Title | A Differential Perspective on Gradient Flows on CAT(K)-Spaces and Applications |

Publication Type | Journal Article |

Year of Publication | 2021 |

Authors | Gigli, N, Nobili, F |

Volume | 31 |

Issue | 12 |

Pagination | 11780 - 11818 |

Date Published | 2021/12/01 |

ISBN Number | 1559-002X |

Abstract | We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$\textsf {CAT} (\kappa )$$-spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-\textsf {E} (y_t)$$one uses in the smooth setting and more precisely that $$y_t'$$selects the element of minimal norm in $$-\partial ^-\textsf {E} (y_t)$$. This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaarâ€“Schoen energy functional on the space of $$L^2$$and CAT(0) valued maps: we define the Laplacian of such $$L^2$$map as the element of minimal norm in $$-\partial ^-\textsf {E} (u)$$, provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$-dense. Basic properties of this Laplacian are then studied. |

URL | https://doi.org/10.1007/s12220-021-00701-5 |

Short Title | The Journal of Geometric Analysis |

## A Differential Perspective on Gradient Flows on CAT(K)-Spaces and Applications

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