8 problems

Rivière's conjecture. There exists $\delta=\delta(C,n,N)>0$ such that

Spencer-polystability conjecture. The ideal sheaf $\mathcal{I}$ is Spencer-polystable if and only if there exists a possibly singular solution $\varphi\in H^0(X,\mathcal{B})$ minim…

Warped-surface extension conjecture. We conjecture that this result extends to warped surfaces for which $\rho$ depends not only on $x$ but also on $\theta$.

Radius-extension conjecture. We conjecture that these results hold whenever

Let $w$ be a minimal solution of the arrival-time problem described above. Non-jump-to-infinity conjecture. A minimal solution $w$ cannot jump to infinity; consequently, $w$ is con…

Let $\Omega$ be a bounded open subset of $\mathbb R^n$ whose boundary $\Sigma=\partial\overline{\Omega}$ is inner embedded, meaning that $\Sigma$ touches itself from the exterior r…

Let $M^{2m+1}$ be a closed pseudohermitian manifold with vanishing pseudohermitian torsion $A_{\alpha\beta}=0$ and Webster Ricci tensor satisfying … Suppose a positive function $f$…

Vacuum constraint extension conjecture. If $(M,g)$ is an extension of $(\gamma,H)$ in $\mathcal P$, then there exists an extension in $\mathcal C_{(\gamma,H)}$; equivalently,