Related papers: New area-minimizing Lawson-Osserman cones

We consider the unconstrained $L_2$-$L_p$ minimization: find a minimizer of $\|Ax-b\|^2_2+\lambda \|x\|^p_p$ for given $A \in R^{m\times n}$, $b\in R^m$ and parameters $\lambda>0$, $p\in [0,1)$. This problem has been studied extensively in…

Tarski's Circle Squaring Problem from 1925 asks whether it is possible to partition a disk in the plane into finitely many pieces and reassemble them via isometries to yield a partition of a square of the same area. It was finally resolved…

We construct two infinite families of algebraic minimal cones in $R^{n}$. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one…

The AdS/CFT correspondence relates Wilson loops in N=4 SYM to minimal area surfaces in $AdS_5\times S^5$ space. Recently, a new approach to study minimal area surfaces in $AdS_3 \subset AdS_5$ was discussed based on a Schroedinger equation…

We consider solutions $u^\varepsilon$ of Poisson problems with the Dirichlet condition on domains $\Omega_\varepsilon$ with holes concentrated at subsets of a domain $\Omega$ non-periodically. We show $u^\varepsilon$ converges to a solution…

We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the…

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m \ge 7$. In this paper, we construct, for each $n=2m$ and $m\ge 3$, a small $n$-gon whose area is the maximal value…

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max…

We settle J. Wetzel's 1970's conjecture and show that a 30{^\circ} circular sector of unit radius can accommodate every planar arc of unit length. Leo Moser asked in 1966 for the smallest (convex) region in the plane that can accommodate…

We prove that in a closed Riemannian manifold with dimension between $3$ and $7$, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case…

We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2\pi$ have minimal area among all nonpositively curved…

We show that every planar convex body is contained in a quadrangle whose area is less than $(1 - 2.6 \cdot 10^{-7}) \sqrt{2}$ times the area of the original convex body, improving the best known upper bound by W. Kuperberg.

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the…

We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension 2 in $\R^3$ are locally $C^{1+\alpha}$-equivalent to minimal cones. The proof is rather elementary, but…

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is…

The notion of tamed Dirichlet space by distributional lower Ricci curvature bounds was proposed by Erbar--Rigoni--Sturm--Tamanini as the Dirichlet space having a weak form of Bakry--\'Emery curvature lower bounds in distribution sense. In…

First we construct minimal hypersurfaces $M\subset\mathbf{R}^{n+1}$ in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone $C\times \mathbf{R}$, for any strictly minimizing strictly stable cone $C$ in…

In this note, we extend diameter bounds of Simon, Topping, and Wu--Zheng to submanifolds with boundary and (potentially non-compact) ambient manifolds with minor curvature restrictions. The bound is dependent on both an integral of mean…

For $\Omega$ a perturbation of the unit ball in $\mathbb{R}^3$, we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy. The sequence converges in $L^1$ to a partition of $\Omega$ whose skeleton is…

We introduce a new family of closed differential forms naturally associated with minimal graphical submanifolds in Euclidean space, defined in arbitrary codimension. For each minimal graph, we construct an explicit closed form whose…