6 problems

Let $n\geqslant 4$ and consider the Yang–Mills Cauchy problem with regular initial data and critical Sobolev norm $\dot{H}_A^{\frac{n-4}{2}}$. Let the conserved norm be the norm as…

High-dimensional fractional Yamabe singularity conjecture. There exists a complete Yamabe metric that is singular on $\Sigma^k$.

Schoen–Yau conjecture. Any distributional solution to the associated partial differential equation would provide a complete metric for the Yamabe problem.

Higher-dimensional singularity conjecture. There exist complete metrics with prescribed singularities of strictly higher dimensions.

Finite-time blowup conjecture. There exists such a compact Riemannian manifold $(M,g)$, of some dimension $d>2$, and such a smooth solution that cannot be smoothly continued to the…

Fractional CR Yamabe solvability conjecture. If