In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.

We show that any complete f -stable translating soliton M admits no codimension one cycle which does not disconnect M . As a corollary, it follows that any two dimensional complete f -stable translating soliton has genus zero.

No abstract