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Register a new userThe space of metrics of positive scalar curvature
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Bernhard Hanke
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We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements of infinite order in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (observer) moduli space of such metrics. Along the way we construct smooth fiber bundles over spheres whose total spaces have non-vanishing A-hat-genera, thus establishing the non-multiplicativity of the A-hat-genus in fibre bundles with simply connected base.
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We prove that a closed $n$-manifold $M$ with positive scalar curvature and abelian fundamental group admits a finite covering $M$ which is strongly inessential. The latter means that a classifying map $u:Mto K(pi_1(M),1)$ can be deformed to the $(n-2)$-skeleton. This is proven for all $n$-manifolds with the exception of 4-manifolds with spin universal coverings.
Extending Aubins construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, tinmathbb{R}$. In particular, there are no topological obstructions for metrics with $varepsilon$-pinched Weyl curvature and negative scalar curvature.
The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of B(pi_0 Diff(M)) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.
In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $(Sigma,gamma,H)$. We prove that given a metric $gamma$ on $mathbf{S}^{n-1}$ ($3leq nleq 7$), $(mathbf{S}^{n-1},gamma,H)$ admits no fill-in of NNSC metrics provided the prescribed mean curvature $H$ is large enough (Theorem ref{Thm: no fillin nonnegative scalar 2}). Moreover, we prove that if $gamma$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $mathbf{S}^{n-1}$, then the much weaker condition that the total mean curvature $int_{mathbf S^{n-1}}H,mathrm dmu_gamma$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P.,23 in cite{Gromov4}). In the second part of this paper, we investigate the $theta$-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.
Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^infty)$ on a compact manifold $M^n$ ($nge 3$) with negative Yamabe invariant $sigma(M)$. It is well-known that if $g$ is a smooth metric on $M$ with unit volume and with scalar curvature $R(g)ge sigma(M)$, then $g$ is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles $leq 2pi$ along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of $M$ attains its minimum, then the same is true for $L^infty$ metrics with isolated point singularities.
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