1,743 research outputs found
Positive scalar curvature and product formulas for secondary index invariants
No full textWe introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature (upsc) outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external product formula for partial secondary invariants, from which we deduce product formulas for the higher.-invariant of a metric with upsc as well as for the higher relative index of two metrics with upsc. Our methods yield a new conceptual proof of the secondary partitioned manifold index theorem and a refined version of the delocalized Atiyah-Patodi-Singer (APS)-index theorem of Piazza-Schick for the spinor Dirac operator in all dimensions. We establish a partitioned manifold index theorem for the higher relative index. We also show that secondary invariants are stable with respect to direct products with aspherical manifolds that have fundamental groups of finite asymptotic dimension. Moreover, we construct examples of complete metrics with upsc on non-compact spin manifolds that can be distinguished up to concordance relative to subsets which are coarsely negligible in a certain sense. A technical novelty in this paper is that we use Yu's localization algebras in combination with the description of K-theory for graded C -algebras due to Trout. This formalism allows direct definitions of all the invariants we consider in terms of the functional calculus of the Dirac operator and enables us to give concise proofs of the product formulas.German Research Foundation (DFG) [1493
Coarse median structures and homomorphisms from Kazhdan groups
No full textWe study Bowditch's notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan's property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock-Drutu-Sapir on the mapping class groups of orientable surfaces. We discuss a metric approximation property of finite subsets in coarsemedian spaces extending the classical result on approximation of Gromov hyperbolic spaces by trees
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
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
Slant products on the Higson–Roe exact sequence
Full text linkWe construct a slant product /: S-p(X x Y) x K1-q (c(red)Y) -> Sp-q(X) on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map mu* : K1-* (c(red)Y) -> K* (Y). We obtain such products on the entire Higson-Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete spine-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona
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
Rudolf Otto filosofo della religione
No full textLo straordinario successo del Sacro (1917), che ha reso celebre Rudolf Otto, ha provocato, per contraccolpo, la diffusione di una figura stilizzata dell’autore, impoverita dall’oblio toccato al resto della sua produzione e da letture parziali e semplificative. Una ricostruzione genetica del modo in cui Otto impone la nozione di «heilig» nel lessico tedesco specializzato, muovendo da Lutero e opponendo al neokantismo una lettura friesiana del trascendentale, fa emergere la qualità filosofica di una teoria del religioso che rivela convergenze insospettate con la fenomenologia husserliana. The extraordinary success of The Holy (1917) made Rudolf Otto famous, but it also contributed to propagate a simplified figure of the author. The rest of his work was condemned to oblivion, which caused misleading interpretations of The Holy itself. A genetic reconstruction of the way in which Otto established the term «heilig» in the technical German proves to be fruitful. When considering his interpretation of Luther and his Fries-based opposition to a neoKantian approach to the concept of «transcendental», the philosophical relevance of Otto’s theory of religion can be fully recognized, as well as unsuspected links to Husserlian phenomenology
An instrument to measure adherence to weight loss programs : the Compliance Praxis Survey-Diet (COMPASS-Diet)
No full textAdherence to behavioral weight loss strategies is important for weight loss success. We aimed to examine the reliability and validity of a newly developed compliance praxis-diet (COMPASS-diet) survey with participants in a 10-week dietary intervention program. During the third of five sessions, participants of the “slim-without-diet” weight loss program (n = 253) completed the COMPASS-diet survey and provided data on demographic and clinical characteristics, and general self-efficacy. Group facilitators completed the COMPASS-diet-other scale estimating participants’ likely adherence from their perspective. We calculated internal consistency, convergent validity, and predictive value for objectively measured weight loss. Mean COMPASS-diet-self score was 82.4 (SD 14.2) and COMPASS-diet-other score 80.9 (SD 13.6) (possible range 12–108), with lowest scores in the normative behavior subscale. Cronbach alpha scores of the COMPASS-diet-self and -other scale were good (0.82 and 0.78, respectively). COMPASS-diet-self scores (r = 0.31) correlated more highly with general self-efficacy compared to COMPASS-diet-other scores (r = 0.04) providing evidence for validity. In multivariable analysis adjusted for age and gender, both the COMPASS-diet-self (F = 10.8, p < 0.001, r2 = 0.23) and other (F = 5.5, p < 0.001, r2 = 0.19) scales were significantly associated with weight loss achieved at program conclusion. COMPASS-diet surveys will allow group facilitators or trainers to identify patients who need additional support for optimal weight loss
Rudolf Mayer
No full textThe bachelor thesis deals with the life and work of Rudolf Mayer. In the first part, attention is given to the author and the reception of his work presented in period magazines and newspapers from the poet´s death in 1945. During the second part of his work is examined in terms of literary discursivity the subjective romanticism
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