Recall that the Whitehead manifold is a contractible 3-manifold which is not homeomorphic to Euclidean 3-space (Wikipedia entry).
- It was proven by Chang-Weinberger-Yu (link) that the Whitehead manifold can not admit a complete metric of uniformly positive scalar curvature.
- Last year their result was strengthened by Jian Wang in arXiv:1805.03544 to the statement that the Whitehead manifold can not admit a complete metric of positive scalar curvature that decays slowly enough (I have blogged about this result here: link).
- Now it seems that Jian Wang could improve the result even further to the desired one, namely that the Whitehead manifold does not admit any complete metric of positive scalar curvature (arXiv:1901.04605).
In the proof Wang uses minimal surfaces (as in the proof of the earlier result with the decay condition) and again the idea of filling curves by minimal discs. But this time there is a sequence of such curves and discs, and one has to discuss convergence issues. The connection to scalar curvature happens by the Cohn-Vossen inequality (Wikipedia entry) of which Wang proves a suitable version adapted to our needs here.