We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in R n for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close to the flat metric.
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