Quantitative gradient estimates for harmonic maps into singular spaces

Abstract

In this paper, we will show the Yau's gradient estimate for harmonic maps into a metric space (X, d X ) with curvature bounded above by a constant κ, κ 0, in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of S. Y. Cheng [4] and H. I. Choi [5] to harmonic maps into singular spaces.2010 Mathematics Subject Classification. 58E20.

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

“…Once local Lipschitz continuity has been established, the Bochner-Eells-Sampson inequality with Hessian type term (1.4) will be proved via bootstrap. The main idea, borrowed from the recent [99], is to run the same arguments above changing the squared distance d 2 (x, y) with any power d p (x, y) for 1 < p < ∞ and then to let p → ∞. 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].…”

“…The main idea, borrowed from the recent [99], is to run the same arguments above changing the squared distance d 2 (x, y) with any power d p (x, y) for 1 < p < ∞ and then to let p → ∞. 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.…”

“…To the best of our knowledge, Theorem 1.2 is the first instance of a Bochner-Eells-Sampson inequality with Hessian-type term for harmonic maps when both the source and the target spaces are non smooth. We remark that the appearance of the Hessian-type term in (1.4) is expected to have fundamental importance in the future developments of the theory, see for instance the discussion in the introduction of [99].…”

“…We refer to the very recent work [99] by Zhang-Zhong-Zhu for an analogous statement under the assumption that the source space is a smooth N -dimensional Riemannian manifold with Ricci curvature bounded from below by K and to [32,33] for previous instances of Bochner-Eells-Sampson formulas (without Hessian-type terms) for maps with values into CAT(k) spaces and with source smooth Riemannian manifolds and polyhedra, respectively.…”

Some regularity results for p-harmonic mappings between Riemannian manifolds

“…The dependence of the constant C was further improved by Zhang, Zhong and Zhu in their very recent work [46] 1 . Concerning the quantitative gradient estimate for stationary p-harmonic mappings, Duzaar and Fuchs proved in [6, Theorem 2.1] that, there exist ε and C depending only on n, p and the curvature bound of N , such that if…”