifold with totally geodesic outer boundary at r = r1.