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We consider wave maps on $(1+d)$-dimensional Minkowski space. For eachdimension $d\geq 8$ we construct a negatively curved, $d$-dimensional targetmanifold that allows for the existence of a self-similar wave map whichprovides a stable blowup mechanism for the corresponding Cauchy problem.

On the existence and stability of blowup for wave maps into a negatively curved target