Blow-up in the Parabolic Scalar Curvature Equation

TL;DR

This paper investigates finite-time blow-up phenomena in solutions to the parabolic scalar curvature equation, which models prescribed scalar curvature metrics, and explores conditions under which solutions extend continuously despite blow-up.

Contribution

It provides an analysis of blow-up behavior in the parabolic scalar curvature equation and conditions for continuous extension of solutions at finite blow-up time.

Findings

Finite-time blow-up can occur for positive scalar curvature R.

Solutions may extend continuously to the boundary despite blow-up.

Conditions for the extendibility of the metric at the blow-up time are identified.

Abstract

The \textit{parabolic scalar curvature equation} is a reaction-diffusion type equation on an $(n-1)$-manifold $\Sigma$, the time variable of which shall be denoted by $r$. Given a function $R$ on $[r_0,r_1)\times\Sigma$ and a family of metrics $\gamma(r)$ on $\Sigma$, when the coefficients of this equation are appropriately defined in terms of $\gamma$ and $R$, positive solutions give metrics of prescribed scalar curvature $R$ on $[r_0,r_1)\times\Sigma$ in the form \[ g=u^2dr^2+r^2\gamma.\] If the area element of $r^2\gamma$ is expanding for increasing $r$, then the equation is parabolic, and the basic existence problem is to take positive initial data at some $r=r_0$ and solve for $u$ on the maximal interval of existence, which above was implicitly assumed to be $I=[r_0,r_1)$; one often hopes that $r_1=\infty$. However, the case of greatest physical interest, $R>0$, often leads to blow-up in finite time so that $r_1<\infty$. 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 $\bar M=[r_0,r_1]\times\Sigma$ as a manifold with totally geodesic outer boundary at $r=r_1$.