The logarithmic Sobolev inequality for a submanifold in manifolds with asymptotically nonnegative sectional curvature

In this note, we prove a logarithmic Sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional curvature. Like the Michale-Simon Sobolev inequality, this inequality contains a term involving the mean curvature.