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43
METRICS OF POSITIVE SCALAR CURVATURE AND GENERALISED MORSE FUNCTIONS, PART 1
, 2008
"... It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, thi ..."
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It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in the case of closed simply connected manifolds of dimension at least five.
Geometry of Spin and Spinc structures in the Mtheory partition function
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SMOOTH YAMABE INVARIANT AND SURGERY
, 804
"... Abstract. We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M, such that σ(N) ≥ min{σ(M), Λn}. ..."
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Abstract. We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M, such that σ(N) ≥ min{σ(M), Λn}.
The Yamabe invariant for nonsimply connected manifolds
 J. Differential Geom
, 2001
"... The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension # 5. We extend this to show that Yamabe invariant is ..."
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Cited by 8 (2 self)
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The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension # 5. We extend this to show that Yamabe invariant is nonnegative for all closed manifolds of dimension # 5 with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed manifolds of dimension # 5 with fundamental group of odd order having all Sylow subgroups elementary abelian, with nonspin universal cover, admit metrics of positive scalar curvature, once one restricts to the "complement" of manifolds whose homology classes are "toral." The exceptional toral homology classes only exist in dimensions not exceeding the "rank" of the fundamental group, so this proves important cases of the GromovLawsonRosenberg Conjecture once the dimension is su#ciently large. 1
On the σ2scalar curvature
 J. Differential Geom
, 2010
"... Abstract. In this paper, we establish an analytic foundation for a fully nonlinear equation σ2 σ1 = f on manifolds with positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equa ..."
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Abstract. In this paper, we establish an analytic foundation for a fully nonlinear equation σ2 σ1 = f on manifolds with positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equation without any condition on the sign of f and it is elliptic not only for f> 0 but also for f < 0. By defining a Yamabe constant Y2,1 with respect to this equation, we show that a positive scalar curvature manifold admits a conformal metric with positive scalar curvature and positive σ2scalar curvature if and only if Y2,1> 0. We give a complete solution for the corresponding Yamabe problem. Namely, let g0 be a positive scalar curvature metric, then in its conformal class there is a conformal metric with σ2(g) = κσ1(g), for some constant κ. Using these analytic results, we give a rough classification of the space of manifolds with positive scalar curvature metrics. 1.
LARGE AND SMALL GROUP HOMOLOGY
, 2009
"... For several instances of metric largeness like enlargeability or having hyperspherical universal covers, we construct nonlarge subvectorspaces in the rational homology of finitely generated groups. The functorial properties of this construction imply that the corresponding largeness properties of ..."
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For several instances of metric largeness like enlargeability or having hyperspherical universal covers, we construct nonlarge subvectorspaces in the rational homology of finitely generated groups. The functorial properties of this construction imply that the corresponding largeness properties of closed manifolds depend only on the image of their fundamental classes under the classifying map. This is applied to construct, amongst others, examples of essential manifolds whose universal covers are not hyperspherical, thus answering a question of Gromov (1986).
Perelman’s λfunctional and the SeibergWitten equations
, 2006
"... In this paper we study the supremum of Perelman’s λfunctional λM(g) on Riemannian 4manifold M by using the SeibergWitten equations. We prove among others that, for a compact KählerEinstein complex surface (M, J, g0) with negative scalar curvature, (i) If g1 is a Riemannian metric on M with λM( ..."
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Cited by 6 (3 self)
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In this paper we study the supremum of Perelman’s λfunctional λM(g) on Riemannian 4manifold M by using the SeibergWitten equations. We prove among others that, for a compact KählerEinstein complex surface (M, J, g0) with negative scalar curvature, (i) If g1 is a Riemannian metric on M with λM(g1) = λM(g0), then Volg1(M) ≥ Volg0(M). Moreover, the equality holds if and only if g1 is also a KählerEinstein metric with negative scalar curvature. (ii) If {gt}, t ∈ [−1, 1], is a family of Einstein metrics on M with initial metric g0, then gt is a KählerEinstein metric with negative scalar curvature.
Positive scalar curvature, diffeomorphisms and the Seiberg–Witten invariants
, 2001
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