Christian Bär (UP)
Gromov conjectured that given a closed Riemannian manifold \(M\) and a constant \( \sigma\in\mathbb{R}\) there is a constant \(C(M,\sigma)>0\) such that the following holds: if \(X\) is a compact Riemannian manifold with boundary \(M\) and scalar curvature \(\ge\sigma\), then
\[ \int_M H \le C(M,\sigma). \]
Here \(H\) is the mean curvature of the boundary induced by \(X\). The point is that the upper bound is not allowed to depend on \(X\) apart from its lower scalar curvature bound.
This conjecture has been proved in large generality but one either assumes in addition that \(H>0\) or one has no additional assumption but the constant \(C\) also depends on a lower bound on \(H\). It is strange that a lower bound on \(H\) is needed to obtain an upper bound on its integral.
We show that in dimension 2 no such lower bound on \(H\) is needed. One might be tempted to think that this is an easy consequence of the Gauss-Bonnet theorem but it turns out to be quite subtle and interesting.
