Combinatorial Harmonic Maps and Discrete-group Actions on Hadamard Spaces

Izeki, Hiroyasu; Nayatani, Shin

doi:10.1007/s10711-004-1843-y

Cited by 59 publications

(103 citation statements)

References 27 publications

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“…On a complete CAT( )-space as complete -uniformly convex space, the limit T E t (u) := lim l→∞ (J E t/l ) l (u) exists and call it (non-linear) semigroup or gradient ow associated to E, which was proved by Uwe-Mayer [22] (see also Jost [12]). The gradient ow studied in [22] has in uenced the theory of harmonic maps between geometric singular spaces, for example, it was e ectively applied to prove a xed point theorem in terms of discrete groups acting on spaces and combinatorial harmonic maps between them (see Izeki-Nayatani [9]). The gradient ows in [22] were generalized in Ambrosio-Gigli-Savaré [1] as p-curves of maximal slope for coercive proper lower semicontinuous function E having a p-uniform λ-convexity along a continuous curve on a complete metric space in order to construct gradient ows for functionals on L p -Wasserstein space over a Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Resolvent Flows for Convex Functionals and p-Harmonic Maps

Kuwae

2015

Analysis and Geometry in Metric Spaces

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Abstract:We prove the unique existence of the (non-linear) resolvent associated to a coercive proper lower semicontinuous function satisfying a weak notion of p-uniform λ-convexity on a complete metric space, and establish the existence of the minimizer of such functions as the large time limit of the resolvents, which generalizing pioneering work by Jost for convex functionals on complete CAT( )-spaces. The results can be applied to L p -Wasserstein space over complete p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT( )-spaces in terms of Cheeger type p-Sobolev spaces.

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Section: Introductionmentioning

confidence: 99%

Resolvent Flows for Convex Functionals and p-Harmonic Maps

Kuwae

2015

Analysis and Geometry in Metric Spaces

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“…The Wang invariant plays a crucial role in the theory of rigidity of groups ( [15], [7], [10]). By generalizing the formula (1.2), we obtain the definition of another nonlinear spectral gap which was defined by Gromov [4].…”

Section: Definition 11 (Wang Invariant)mentioning

confidence: 99%

Fixed point property for a CAT(0) space which admits a proper cocompact group action

Toyoda¹

2016

Kodai Math. J.

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Abstract. We prove that if a geodesically complete CAT(0) space X admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of X is less than 1. Let G be a finite connected graph, µ 1 (G) be the linear spectral gap of G, and λ 1 (G, X) be the nonlinear spectral gap of G with respect to such a CAT(0) space X. Then, the result implies that the ratio λ 1 (G, X)/µ 1 (G) is bounded from below by a positive constant which is independent of the graph G. It follows that any isometric action of a random group of the graph model on such X has a global fixed point. In particular, any isometric action of a random group of the graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.

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“…In the case of a Hadamard space, in [7], they prove that if the assumptions in Theorem 1 hold, then there exists a fixed point of the image of the original homomorphism ρ in the original space. Let L be a family of global Busemann nonpositive curvature spaces which is stable under ultralimit (Definition 2.3).…”

Section: Theorem 1 Let ρ Be a Homomorphism From γ Into The Full Isommentioning

confidence: 99%

“…One of the purposes of this paper is to generalize results in [6] and [7] for Hadamard spaces to global Busemann nonpositive curvature spaces: For a family of global Busemann nonpositive curvature spaces which is stable under ultralimit, we investigate whether any isometric action of a finitely generated group on any space in the family has a fixed point, in terms of the energy of equivariant maps from the group into spaces in the family.…”

Section: Introductionmentioning

confidence: 99%

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The energy of equivariant maps and a fixed-point property for Busemann nonpositive curvature spaces

Tanaka

2011

Trans. Amer. Math. Soc.

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Abstract. For an isometric action of a finitely generated group on the ultralimit of a sequence of global Busemann nonpositive curvature spaces, we state a sufficient condition for the existence of a fixed point of the action in terms of the energy of equivariant maps from the group into the space. Furthermore, we show that this energy condition holds for every isometric action of a finitely generated group on any global Busemann nonpositive curvature space in a family which is stable under ultralimit, whenever each of these actions has a fixed point.We also discuss the existence of a fixed point of affine isometric actions of a finitely generated group on a uniformly convex, uniformly smooth Banach space in terms of the energy of equivariant maps.

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Combinatorial Harmonic Maps and Discrete-group Actions on Hadamard Spaces

Cited by 59 publications

References 27 publications

Resolvent Flows for Convex Functionals and p-Harmonic Maps

Resolvent Flows for Convex Functionals and p-Harmonic Maps

Fixed point property for a CAT(0) space which admits a proper cocompact group action

The energy of equivariant maps and a fixed-point property for Busemann nonpositive curvature spaces

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