Related papers: Lipschitz regularity of energy-minimal mappings be…

Let X be a finite CW complex or compact Lipschitz neighborhood retract with universal cover Z; let M be a compact orientable manifold of dimension at least 2 and nonempty boundary. We establish the existence of an isoperimetric profile for…

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…

We show that for every complete metric space $M$ there exists another complete metric space $N$ of the same density character such that the curve-flat quotient of $N$ is isometric to $M$. Moreover, we show that if $M$ is compact and…

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\widehat{f}$ of a Lipschitz map $f:M\to N$ is Dunford-Pettis if and only if it…

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(\omega , \omega )> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $\omega$ is a…

Let $M$ be a compact, orientable, $n$-dimensional Riemannian manifold, $n\geq2$, and let $F$ be the energy functional acting on the space $\Xi (M)$ of $C^{\infty }$ vector fields of $M$, \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Vert…

We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, \alpha}$ for…

We show that any smooth bi-Lipschitz $h$ can be represented exactly as a composition $h_m \circ ... \circ h_1$ of functions $h_1,...,h_m$ that are close to the identity in the sense that each $\left(h_i-\mathrm{Id}\right)$ is Lipschitz, and…

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…

In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form \[ \mathfrak{F}( v, \Omega )= \int_{\Omega} \! F(x, Dv(x)) \, dx, \] where, for ${n > 2}$ and $N\ge 1$, $\Omega$ is a bounded…

Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing"…

We give characterizations of the bounded subanalytic $\mathscr{C}^\infty$ submanifolds $M$ of $\mathbb{R}^n$ for which the space of Neumann regular functions is dense in Sobolev spaces. By ``Neumann regular function'', we mean a function…

We study $p$--harmonic maps with Dirichlet boundary conditions from a planar domain into a general compact Riemannian manifold. We show that as $p$ approaches $2$ from below, they converge up to a subsequence to a minimizing singular…

Loewner's equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy…

We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichm\"u{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive…

Let $\pi:\mc{X}\to \mc{T}$ be Teichm\"uller curve over Teichm\"uller space $\mc{T}$, such that the fiber $\mc{X}_z=\pi^{-1}(z)$ is exactly the Riemann surface given by the complex structure $z\in \mc{T}$. For a fixed Riemannian manifold $M$…

We generalize the notion of calibrated submanifolds to smooth maps and show that the several examples of smooth maps appearing in the differential geometry become the examples of our situation. Moreover, we apply these notion to give the…

We investigate the continuity of boundary operators, such as the Neumann-to-Dirichlet map, with respect to the coefficient matrices of the underlying elliptic equations. We show that for nonsmooth coefficients the correct notion of…

We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense.…

We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below…