6 problems

Let ${\Bbb D}$ be the unit disk and ${\Bbb S}=\partial{\Bbb D}$ its boundary. Suppose ${\cal A}:[1,\infty)\to[1,\infty)$ is convex and increasing, and for a mapping…

Let $X:[0,t_0]\to\cX^{in}$ be a smooth path of inward-pointing vector fields. A path is geometrically exponentiable if there exists an annulus with a framing $h$ satisfying…

Let $X:[0,t_0]\to\cX^{in}$ be a smooth path of inward-pointing vector fields, and set $z=t+i\theta$. The associated degenerate Beltrami equation is … where $\mu=(X-i)/(X+i)$. Degen…

Let $s\in\mathbb{R}$, write $s=n_s+\{s\}$ with $n_s\in\mathbb{Z}$ and $0<\{s\}\leq 1$, and let $\Omega$ be a bounded $F^{s+1-1/p}_{p,p}$-domain for some $2<p\{s\}<\infty$. Let…

Let $0<s<2$ and $1<p<\infty$ satisfy $s<\frac{2}{p}$. Let $\mu,\nu\in F^s_{p,q}\cap L^\infty_c$ satisfy the ellipticity condition, and let $f$ be the principal solution to the Belt…

Nonvanishing imaginary-part conjecture. Either $\operatorname{Im}(f_z)$ is constant, or