2 problems

Discreteness conjecture. Under either of these alternatives, the spectrum of the Laplace–Beltrami operator on $M$ is discrete.

Completeness conjecture. There exists such a metric $G^k$ for which: (a) $(\operatorname{Imm}^k(M,N),G^k)$ is metrically complete; (b) $(\operatorname{Imm}^l(M,N),G^k)$ is geodesic…