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

Mondino, Andrea; Semola, Daniele

doi:10.48550/arxiv.2202.01590

Cited by 3 publications

(3 citation statements)

References 82 publications

(248 reference statements)

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“…Very recently, Gigli [17] has shown a quantitative Lipschitz estimate for such harmonic maps, and produced a Cheng type Liouville theorem ( [7]) based on [10], [18], [22] (see also [41], [42]). Mondino-Semola [36] have also obtained a similar result, independently.…”

supporting

confidence: 56%

Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ space

Sakurai¹

2022

Preprint

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supporting

confidence: 56%

Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ space

Sakurai¹

2022

Preprint

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“…When the work on this paper was nearly completed I got knowledge of the related independent work [66] containing partially overlapping results.…”

Section: Some Comments Are In Ordermentioning

confidence: 99%

On the regularity of harmonic maps from ${\sf RCD}(K,N)$ to ${\sf CAT}(0)$ spaces and related results

Gigli¹

2022

Preprint

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For an harmonic map u from a domain U ⊂ X in an RCD(K, N ) space X to a CAT(0) space Y we prove the Lipschitz estimatewhere r ∈ (0, R) is the radius of B. This is obtained by combining classical Moser's iteration, a Bochner-type inequality that we derive (guided by recent works of Zhang-Zhu) together with a reverse Poincaré inequality that is also established here. A direct consequence of our estimate is a Lioville-Yau type theorem in the case K = 0. Among the ingredients we develop for the proof, a variational principle valid in general RCD spaces is particularly relevant. It can be roughly stated as: if (X, d, m) is RCD(K, ∞) and f ∈ C b (X) is so that ∆f ≤ C for some constant C > 0, then for every t > 0 and m-a.e. x ∈ X there is a unique minimizer Ft(x) for y → f (y) +and the map Ft satisfiesHere existence is in place without any sort of compactness assumption and uniqueness should be intended in a sense analogue to that in place for Regular Lagrangian Flows and Optimal Maps (and is related to both these concepts). Finally, we also obtain a Rademacher-type result for Lipschitz maps between spaces as above.

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“…For sake of completeness, let us mention that in the recent years the study of harmonic maps has been generalized not only by relaxing the assumptions on the target, but also on the source space. In this respect, it is worth citing the two groundbreaking papers [14,28], where regularity of harmonic maps from an RCD(K , N ) space into a CAT(0) one is established.…”

Section: Introductionmentioning

confidence: 99%

Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

Lavenant,

Monsaingeon,

Tamanini

et al. 2024

Calc. Var.

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If $$u: \Omega \subset \mathbb {R}^d \rightarrow \textrm{X}$$ u : Ω ⊂ R d → X is a harmonic map valued in a metric space $$\textrm{X}$$ X and $$\textsf{E}: \textrm{X}\rightarrow \mathbb {R}$$ E : X → R is a convex function, in the sense that it generates an $$\textrm{EVI}_0$$ EVI 0 -gradient flow, we prove that the pullback $$\textsf{E}\circ u: \Omega \rightarrow \mathbb {R}$$ E ∘ u : Ω → R is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on $$\textrm{X}$$ X , while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the $$L^q$$ L q norm of $$\textsf{E}\circ u$$ E ∘ u on $$\partial \Omega $$ ∂ Ω controls the $$L^p$$ L p norm of $$\textsf{E}\circ u$$ E ∘ u in $$\Omega $$ Ω for some well-chosen exponents $$p \ge q$$ p ≥ q , including the case $$p=q=+\infty $$ p = q = + ∞ . In particular, our results apply when $$\textsf{E}$$ E is a geodesically convex entropy over the Wasserstein space, and thus settle some conjectures of Brenier (Optimal transportation and applications (Martina Franca, 2001), volume 1813 of lecture notes in mathematics, Springer, Berlin, pp 91–121, 2003).

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

Cited by 3 publications

References 82 publications

Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ space

Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ space

On the regularity of harmonic maps from ${\sf RCD}(K,N)$ to ${\sf CAT}(0)$ spaces and related results

Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

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