Using Gromov's h-Principles for curvature inequalities on open manifolds

From Gromov's h-principle for diffeomorphism-invariant open partial differential relations on open manifolds, we derive existence theorems on (pseudo-)Riemannian metrics whose curvatures satisfy certain inequalities. Some typical results: Let $M$ be an open manifold of dimension $n\geq2$. For every real-valued continuous function $f$ on $M$, there is a (possibly incomplete) Riemannian metric with Ricci curvature greater than $f$. Let $\lambda1\leq\dots\leq\lambda{n(n-1)/2}$ and $\eps>0$ be real numbers. If $M$ is parallelizable, then $M$ admits a Riemannian metric $g$ such that pointwise the eigenvalues $\sigma1\leq\dots\leq\sigma{n(n-1)/2}$ of the curvature operator of $g$ satisfy $\lambdai<\sigmai<\lambda_i+\eps$. This is joint work with Marc Nardmann.