1,721,074 research outputs found
Callias-type operators in C∗-algebras and positive scalar curvature on noncompact manifolds
No full text
Periodic Solutions of Semilinear Multivalued and Functional Evolution Equations in Banach Spaces
No full textThis paper deals with the semilinear multivalued evolution equationx'(t) + A(t)x(t) Є F(t, x(t)), t Є [a, b] and x Є E,in an arbitrary Banach space E.The linear operators {A(t) : t Є [a, b]} are densely defined on a common domain in E and generate a strongly continuous evolution system. We discuss the existence of mild periodic solutions, also in the case when the nonlinear term F depends on aretarded argument. We also show that in both cases the solutions set is compact. The proofs are based on topological arguments and make use of the theory of condensing multimaps
The positive mass theorem and distance estimates in the spin setting
No full textLet E \mathcal {E} be an asymptotically Euclidean end in an otherwise arbitrary connected Riemannian spin manifold ( M , g ) (M,g) . We show that if E \mathcal {E} has negative ADM-mass, then there exists a constant R > 0 R > 0 , depending only on E \mathcal {E} , such that M M must become incomplete or have a point of negative scalar curvature in the R R -neighborhood around E \mathcal {E} in M M . This gives a quantitative answer to Schoen and Yau’s question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau [ Positive scalar curvature on noncompact manifolds and the liouville theorem , 2020; The positive mass theorem with arbitrary ends , 2021] without the spin condition in dimensions ≤ 7 \leq 7 assuming Schwarzschild asymptotics on the end E \mathcal {E} . We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end E \mathcal {E} . Here we obtain refined constants reminiscent of Gromov’s metric inequalities with scalar curvature
A long neck principle for Riemannian spin manifolds with positive scalar curvature
No full textWe develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if scal(X)≥n(n−1) and there is a nonzero degree map into the sphere f:X→Sn which is strictly area decreasing, then the distance between the support of df and the boundary of X is at most π/n. This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if scal(X)>σ>0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of ∂X is at most π(n−1)/(nσ)−−−−−−−−−−√. Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to N×[−1,1], with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if scal(V)≥σ>0, then the distance between the boundary components of V is at most 2π(n−1)/(nσ)−−−−−−−−−−√. This last constant is sharp by an argument due to Gromov.Georg-August-Universität Göttingen (1018
Nonnegative scalar curvature on manifolds with at least two ends
No full textLet
be an orientable connected
-dimensional manifold with
and let
be a two-sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of
and
are either both spin or both nonspin. Using Gromov's
-bubbles, we show that
does not admit a complete metric of psc. We provide an example showing that the spin/nonspin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension 7, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension 2. We deduce as special cases that, if
does not admit a metric of psc and
, then
does not carry a complete metric of psc and
does not carry a complete metric of uniformly psc, provided that
and
, respectively. This solves, up to dimension 7, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.Studienstiftung des Deutschen Volkes http://dx.doi.org/10.13039/501100004350Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/50110000165
Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields
Full text linkWe prove a positive mass theorem for spin initial data sets that
contain an asymptotically flat end and a shield of dominant energy (a subset of
on which the dominant energy scalar has a positive lower bound).
In a similar vein, we show that for an asymptotically flat end
that violates the positive mass theorem (i.e. ),
there exists a constant , depending only on , such that any
initial data set containing must violate the hypotheses of
Witten's proof of the positive mass theorem in an -neighborhood of
. This implies the positive mass theorem for spin initial data
sets with arbitrary ends, and we also prove a rigidity statement. Our proofs
are based on a modification of Witten's approach to the positive mass theorem
involving an additional independent timelike direction in the spinor bundle.Comment: 18 page
Lipschitz rigidity for scalar curvature
No full textLet M be a closed smooth connected spin manifold of even dimension n , let g be a Riemannian metric of regularity W^{1,p} , p > n , on M whose distributional scalar curvature in the sense of Lee–LeFloch is bounded below by n(n-1) , and let f \colon (M,g) \to \mathbb{S}^n be a 1 -Lipschitz continuous (not necessarily smooth) map of nonzero degree to the unit n -sphere. Then f is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his Four lectures . Our proof is based on spectral properties of Dirac operators for low regularity Riemannian metrics and twisted with Lipschitz bundles. We argue that the existence of a nonzero harmonic spinor field forces f to be quasiregular in the sense of Reshetnyak, and in this way connect the powerful theory for quasiregular maps to the Atiyah–Singer index theorem
Boîte à outils. Promouvoir l’égalité : la contribution des politiques sociales en Amérique latine et dans les Caraïbes
Full text linkL’objectif principal de cette boîte à outils est de faciliter le diagnostic des nombreuses facettes de l’inégalité sociale qui prévaut en Amérique latine et dans les Caraïbes et de fournir des informations pertinentes sur les politiques sociales mises en œuvre dans différents pays de la région qui se sont avérées efficaces dans la réduction de ces inégalités. Il s’agit également d’apporter des données sur les principales normes internationales propres à faciliter, dans le cadre d’une approche fondée sur les droits, la formulation et la mise en œuvre de politiques sociales susceptibles d’atténuer les inégalités en question. Le diagnostic ainsi posé permet de cerner les défis qui doivent être relevés par les politiques sociales, notamment pour pouvoir progresser dans l’amélioration des conditions de vie des groupes de population restés à la traîne. Cet ensemble d’expériences constitue un point de départ pour explorer et élargir le cadre des possibilités de réponse face aux inégalités.Résumé .-- I. Introduction Contexte régional et objectifs de la boîte à outils .-- II. Cadre conceptuel .-- III. Vers plus d’égalité socioéconomique .-- IV. Genre .-- V. Enfance et adolescence .-- VI. la jeunesse .-- VII. Âge adulte .-- VIII. Vieillissement et vieillesse .-- IX. Peuples autochtones .-- X. Personnes d’ascendance
africaine .-- XI. Personnes handicapées .-- XII. Migrants .-- XIII. Inégalités territoriales .-- XIV. Les défis des politiques publiques pour atteindre l’égalité
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