论文标题
$ {\ sf cat}（κ）$ - 空间和应用程序的梯度流的差异视角
A differential perspective on Gradient Flows on ${\sf CAT}(κ)$-spaces and applications
论文作者
论文摘要
我们回顾了$ {\ sf cat}（κ）$ - 空间的凸面和较低半连续功能的梯度流的理论，并证明它们可以通过相同的差分包含$ y_t'\ y_t'\ in- \ in- \ in- \ partial^ - partial^ - {\ sf e}（y_t）$中的元素$ y y y y y y y y y y mive y y y y mive y y mive y mive y y y y y y y y y <在$ - \ partial^ - {\ sf e}（y_t）$中。这概括了以前的结果，在这个方向上也假定能量是Lipschitz。 然后，我们将这种结果应用于$ l^2 $和$ {\ sf cat}（0）$估值的korevaar-schoen能量：我们将这种$ l^2 $映射的laplacian定义为$ - \ partial^ - {\ sf e}（\ sf e}（Unept It It It It It It It As），我们将其定义为$ l^2 $映射的laplacian。梯度流的理论可确保承认拉普拉斯的一组地图为$ l^2 $ dense。然后研究该拉普拉斯式的基本特性。
We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on ${\sf CAT}(κ)$-spaces and prove that they can be characterized by the same differential inclusion $y_t'\in-\partial^-{\sf E}(y_t)$ one uses in the smooth setting and more precisely that $y_t'$ selects the element of minimal norm in $-\partial^-{\sf E}(y_t)$. This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar-Schoen energy functional on the space of $L^2$ and ${\sf CAT}(0)$ valued maps: we define the Laplacian of such $L^2$ map as the element of minimal norm in $-\partial^-{\sf E}(u)$, 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.
