Positive scalar curvature on the double of a manifold

Question

Let $(M,g)$ be a compact Riemannian manifold with boundary and assume it has positive scalar curvature.

Question. Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature?

Answer 1

The answer is negative at least if you do not add some sort of convexity hypothesis for the boundary, and at least in dimension $2$. Take a round $2$-sphere with $h\ge 2$ round holes. It has positive curvature, but its double has genus $h-1\ge 1$ and cannot have a positively curved metric. Of course, in dimension $2$ scalar curvature is far more rigid than in higher dimension.

Answer 2

This is a little expansion on Benoît's answer.

His counterexample actually can be generalized. There is the following conjecture which appears as Conjecture 1.24 in Rosenberg's "Manifolds of positive scalar curvature: A progress report" (http://www2.math.umd.edu/~jmr/psc2006.pdf).

Conjecture A manifold $X$ admits positive scalar curvature (psc), if and only if $X\times S^1$ admits positive scalar curvature.

This conjecture is known to fail in dimension four (a counterexample is $K3\#\overline{\mathbb{CP}}{}^2$), but is verified in many other cases, for example in dimensions 2,3,5,6 or for a great class of Spin-manifolds. This follows for example from the band-width estimates of Räde (https://arxiv.org/abs/2104.10120, Theorem 2.25) and Zeidler (https://arxiv.org/abs/1905.08520, Theorem 1.4).

Now take one of these manifolds, for which the conjecture is verified. By Gromov's h-principle, there is a psc-metric on $X\times \mathbb R$ which can be restricted to give a psc-metric on $M:=X\times [0,1]$. The double of $X$ however is given by $X\times S^1$ and hence admits no psc-metric.

However, the answer is also yes in many cases. Gromov-Lawson have shown in "Spin and scalar curvature in the presence of a fundamental group" (Theorem 5.7) that if M admits a psc-metric with positive mean curvature on the boundary, then the double admits a psc-metric. This has recently been generalized by Bär-Hanke (https://arxiv.org/abs/2012.09127, Corollary 34) to many other boundary condition (positive mean curvature, vanishing mean curvature, non-negative mean curvature, nonnegative second fundamental form, positive second fundamental form, vanishing second fundamental form) in the following way:

Theorem A psc-metric $g$ on $M$ satisfying any of the boundary conditions above can be deformed into a psc-metric $g'$ which can be doubled, i.e. $g'\cup g'$ is a smooth psc-metric on $DM$.

Another very recent result on psc-metrics on manifolds with boundary was obtained by Rosenberg-Weinberger (https://arxiv.org/pdf/2201.01263.pdf), where they give a list of manifold with boundary which admit such a doubling metric.