An obstruction to the positivity of relative Yamabe invariants
Let \(W\) be a compact manifold with nonempty boundary \(M\). The author studies the question ``does there exist a positive scalar curvature metric on \(W\) with minimal boundary condition?'' The author presents an obstruction to the existence of such a metric in terms of the geometry of the boundary \(M\) and of \(W\). This obstruction arises from combining the bordism theoretical construction of Stolz and the minimal hypersurface method of Schoen-Yau. The author uses this obstruction to present some examples of compact manifolds with nonempty boundary which do not admit a positive scalar curvature metric. The relative Yamabe invariant of Akutagawa and Botvinnik plays a central role in the discussion.
scientific article; zbMATH DE number 1922487