Has Yau's conjecture been proved?

Question

The Wikipedia article about Yau's conjecture says "The conjecture has recently been claimed by Kei Irie, Fernando Codá Marques, and André Neves in the generic case." https://en.wikipedia.org/wiki/Yau%27s_conjecture

Does this mean they claim it has been proved for all cases? Has it been proved once and for all?

Answer 1

Their theorem is that for every compact 3-dimensional manifold $M$ in the space $Riem(M)$ (consisting of all Riemannian metrics on $M$ equipped with $C^\infty$ topology), there exists a countable union of nowhere dense closed subsets: $$ C= \bigcup_{n\in {\mathbb N}} C_n\subset Riem(M) $$ such that for every metric $g\in U:=Riem(M) \setminus C$, Yau's conjecture holds. The set $U$ can be regarded as a set of "generic" Riemannian metrics on $M$. (For instance, it is dense in $Riem(M)$.)

As an aside, their proof depends on the earlier result by Liokumovich, Marques and Neves proving a "nonlinear version of Weyl's law" (conjectured by Gromov) for minimal surfaces.

I am not sure though how much the terminology I used makes sense to you.

Answer 2

Just in the generic case. But note that, more recently, a student of Marques has proven the conjecture for all metrics. See

https://arxiv.org/pdf/1806.08816.pdf

One can even update the Wikipedia article now!