Related papers: Deformations of Annuli on Riemann surfaces with Sm…

We extend the main results obtained by Iwaniec and Onninen in Memoirs of the AMS (2012). In the paper it is solved the minimization problem of $(\rho,n)$ energy of Sobolev homeomorphisms between two concentric annuli in the Euclidean space…

The conjecture in question concerns the existence of a harmonic homeomorphism between circular annuli A(r,R) and A(r*,R*), and is motivated in part by the existence problem for doubly-connected minimal surfaces with prescribed boundary. In…

Assume that $(\mathcal{N},\hbar)$ and $(\mathcal{M},\wp)$ are two Riemann surfaces with conformal metrics $\hbar$ and $\wp$. We prove that if there is a harmonic homeomorphism between an annulus $\mathcal{A}\subset \mathcal{N}$ with a…

Given two annuli $\mathbf{A}(r,R)$ and $\mathbf{A}(r_\ast, R_\ast)$, in $\mathbf{R}^3$ equipped with the Euclidean metric and the weighted metric $|y|^{-2}$ respectively, we minimize the Dirichlet integral, i.e. the functional…

Let $\mathbb{A}=\{z: r< |z|<R\}$ and $\A^\ast=\{z: r^\ast<|z|<R^\ast\}$ be annuli in the complex plane. Let $p\in[1,2]$ and assume that $\mathcal{H}^{1,p}(\A,\A^*)$ is the class of Sobolev homeomorphisms between $\A$ and $\A^*$, $h:\A\onto…

Let $\mathbb{A}$ and $\mathbb{A_{*}}$ be two non-degenerate spherical annuli in $\mathbb{R}^{n}$ equipped with the Euclidean metric and the weighted metric $|y|^{1-n}$, respectively. Let $\mathcal{F}(\mathbb{A},\mathbb{A_{*}})$ denote the…

As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism $h \colon A(r,R) \to A(r_*, R_*)$ between planar annuli exists if and only if $\frac{R_*}{r_*} \ge {1/2} (\frac{R}{r} + \frac{r}{R})$. We prove this conjecture when the…

Let $M$ and $N$ be doubly connected Riemann surfaces with $\mathscr{C}^{1,\alpha}$ boundaries and with nonvanishing conformal metrics $\sigma$ and $\wp$ respectively, and assume that $\wp$ is a smooth metric with bounded Gauss curvature…

Let $\rho_\Sigma=h(|z|^2)$ be a metric in a Riemann surface $\Sigma$, where $h$ is a positive real function. Let $\mathcal H_{r_1}=\{w=f(z)\}$ be the family of univalent $\rho_\Sigma$ harmonic mapping of the Euclidean annulus…

We consider the so called combined energy of a deformation between two concentric annuli and minimize it, provided that it keep order of the boundaries. It is an extension of the corresponding result of Euclidean energy. It is intrigue…

Let $\mathbb{A}$ and $\mathbb{B}$ be circular annuli in the complex plane and consider the Dirichlet energy integral of $j-$degree mappings between $\mathbb{A}$ and $\mathbb{B}$. Then we minimize this energy integral. The minimizer is a…

Let $N=(\Omega,\sigma)$ and $M=(\Omega^*,\rho)$ be doubly connected Riemann surfaces and assume that $\rho$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. The paper establishes the existence of homeomorphisms…

Let $A_1$ and $A_2$ be two circular annuli and let $\rho$ be a radial metric defined in the annuli $A_2$. We study the existence and uniqueness of the extremal problem for weighted combined energy between $A_1$ and $A_2$, and obtain that…

In a two dimensional annulus $A_\rho=\{x\in \mathbb R^2: \rho<|x|<1\}$, $\rho\in (0,1)$, we characterize $0$-homogeneous minimizers, in $H^1(A_\rho;\mathbb S^1)$ with respect to their own boundary conditions, of the anisotropic energy…

We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that…

In the biharmonic submanifolds theory there is a generalized Chen's conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a…

We study the global Lipschitz character of minimisers of the Dirichlet energy of diffeomorphisms between doubly connected domains with smooth boundaries from Riemann surfaces. The key point of the proof is the fact that minimisers are…

Given two-dimensional Riemannian manifolds $\mathcal{M},\mathcal{N}$, we prove a lower bound on the distortion of embeddings $\mathcal{M} \to \mathcal{N}$, in terms of the areas' discrepancy $V_{\mathcal{N}}/V_{\mathcal{M}}$, for a certain…

We consider the existence and uniqueness of a minimizer of the extremal problem for weighted combined energy between two concentric annuli and obtain that the extremal mapping is a certain radial mapping. Meanwhile, this in turn implies a…

We consider multivalued maps between $\Omega \subset \mathbb{R}^N$ open ($N \ge 2$) and a smooth, compact Riemannian manifold $\mathcal{N}$ locally minimizing the Dirichlet energy. An interior partial H\"older regularity result in the…