Abstract
We extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case of the source space being an mathsf{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) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation.
Highlights
A seminal paper in the study of the regularity of harmonic maps between Riemannian manifolds is the one [12] by Eells and Sampson
A crucial result that they obtain is a local Lipschitz estimate in terms of a lower bound on the Ricci curvature of the source manifold and an upper bound on its dimension under the assumption that the target manifold has non-positive sectional curvature and is connected
It is natural to wonder whether the same Lipschitz estimates can be obtained in the non-smooth setting under the appropriate weak curvature condition: we refer to [22,23,24,28,29,34,35,45,46,50] for a non-exhaustive list of papers studying this issue at various levels of generality
Summary
A seminal paper in the study of the regularity of harmonic maps between Riemannian manifolds is the one [12] by Eells and Sampson. It can be seen that whenever the target space has the appropriate kind of Hilbert-like behaviour on small scales - the relevant concept is that of ‘universally infinitesimally Hilbertian metric spaces’ - S2(du) coincides, up to a dimensional constant, with the squared Hilbert-Schmidt norm |du|2H S of du, as expected For our case this is interesting because in [10] it has been proved that C AT (0) spaces (and more generally spaces that are locally C AT (κ)) are universally infinitesimally Hilbertian (see Theorem 6.5 for the rigorous meaning of this) and the energy of a Sobolev map u from an RC D(K , N ) space to a C AT (0) space can be written as E(u) = c(d) |du|2H S dm, providing a close analogue of the defining formula (1.1), where here c(d) is a dimensional constant and d the dimension of the source space when seen as a strongly rectifiable space. We mention that building on top of the content of this paper, in [49] it has been defined a suitable notion of ‘Laplacian’ for maps from (open subsets of) RC D(K , N ) to C AT (0) spaces
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
