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Let $\Phi : S \times [0,\infty) \to M$ be Struwe's weak global solution to the heat equation with smooth initial data $\phi : S \to M$, where $S$ is a compact surface and $M$ is a compact three-dimensional Riemannian manifold. Except at finitely many points in space-time, $\Phi$ is smooth. At these singular points, non-constant harmonic maps $S^2 \to M$ "bubble off" in the precise sense described by Struwe in 1985.

My question is about the stability of these bubbles under small perturbations of the initial data. Specifically, let $\phi_k : S \to M$ be a sequence of maps such that $\phi_k \to \phi$ in the $C^\infty$ norm. Suppose that $\Phi_k : S \times [0,\infty) \to M$ is a sequence of weak solutions with $\Phi_k(\cdot,0) = \phi_k(\cdot)$ and that $(x_k,t_k) \in S \times (0,T]$ are singularities of the respective $\Phi_k$ at which bubbles $u_k : S \to M$ form. Suppose also that the $u_k$ converge in $C^\infty$ to a harmonic map $u$. If $(x_k,t_k) \to (x,T)$, is it true that $\Phi$ has a bubble of $u$ at $(x,T)$?

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    $\begingroup$ Whenever you have monotonicity and epsilon regularity, the limit of singular points will be singular. If $X_j$ is singular, then epsilon regularity implies that $\Theta_{\Phi_k}(X_j)>\epsilon$ (where $\Theta$ is the monotone quantity: in this case, Struwe's quantity) so $\Theta_{\Phi_k}(X_j,r) >\epsilon$ for all $r>0$ by monotonicity. This passes to the limit, so $\Theta_{\Phi}(X,r)>\epsilon$ for all $r>0$. This implies that $X$ is a singular point. $\endgroup$
    – Otis Chodosh
    Commented Apr 9, 2020 at 22:50
  • $\begingroup$ Thank you for the response. I do see from this that there is a singularity at $(x,T)$. I'm also concerned about the form of the singularity, though, and whether the bubbles converge to a bubble in the limit. I suspect they should. $\endgroup$
    – James Dibble
    Commented Apr 10, 2020 at 14:10

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