Sobolev inequality and isoperimetric inequality for submanifolds in a smooth metric measure space

Brendle recently proved a sharp Sobolev inequality and logarithmic Sobolev inequality for submanifolds in Euclidean space. From the sharp Sobolev inequality, he achieved a breakthrough in the conjecture of isoperimetric inequality for minimal submanifolds. In this paper, we extend Brendle's results to submanifolds in a smooth metric measure space. As an application, we prove some new isoperimetric-type inequalities in some smooth metric measure spaces. For example, we obtain a new isoperimetric-type inequality for self-expander.