Transformation of $p$-gradient flows to $p^{\mathrm{\prime }}$-gradient flows in metric spaces

We explicitly construct parameter transformations between gradient flows in metric spaces, called curves of maximal slope, having different exponents when the associated function satisfies a suitable convexity condition. These transformations induce the uniqueness of gradient flows for all exponents under a natural assumption which is satisfied in many examples. We also prove the regularizing effects of gradient flows. To establish these results, we directly deal with gradient flows instead of using variational discrete approximations which are often used in the study of gradient flows.