A beautiful argument by Nabutovsky and Weinberger (see http://math.uchicago.edu/~shmuel/fractal.ps) shows that, if $M$ is any smooth compact manifold of dimension $\ge 5$, then the diameter functional has infinitely local minima on the space of Riemannian metrics on $M$ with sectional curvature bounded by $\vert K\vert\le 1$. The book "Computers, rigidity, and moduli" (http://press.princeton.edu/titles/7903.html) covers this, and its extensions, in detail.
Surprisingly, the proof crucially uses computability theory - specifically, Nabutovsky and Weinberger show how to computably construct for each $e$ a homology sphere $S_e$ such that $S_e$ is actually a sphere iff $\Phi_e(e)$ halts. They then use this to show that if the diameter functional had only finitely many minima, the halting problem would be computable (this of course takes substantial work).
My question is about the current role of computability theory in this area. A paper in the Bulletin of Symbolic Logic by Soare http://projecteuclid.org/euclid.bsl/1102083758, about these results, states that:
Without using computability theory there is no known proof that local minima exist even for simple manifolds like the torus $T^5$.
Not being a geometer in the slightest, my question is: is this still true?
