arXiv research

New minimal surfaces found using Toda lattice and integrable systems.

Paper finds minimal number of curves in surface systems.

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal sur…

Unified representation for minimal and constant mean curvature surfaces.

A minimal space-like surface in Minkowski space-time is said to be of general type if it is free of degenerate points. The fact that minimal space-like surfaces of general type in Minkowski space-time admit canonical parameters of the first (second) type implies that any minimal space-like surface is determined uniquel…

New limits of minimal surface systems have surprising large interior parts.

The paper finds a Weierstrass representation for a specific type of Lorentzian minimal surface.

New approach to solving minimal surface system Dirichlet problem on smooth domains.

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for st…

We apply the complex analysis over the double numbers $D$ to study the minimal time-like surfaces in $R^4_2$. A minimal time-like surface which is free of degenerate points is said to be of general type. We divide the minimal time-like surfaces of general type into three types and prove that these surfaces admit specia…

The paper classifies affine minimal translation surfaces and finds their properties.

Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type locally admits geometrically determined parameters - canonical parameters. In such pa…

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of …

We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.

We solve the Cauchy-Dirichlet problem for the minimal surface system in arbitrary dimension and codimension assuming a condition on the variation of the initial submanifold .

We consider a surface $M$ immersed in $\mathbb{R}^3$ with induced metric $g=ψδ_2$ where $δ_2$ is the two dimensional Euclidean metric. We then construct a system of partial differential equations that constrain $M$ to lift to a minimal surface via the Weierstrauss-Enneper representation demanding the metric is of the a…

Minimal crossing number found in arithmetic curve systems.

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 results for the minimal surface equation can be generalized. We also conjecture…

Study minimizes crossing points of up to 12 curves on a genus 2 surface.

The Ryu-Takayanagi conjecture connects the entanglement entropy in the boundary CFT to the area of open co-dimension two minimal surfaces in the bulk. Especially in AdS(4), the latter are two-dimensional surfaces, and, thus, solutions of a Euclidean non-linear sigma model on a symmetric target space that can be reduced…

Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.

Minimal surfaces found in 4D space.

Given a reductive representation $ρ: π_1(S)\rightarrow G$, there exists a $ρ$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $Σ$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the…

Paper classifies minimal graph transformations into new families of surfaces.

We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.

The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as th…

Minimal surfaces can be transformed into others with unchanged bending content.

Extends Newton's minimal resistance problem to Riemannian surfaces.

The paper studies how to transform a sequence of cmc planes into a minimal surface.

Using techniques of integrable systems, we study a Weierstrass representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in Minkowski 3-spa…

This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces. We shall mainly focus on two directions: (1) Further systematic developments after Lawson-Osserman's paper \cite{l-o} on the Dirichlet problem for minimal graphs of high codi…

New integrable systems for marginally trapped surfaces in 4D Lorentz-Minkowski space.

In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian minimal surfaces can be defined as the critical points of the '`horizontal'' ar…

This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some results, mostly obtained by the authors, about three important classes of surfaces in Laguerre geometry, namely L-isothermic, L-minimal, and generalized L-minimal surfaces. The quadric model of Lie sphere geometry is a…

We extend the work of Simon and Wickramasekera, who constructed a large class of $C^{1,μ}$ multivalued solutions to the minimal surface equation, to produce $C^{1,μ}$ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the La…

We present a connection between minimal surfaces of index one and General Relativity. First, we show that for a certain class of (electro)static systems, each of its unstable horizons is the solution of a one-parameter min-max problem for the area functional, in particular it has index one. We also obtain an inequality…

The aim of this paper is to give a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a d…

Minimal energy local systems on curves are compact components of character varieties.

In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of minimal…

Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper ge…

Explains Bernstein theorems for various geometric PDEs.

Study harmonic surfaces in 3D space, proving superposition principle.

The paper extends Bernstein Theorem for minimal spacelike surfaces in 4D Minkowski space.

In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold $Σ$ is the graph of a (strictly) distance-decreasing map, then $…

The paper finds surfaces closest to being flat that span a given contour.

We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or integrable systems principles. Finally, we state the shared properties that those…