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?
> 
> 
> 



Positive scalar curvature on the double of a manifold
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.


