Positive scalar curvature metrics on manifolds with boundary

Usually when I blog here about positive scalar curvature, the manifolds I consider are assumed to have no boundary. In this post I want to explain the basics of what happens when the manifolds actually do have a boundary.

So first of all, one has to mention that any compact manifold with boundary admits a Riemannian metric of positive scalar curvature! The argument for this is easy if one takes the following result of Kazdan and Warner for granted:

If \(W\) is any closed manifold and \(f\) any function on \(W\) which is negative somewhere, then there exists a Riemannian metric \(g\) on \(W\) such that the scalar curvature of \((W,g)\) is exactly \(f\).

So let now \(M\) be any manifold with boundary. We consider its double \(W := M \cup_{\partial M} M\) and denote the two halves of it by \(M^+\) and \(M^-\) (which are both canonically identified with \(M\)). We choose any function \(f\) on \(W\) which is negative somewhere on \(M^-\) and is positive on \(M^+\), and invoke the result of Kazdan-Warner to get a corresponding Riemannian metric \(g\) on \(W\). Then \(g|_{M^+}\) is a Riemannian metric of positive scalar curvature on \(M\).

Actually, one can strengthen the above result considerably. The authors of arXiv:2007.06756 propose an argument that every Riemannian metric on \(\partial M\) can be extended to a Riemannian metric of positive scalar curvature on \(M\).

Since the question about the existence of Riemannian metrics of positive scalar curvature on (compact) manifolds with boundary is completely answered, one can try to find the next harder question. For example, one can impose boundary conditions, and a natural quantity to consider here is the mean curvature of the boundary. An important basic result in this direction is the following one of Gromov and Lawson:

If a compact Riemannian manifold with boundary carries a Riemannian metric of positive scalar curvature and such that the boundary has positive mean curvature, then its double carries a Riemannian metric of positive scalar curvature.

Since there are obstruction to the existence of positive scalar curvature metrics on closed manifolds, the above theorem provides restrictions on the existence of positive scalar curvature metrics on compact manifolds with positive mean curvature on the boundary.

(If we only ask about non-negative scalar curvature and non-negative mean curvature on the boundary, then there are also results like the following one to rule out such metrics: link.)

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