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It has been proved that, If $\lambda_1,\,\lambda_2,\cdots,\lambda_n$ are real analytic functions from $\mathbb{R}^n$ to $\mathbb{R}$, such that $\lambda_i(0)\neq \lambda_j(0)$ for $i\neq j$, then ther …

It can be proved that all $\mathbb{R^2}\rightarrow\mathbb{R^2}$ mappings with constant singular values are affine. In three dimensions, however, there are non-trivial examples, like $$ \begin{align} …

It can be shown that there are (smoothly bounded, Jordan) domains $E\subset \mathbb{R}^2$ which are $\textit{not}$ images of mappings $f$ from the unit disk (or any other planar domain), such that $\m …

Let $\left(\mathcal{M}^2,g_\mathcal{M};X\right)$ and $\left(\mathcal{N}^2,g_{\mathcal{N}};Y\right)$ be two smooth two-dimensional, simply connected Riemannian manifolds (with or without boundary), equ …

Let $D\subset\mathbb{R}^2$ be a planar domain (maybe simply connected) and consider all the mappings $f:D\to\mathbb{R}^2$ with constant, fixed, positive singular values. Let $E=(E_1,E_2)$ be the ortho …

Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function o …

Consider mappings $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ with differential \begin{align} \mathsf{d} f= \begin{pmatrix} \cos\psi(x) &\cos\phi(y) \\ \sin \psi(x)& \sin\phi(y) \end{pmatrix}, \e …

I have the following PDE in two dimensions $$ 2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0, $$ with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively $$ 2\p …

Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. …

Did you find out the answer to the original question ? I came across this* work (pg. 775, conjecture 7.1) where precisely that question is formulated as a conjecture (I ignore whether or not the autho …

Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 …

Think of a beach ball on an pool of water or sand. Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a …

Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\gamma \r …