Related papers: Area minimizing polyhedral surfaces are saddle
We show that there exists a geodesic spanner with almost linear number of edges.
We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical…
Given a tiling $\mathcal{T}$ of the plane by straight edge polygons, which is invariant by two independent translations, we construct a family of embedded triply periodic minimal surfaces which desingularizes $\mathcal{T}\times\mathbb{R}$.…
In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal…
For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of…
Generalizing tensor-product splines to smooth functions whose control nets outline topological polyhedra, bi-cubic polyhedral splines form a piecewise polynomial, first-order differentiable space that associates one function with each…
The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric…
We introduce and study a notion of relative 1-homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local H\"older regularity of area…
We prove that singular minimal surfaces with constant Gauss curvature are planes, spheres and cylindrical surfaces. We also classify all singular minimal surfaces with a constant principal curvature and singular minimal surfaces with…
We prove that the regular octahedron has the minimal surface area among 3-polytopes of given volume and having at most six vertices.
We prove the existence of foliations by area-minimizing hypersurfaces in asymptotically flat (AF) manifolds with arbitrary dimension and arbitrary ends. Also we provide behaviors of those hypersurfaces near the infinity of AF ends and…
Let $\mathbb{M}^{2}$ be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$. First, using a…
We give a complete topological classification of minimal surfaces in Euclidian three-space.
We give examples of proper minimal immersions in Euclidean space with very rapid area growth. The first is a proper embedding into $\bf{R}^4$ that yields a stable minimal surface, while the second is a proper immersion into $\bf{R}^3$.…
We prove that a flow on a compact surface is expansive if and only if the singularities are of saddle type and the union of their separatrices is dense. Moreover we show that such flows are obtained by surgery on the suspension of minimal…
We give cohomological criteria for logarithmic good reduction of elliptic surfaces up to modification. Along the way, we prove several more general results about such surfaces in positive characteristic, as well as about log smooth…
We give a characterization of completely regular topological spaces. Applying some recent results for supinf problems in completely regular topological spaces we establish a variational principle for saddle points. Well-posedness of saddle…
we construct a properly embedded minimal surface in the flat product R^2*S^1 which is quasi-periodic but is not periodic.
We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic…
Recently, the properties of supercooled liquids have been studied by mapping instaneous configurations onto minima of the gradient squared. It was originally suggested that this mapping would probe higher-order saddle points of the…