Sobolev mappings between RCD spaces and applications to harmonic maps: a heat kernel approach

Abstract

We investigate a Sobolev map $f$ 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 $f(X)$ is smooth in a weak sense (which is satisfied if $f_{\sharp}\meas_X$ is absolutely continuous with respect to the Hausdorff measure $\mathcal{H}^N$, or if $(Y, \dist_Y, \mathcal{H}^N)$ is smooth in a weak sense), then the pull-back $f^*g_Y$ of the Riemannian metric $g_Y$ of $(Y, \dist_Y, \mathcal{H}^N)$ is well-defined as an $L^1$-tensor on $X$, the minimal weak upper gradient $G_f$ of $f$ can be written by using $f^*g_Y$, and it coincides with the local slope for $\meas_X$-almost everywhere points in $X$ when $f$ 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 $f$. The energy coincides with the Korevaar-Schoen energy.In order to achieve this, we use a smoothing of $g_Y$ via the heat kernel embedding $\Phi_t:Y \hookrightarrow L^2(Y, \mathcal{H}^N)$, which is established by Ambrosio-Portegies-Tewodrose and the first named author. Moreover we improve the regularity of $\Phi_t$, which plays a key role. We show also that $(Y, \dist_Y)$ is isometric to the $N$-dimensional standard unit sphere in $\mathbb{R}^{N+1}$ and $f$ 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 $f$ 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.