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I have a problem regarding a statement in the paper Smooth compactness of self-shrinkers by Colding and Minicozzi. In the article, they define a surface $\Sigma$ in $\mathbb R^3$ to be a self-shrinker ...

Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of $$ \partial_tu-\Delta ...