Korevaar–Schoen’s energy on strongly rectifiable spaces

Abstract

We extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case of the source space being an $$\mathsf{RCD}$$ RCD space. In this situation it appears that no version of the ‘subpartition lemma’ holds: to obtain both existence of the limit of the approximated energies and the lower semicontinuity of the limit energy we shall rely on: the fact that such spaces are ‘strongly rectifiable’ a notion which is first-order in nature (as opposed to measure-contraction-like properties, which are of second order). This fact is particularly useful in combination with Kirchheim’s metric differentiability theorem, as it allows to obtain an approximate metric differentiability result which in turn quickly provides a representation for the energy density, the differential calculus developed by the first author which allows, thanks to a representation formula for the energy that we prove here, to obtain the desired lower semicontinuity from the closure of the abstract differential. When the target space is $$\mathsf{CAT}(0)$$ CAT ( 0 ) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation.

Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

“…In this paper we establish local Lipschitz continuity of harmonic maps from non smooth metric measure spaces with synthetic Ricci curvature lower bounds in the sense of the RCD theory with values into CAT(0) metric spaces with non-positive sectional curvature. This answers to a question raised several times in the recent literature, see [42,28,44,45,49]. Building on the top of this regularity result, we prove a Bochner-Eells-Sampson type inequality with Hessian term, which is expected to be fundamental for future applications.…”

“…Local Hölder regularity of harmonic maps has been recently obtained by Guo in [49], along the lines of the previous [58,72]. The question of local Lipschitz regularity of harmonic maps from RCD(K, N ) spaces to CAT(0) spaces has been raised several times in the recent literature, starting from [42] and later in [28,44,45,49].…”

“…While [99] considers smooth Riemannian manifolds, following the strategy of [98], in the present context the analysis is possible thanks to the Wasserstein contractivity of the Heat Flow in any Wasserstein space with distance W p for 1 ≤ p ≤ ∞, see [84]. Combined with a version of Kirchheim's metric differentiability theorem [64], recently established in [45], this will lead to the sought Bochner-Eells-Sampson inequality in section 7.…”

“…We recall that RCD(K, N ) metric measure spaces (X, d, m) are strongly rectifiable spaces, according to [45,Definition 2.18] (see also the previous [41]). This condition means that for any ε > 0 there exists a countable covering of X with Borel sets U ε i , away from a set of m-measure 0, such that all these sets are (1 + ε)-biLipschitz to Borel subsets of R n with charts ϕ ε i , for a given n ∈ N independent of i, and it holds…”

“…The role of the lower Ricci curvature bound on the source space for the regularity of harmonic maps in the classical theory, together with the remarkable developments of Geometric Analysis on RCD(K, N ) spaces in recent years, give strong motivations for a theory of harmonic maps from RCD(K, N ) spaces with values into CAT(0) metric spaces with non-positive curvature. A theory of Sobolev maps in this framework has been developed by Gigli-Tyulenev in [45], where existence and uniqueness for solutions of the Dirichlet problem have been achieved too. Local Hölder regularity of harmonic maps has been recently obtained by Guo in [49], along the lines of the previous [58,72].…”