Scalar Curvature and Generalized Callias Operators

Abstract

We show that in every dimension n ≥ 8, there exists a smooth closed manifold M n which does not admit a smooth positive scalar curvature ("psc") metric, but M admits an L ∞ -metric which is smooth and has psc outside a singular set of codimension ≥ 8. This provides counterexamples to a conjecture of Schoen. In fact, there are such examples of arbitrarily high dimension with only single point singularities. In addition, we provide examples of L ∞ -metrics on R n for certain n ≥ 8 which are smooth and have psc outside the origin, but cannot be smoothly approximated away from the origin by everywhere smooth Riemannian metrics of non-negative scalar curvature. This stands in precise contrast to established smoothing results via Ricci-DeTurck flow for singular metrics with stronger regularity assumptions. Finally, as a positive result, we describe a KO-theoretic condition which obstructs the existence of L ∞ -metrics that are smooth and of psc outside a finite subset. This shows that closed enlargeable spin manifolds do not carry such metrics.

Metrics on twisted pluricanonical bundles and finite generation of twisted canonical rings

“…Proof. The proof is the same as the preceding proposition, with a direct use of the result in [9]. Proposition 2.9.…”

“…By the similar diagram chase as (3.7), E contains a nice incompressible hypersurface. It follows from [6] (Theorem 1.1), [9] (Theorem 1.5) and [35] (Theorem 1.7) that E admits no PSC metric.…”

“…Assume there is an incompressible hypersurface Σ in M with depth greater than π, then Σ is also incompressible in M 0 . By [9] (Proposition 4.6) there is a covering M0 of M 0 with π 1 p M0 q " π 1 pΣq, such that Σ separates the open band (for the definition see [9]) M0 , and any hypersurface separating M0 admits no PSC metric. Let V be as (2.3), Then any hypersurface Σ 1 separating V also seperates M0 .…”

Nonnegative scalar curvature on manifolds with at least two ends

“…Remark This notion of a model space arises in the scalar and mean curvature comparison theory surrounding Conjecture 1.3. In [3, 25], a compact Riemannian band $$(X,g)$$ is compared, in scalar curvature, mean curvature and width, to a warped product $$(M,g_\varphi )$$ over an arbitrary scalar flat manifold $$(N,g_N)$$ with strictly $$\log$$‐concave warping function. It turns out that, if $$X$$ has Property A, $$\mathrm{scal}(X,g)\geqslant \mathrm{scal}(M,g_\varphi )$$ and $$\mathrm{H}(\partial X,g)\geqslant \mathrm{H}(\partial M,g_\varphi )$$, then $$\mathrm{width}(X,g)\leqslant \mathrm{width}(M,g_\varphi )$$.…”

“…The first established cases of Conjecture 1.3 by Gromov [14, Section 2] used the classical minimal hypersurface method. Subsequently, also a Dirac operator approach to Conjecture 1.3 was developed by Cecchini and Zeidler [2, 3, 36, 37].…”