arXiv research

Study minimal surfaces in Kropina 3D space, finding only planes as minimal translation surfaces.

Lie minimal surfaces are characterized by differential equations of principal curvatures.

An infinite sequence of commuting nonpolynomial contact symmetries of the two-dimensional minimal surface equation is constructed. Local and nonlocal conservation laws for $n$-dimensional minimal area surface equation are obtained by using the Noether identity.

The study characterizes minimal surfaces in a specific three-dimensional metric space and finds that planes are the only minimal surfaces.

Sharp bounds found for minimal surface solutions.

This paper solves minimal surface equations near Hardt-Simon foliations.

Paper classifies minimal graph transformations into new families of surfaces.

Most known examples of doubly periodic minimal surfaces in $\mathbb{R}^3$ with parallel ends limit as a foliation of $\mathbb{R}^3$ by horizontal noded planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a correspo…

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

We provide a congruence theorem for minimal surfaces in $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in $S^5$ with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we wi…

New minimal surfaces derived from helicoids.

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…

Study finds minimum growth rate for surface solutions.

Unified treatment of spacelike and timelike minimal surfaces via Liouville equation.

Study determines a minimal surface in a Riemannian manifold from boundary data.

Minimal surface equation results in constant solutions on RCD spaces.

Minimal graphs over simply connected domains grow at most exponentially.

Equations for minimal surfaces from rigid motions in high dimensions.

Minimal surfaces found in 4D space.

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…

We construct nonlinear entire solutions in $\mathbb{R}^6$ to equations of minimal surface type that correspond to parametric elliptic functionals.

Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.

Minimal surfaces are the only biharmonic in Sol3.

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…

Study determines minimal surfaces from boundary data, proving topological and conformal recoverability.

Study on spheres with minimal equators.

Corrected a mistake in a paper about minimal surfaces.

For $n\geq2,$ we obtain Liouville type theorems for minimal surface equations in half space $\mathbf R^n_+$ with affine Dirichlet boundary value or constant Neumann boundary value.

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…

Paper proves existence of solutions for complex surface diffusion equation.

The Björling problem is explored for Born-Infeld solitons.

Study minimizes CR surfaces in Heisenberg group with rotational symmetry.

We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in CH2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameter…

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is an $n$-dimensional space form. Then, we use this equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}$ It…

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^3(c)\times\mathbb{R}$, where $M^3(c)$ is a 3-dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.

New minimal surfaces in 4D space derived from parametric equations.

Study shows zero level sets of solutions to Allen-Cahn equation are minimal surfaces with zero mean curvature.

In a 2004 paper, Lindblad demonstrated that the minimal surface equation on $\mathbb{R}l^{1,1}$ describing graphical time-like minimal surfaces embedded in $\mathbb{R}^{1,2}$ enjoy small data global existence for compactly supported initial data, using Christodoulou's conformal method. Here we give a different, geometr…

The change in Holographic entanglement entropy (HEE) for small fluctuations about pure anti De Sitter (AdS) is obtained by a perturbative expansion of the area functional in terms of the change in the bulk metric and the embedded extremal surface. However, it is known that change in the embedding appears in second orde…

Minimal graph theorem proven for convex domains.

New minimal surfaces found using Toda lattice and integrable systems.

Let $Ω\subset\r^n$ be a bounded mean convex domain. If $α<0$, we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $Ω$ for the $α$-singular minimal surface equation with arbitrary continuous boundary data.

When using Traizet's regeneration technique to construct minimal surfaces, the simplest nontrivial configurations are given as the roots of polynomials that satisfy a hypergeometric differential equation. We exhibit examples of simple minimal surfaces exhibiting the same behavior.

This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or by having a flat structure 3-web. We observe that the structure equation un-coup…

Minimal surfaces in harmonic conformally flat space are studied.

Minimal volume vector fields on surfaces via calibrations.

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere S^3. After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known examples of embedded minimal surfaces in S^3. Besides the equator and the Clifford t…