  We derive sharp estimates on the modulus of continuity for solutions of a
large class of quasilinear isotropic parabolic equations on smooth metric
measure spaces (with Dirichlet or Neumann boundary condition in case the
boundary is non-empty). We also derive optimal lower bounds for the first
Dirichlet eigenvalue of a class of homogeneous quasilinear operators, which
include non-variational operators. The main feature is that this class of
operators have corresponding one-dimensional operators, which allow sharp
comparisons with solutions of one-dimensional equations.
