arXiv research

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of fin…

Study on minimal submanifolds with finite curvature in Euclidean space.

In this short note, we use classic computations for Kähler-Ricci flow to achieve scalar curvature bound for minimal manifold of general type.

Paper estimates curvature of minimal surfaces in a specific geometric space.

Paper estimates Gaussian curvature of minimal graphs in a specific manifold.

A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…

Study elliptic Weingarten surfaces in warped product space with specific curvature conditions.

Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.

Study minimal time-like surfaces in 4D space-time, proving conditions for existence.

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.

Study classifies zero mean curvature surfaces with planar curvature lines.

We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with …

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

Paper constructs a minimal surface with specific ends and curvature.

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

Extends Choi-Wang inequality to Li-Xia affine connections.

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…

The paper improves estimates of Gaussian curvature for minimal graphs over a unit disk.

We construct a complete, embedded minimal surface in euclidean 3-space which has unbounded Gaussian curvature. It has infinite genus, infinitely many catenoidal type ends and one limit end.

Paper constructs flows converging to cones and foliations.

Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.

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 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 …

Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.

Study on 2-ruled hypersurfaces in a Walker 4-manifold.

This paper connects Laguerre minimal surfaces to Weierstrass representations.

Study on minimizing singular capillary cones with stability and instability results.

In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H^2xR. We also prove that a minimal …

Study on stable minimal hypersurfaces under Ricci curvature constraints.

Paper solves long-standing Gaussian curvature conjecture for minimal graphs.

This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4πmust be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.

Survey on discrete minimal surfaces and their properties.

Sharp curvature bounds for minimal graphs over unit disk.

Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.

We obtain new curvature estimates and Bernstein type results for minimal $n-$submanifolds in $\ir{n+m},\, m\ge 2$ under the condition that the rank of its Gauss map is at most 2. In particular, this applies to minimal surfaces in Euclidean spaces of arbitrary codimension.

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

In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all q…

New 5-manifold found with zero Ricci curvature.

Study confirms conjecture about Kähler metrics on smooth minimal models.

The study classifies weakly almost Fuchsian manifolds and proves geometric properties.

The study characterizes hypersurfaces in spheres with constant scalar curvature.

We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

We construct examples of spherical space forms $(S^3/Γ,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/Γ,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in d…

A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.

Paper tackles Santaló's convex surface problem in hyperbolic 3-space.

Paper proves Liouville-type theorems for minimal graphs with capillary boundary.

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies apriori area and curvature estimates for these minimal surfaces in terms of the …

Study classifies minimal surfaces and solitons in hyperbolic 3-space as translation surfaces.