Homomorphisms from mapping class groups

This paper concerns rigidity of the mapping class groups. We show that any homomorphism $\varphi :{\mathrm{M}\mathrm{o}\mathrm{d}}_g\to {\mathrm{M}\mathrm{o}\mathrm{d}}_h$ between mapping class groups of closed orientable surfaces with distinct genera $g>h$ is trivial if $g\ge 3$ and has finite image for all $g\ge 1$. Some implications are drawn for more general homomorphs of these groups.