| Abstract: |
| In 1964, Eells and Sampson proved the celebrated long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. In 1992, Gromov and Schoen initiated the study of harmonic maps into CAT(0) metric spaces. It naturally motivates the study of the harmonic map heat flow into singular metric spaces.
In the 1990s, Mayer and Jost independently studied convex functionals on CAT(0) spaces and extended Crandall-Liggett`s theory of gradient flows from Banach spaces to CAT(0) spaces to obtain the weak solutions for the harmonic map heat flow into CAT(0) spaces. The weak solutions enjoy the favorable long-time existence, uniqueness and well-established long-time behaviors. It is a long-standing open question whether the weak solutions possess Lipschitz regularity. Very recently, by using elliptic approximation method, Lin, Segatti, Sire, and Wang proved the weak solutions are Lipschitz in space and (1/2)-H\older continuous in time, for a wide class of CAT(0) spaces.
In this talk, we will introduce a complete answer to the question, showing that every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time. We also establish an Eells-Sampson-type Bochner inequality. This is based on a joint work with Xi-Ping Zhu. |
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