23 problems

Let $A_1$ and $A_2$ be annuli with inner and outer radii encoded by $r$ and $R$. Nitsche's conjecture. A harmonic homeomorphism between $A_1$ and $A_2$ exists if and only if … The…

Kalaj's $\rho$-Nitsche conjecture. If there exists a $\rho$-harmonic homeomorphism of $A_1$ onto $A_2$, then

Consider an annulus of thickness $\tau$, and write $\delta=A/|\Omega|$ for the prescribed area fraction. Let $D=D_{\alpha,A}$ be an optimal configuration, and let symmetry breaking…

Dimension bound conjecture. One has

Juxtaposition conjecture. All proper holomorphic maps between annuli are juxtapositions of homogeneous maps.

Karpukhin–Medvedev conjecture. If $T>T_1$, the value of $\Sigma_1(\mathbb{A},[g_T])$ is achieved by an explicitly described rotationally symmetric metric, whereas if $T<T_1$, the…

Let $\operatorname{Ann}$ be the annulus semigroup and let … be the stated central extension. Diffeological universality conjecture. This extension is a universal central extension…

Max-spectrum conjecture. For any path $X\in\cP$, the max-spectrum of the ring of $X$-holomorphic functions is an annulus. This gives an algebraic reconstruction of the annulus asso…

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…

Universality conjecture. This central extension is a universal central extension. The claim concerns universality in the category of central extensions of the annulus semigroup. Th…

Let $\operatorname{Witt}^{in}$ be the cone of inward-pointing vector fields on $S^1$, and let $\cP$ be the set of smooth paths $[0,1]\to\operatorname{Witt}^{in}$. A path is geometr…

Nitsche-threshold conjecture. One has $r_\diamond<R$, and for every $r>r_\diamond$ there exists a minimizing $\rho$-harmonic diffeomorphism from $\mathbb{A}(r,1)$ onto…

Kalaj's conjecture. The energy $\mathcal{E}[h]$ among mappings in $\mathcal{F}(\mathbb{A},\mathbb{A}_{*})$ achieves its minimum for certain generalized radial diffeomorphisms betwe…

Monotonicity conjecture. The quantity $\sigma_1^t$ is monotone decreasing as a function of $t$.

Let $a>0$ and consider the buckling eigenvalue problem in the annulus. The eigenfunctions associated with the eigenvalues given by the first zeroes of the equations denoted by and…

Let $\mathbb{A}(r,R)$ and $\mathbb{A}(r_\ast,R_\ast)$ be spherical rings in $\mathbf{R}^n$, and let $\mathcal{F}$ be the family of homeomorphisms between them belonging to…

Let $\mathbb{A}$ and $\mathbb{B}$ be the annular domains appearing in the class $\mathcal{F}(\mathbb{A},\mathbb{B})$ of admissible mappings, and let…

Nitsche bound. Then

Hyperbolic Nitsche conjecture. If a hyperbolic harmonic mapping from $A(r,1)$ onto $A(\tau,\sigma)$ exists, then

The regular-metric annulus conjecture. Under these hypotheses,

Conjecture 1. If there exists a Riemann harmonic mapping of the annulus $A(r,1)$ onto the annulus $A(\tau,\sigma)$, then

Let $\mathbb A=A(r,R)$ and $\mathbb A^\ast=A(r_\ast,R_\ast)$ be circular annuli, and let $h\colon\mathbb A\overset{\textnormal{\tiny{onto}}}{\longrightarrow}\mathbb A^\ast$ be a…