arXiv research

Study on existence and structure of P-area surfaces in Heisenberg group.

We prove the existence of a continuous $BV$ minimizer with $C^{0}$ boundary value for the $p$-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from $BV$ functions to vector-valued measures. Our main purpose is to study the first and second v…

Formula for Heisenberg group surface areas derived.

We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg group. The generalized $p$-area of a graph defined by $u$ reads $\int |\nabla u+\vec{F}| + Hu$. If $u$ and $v$ are two minimizers for the generalized $p$-area satisfying the same Dirichlet boundary condition, then we can only get $N_{\vec{F}}…

In \cite{CHMY04}, we studied $p$-mean curvature and the associated $p$-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized $p$-area and associated ($p$-) minimizers in general di…

In this paper we study areas (called p-areas) and volumes for parametric surfaces in the 3D-Heisenberg group $\mathbb{H}_1$, which is considered as a flat model of pseudo-hermitian manifolds. We derive the formulas of p-areas and volumes for parametric surfaces in $\mathbb{H}_1$ and show that the classical result of Pa…

We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is prove…

In this paper, we study the structure of the singular set for a $C^{1}$ smooth surface in the $3$-dimensional Heisenberg group $\boldsymbol{H}_{1}$. We discover a Codazzi-like equation for the $p$-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation …

We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We establish the existence of convex body with prescribed $k$-th even $p$-area measure on $\mathbb S^n$, under an appropriate assumption on the prescribed function. We co…

Study on minimal surfaces in a 3D space with 2m-norm.

This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.

Study proves surfaces with constant curvature are simple shapes.

Minimal surfaces in 3-sphere created by reflections from polygons, with new examples based on pentagons.

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

Paper proves stable minimal surfaces in 3D are flat.

New inequality helps map stability in minimal surfaces.

Minimal surfaces help compare geometric shapes.

New minimal surfaces found using Toda lattice and integrable systems.

This paper connects Laguerre minimal surfaces to Weierstrass representations.

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…

New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.

Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.

Unstable minimal surfaces in n-space link to hyperbolic products.

The study examines minimal surfaces in Riemannian products of surfaces.

New theorem for nonlocal minimal surfaces in any dimension.

A general study of minimal surfaces of the Riemannian product of two spheres S^2xS^2 is tackled. We stablish a local correspondence between (non-complex) minimal surfaces of S^2xS^2 and certain pair of minimal surfaces of the sphere S^3. This correspondence also allows us to link minimal surfaces in S^3 and in the Riem…

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

Paper studies singularities of timelike minimal surfaces in Minkowski 3-space.

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 …

We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. …

Complex analysis aids in studying minimal surfaces.

Minimal surfaces are the only biharmonic in Sol3.

Study on minimal surfaces in Heisenberg group with duality formula.

New minimal surfaces found with Cantor ends in convex domains.

Crochet patterns for minimal surfaces created using trigonometry.

We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal …

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

The paper proves cylindrical nature of singular minimal ruled surfaces.

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 surfaces.

The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.

New minimal surfaces grow area very quickly.

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau prob…

Sharp upper bound found for stable minimal surfaces.

Characterizes stable minimal capillary surfaces with specific angles.

Construct minimal Lagrangian surfaces in complex projective plane via loop group method.

We classify ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is showed that there is no noncylindrical ruled minimal surface and there is a family of cylindrical ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is also proved that all translation minimal surfaces are ruled.

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

Minimal surfaces with negative curvature found in large spheres.