Improvements of upper curvature bounds

Lytchak, Alexander; Stadler, Stephan

doi:10.1090/tran/8123

Cited by 7 publications

(4 citation statements)

References 34 publications

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“…This is generalized to the present setting in [40,Theorem 4.1]. See also [74] for the case of maps with Euclidean source and CAT(0) target.…”

Section: Energy and Non-linear Harmonic Mapsmentioning

confidence: 95%

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

Mondino¹,

Semola²

2022

Preprint

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We establish Lipschitz regularity of harmonic maps from RCD(K, N ) metric measure spaces with lower Ricci curvature bounds and dimension upper bounds in synthetic sense with values into CAT(0) metric spaces with non-positive sectional curvature. Under the same assumptions, we obtain a Bochner-Eells-Sampson inequality with a Hessian type-term. This gives a fairly complete generalization of the classical theory for smooth source and target spaces to their natural synthetic counterparts and an affirmative answer to a question raised several times in the recent literature. The proofs build on a new interpretation of the interplay between Optimal Transport and the Heat Flow on the source space and on an original perturbation argument in the spirit of the viscosity theory of PDEs.

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“…This is generalized to the present setting in [40,Theorem 4.1]. See also [74] for the case of maps with Euclidean source and CAT(0) target.…”

Section: Energy and Non-linear Harmonic Mapsmentioning

confidence: 95%

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

Mondino¹,

Semola²

2022

Preprint

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“…A nonsmooth version of (4.8) has already been addressed in [26] (see Theorem 1.2 there) for maps with Euclidean source domain and CAT(0)-target. Nevertheless, as we are going to show in Theorem 4.18, the discussion generalizes to our framework: the main stumbling block to overcome being the absence of Lipschitz vector field on a RCD-space.…”

Section: Hsmentioning

confidence: 99%

“…Also, for the very same reason, we shall drop the subscriptȳ from Y when is bounded as the L 2 -integrability depends no more on the particular chosen pointȳ ∈ Y. Compare the proof with [26,Lemma 3.1]. Lemma 4.17 Let (X, d, m) be a RCD(K , N ) space, Y CAT(0)-space and ⊂ X open and bounded.…”

Section: Hsmentioning

confidence: 99%

A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

Gigli

Nobili

2021

J Geom Anal

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We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$\textsf {CAT} (\kappa )$$ CAT ( κ ) -spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-\textsf {E} (y_t)$$ y t ′ ∈ - ∂ - E ( y t ) one uses in the smooth setting and more precisely that $$y_t'$$ y t ′ selects the element of minimal norm in $$-\partial ^-\textsf {E} (y_t)$$ - ∂ - E ( y t ) . This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of $$L^2$$ L 2 and (0) valued maps: we define the Laplacian of such $$L^2$$ L 2 map as the element of minimal norm in $$-\partial ^-\textsf {E} (u)$$ - ∂ - E ( u ) , provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$ L 2 -dense. Basic properties of this Laplacian are then studied.

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“…For the second claim, we use the strong maximum principle to conclude that ' ı u is constant. The claim follows from [32,Corollary 1.6].…”

Section: Basic Propertiesmentioning

confidence: 84%

The structure of minimal surfaces in CAT(0) spaces

Stadler

2021

J. Eur. Math. Soc.

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Improvements of upper curvature bounds

Abstract: We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.2010 Mathematics Subject Classification. 53C20, 53C23, 58E20.

Cited by 7 publications

References 34 publications

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

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

A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

The structure of minimal surfaces in CAT(0) spaces

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