Sobolev mappings between RCD spaces and applications to harmonic maps: a heat kernel approach
arXiv:2105.08578
Abstract
We investigate a Sobolev map from a finite dimensional RCD space $(X, \dist_X, \meas_X)$ to a finite dimensional non-collapsed compact RCD space $(Y, \dist_Y, \mathcal{H}^N)$. If the image is smooth in a weak sense (which is satisfied if $f_{\sharp}\meas_X$ is absolutely continuous with respect to the Hausdorff measure , or if $(Y, \dist_Y, \mathcal{H}^N)$ is smooth in a weak sense), then the pull-back of the Riemannian metric of $(Y, \dist_Y, \mathcal{H}^N)$ is well-defined as an -tensor on , the minimal weak upper gradient of can be written by using , and it coincides with the local slope for $\meas_X$-almost everywhere points in when is Lipschitz. In particular the last statement gives a nonlinear analogue of Cheeger's differentiability theorem for Lipschitz functions on metric measure spaces. Moreover these results allow us to define the energy of . The energy coincides with the Korevaar-Schoen energy.In order to achieve this, we use a smoothing of via the heat kernel embedding , which is established by Ambrosio-Portegies-Tewodrose and the first named author. Moreover we improve the regularity of , which plays a key role. We show also that $(Y, \dist_Y)$ is isometric to the -dimensional standard unit sphere in and is a minimal isometric immersion if and only if $(X, \dist_X, \meas_X)$ is non-collapsed up to a multiplication of a constant to $\meas_X$, and is an eigenmap whose eigenvalues coincide with the essential dimension of $(X, \dist_X, \meas_X)$, which gives a positive answer to a remaining problem from a previous work by the first named author.