Classification of threefold canonical thresholds

We show that the set ${\mathcal{T}}_{3,\mathrm{s}\mathrm{m}}^{\mathrm{c}\mathrm{a}\mathrm{n}}$ of smooth threefold canonical thresholds coincides with ${\mathcal{T}}_{2,\mathrm{s}\mathrm{m}}^{\mathrm{l}\mathrm{c}}={\mathcal{H}\mathcal{T}}_2$, where ${\mathcal{H}\mathcal{T}}_2$ is the $2$-dimensional hypersurface log canonical thresholds characterized by Kuwata \cite{K99a, K99b}. We classify the set ${\mathcal{T}}_3^{\mathrm{c}\mathrm{a}\mathrm{n}}$ of threefold canonical thresholds. More precisely, we prove ${\mathcal{T}}_3^{\mathrm{c}\mathrm{a}\mathrm{n}}=\{0\}\cup \{\frac{4}{5}\}\cup {\mathcal{T}}_{3,\mathrm{s}\mathrm{m}}^{\mathrm{c}\mathrm{a}\mathrm{n}}$.