Higher order intrinsic weak differentiability and Sobolev spaces between manifolds

Abstract

We define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N . When M and N are endowed with Riemannian metrics, p ≥ 1 and k ≥ 2, this allows us to define the intrinsic higher-order homogeneous Sobolev spaceẆ k,p (M, N ). We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic spaceẆ k,p (M, N ) is that π ⌊kp⌋ (N ) ≃ {0}. We investigate the chain rule for higher-order differentiability in this setting.

Some regularity results for p-harmonic mappings between Riemannian manifolds

“…Let us recall the following definition of Sobolev spaces from [4]. A mapping u : M → N is said to be colocally weakly differentiable if u is measurable and f • u is weakly differentiable for every smooth compactly supported function f ∈ C 1 0 (N, R).…”

Weak geodesics on prox-regular subsets of Riemannian manifolds

“…where DV : [11,12]. Now let S be a closed and connected prox-regular subset of M and U be an open neighborhood of S on which P S is single-valued and locally Lipschitz.…”

Reshetnyak Rigidity for Riemannian Manifolds

“…Finally, note that for every p ≥ 1 (including p ≤ d), there is a notion of weak derivative du of u ∈ W 1,p (M; N) (and not only of ι • u), which is measurable as a function TM → TN [CS16]. This implies, using local coordinates, that our energy density x → dist (g x ,h u(x) ) (du(x), SO(g x , h u(x) )) is indeed a measurable function for every u ∈ W 1,p (M; N).…”