arXiv research

The class of second order ODE's cubic with respect to the first order derivative is considered. Using geometric structures associated with these equations, the subclasses of umbilical equations, zero mean curvature equations, and zero Gaussian curvature equations are defined. Zero mean curvature equations are studied w…

On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this …

The paper explores connections between three equations via Wick rotations and symmetries.

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

Classifies zero mean curvature surfaces in Lorentz-Minkowski space.

In this work, we study a class of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}_2^4$ whose profile curves lie in two-dimensional planes. We solve the differential equation that characterizes the rotational surfaces with zero mean curvature to determine the profile curves of such rotational surfaces. The…

Equations link metrics with tensors, revealing curvature constraints.

The paper shows how Scherk-type surfaces can be decomposed into helicoids.

Constructs all real analytic germs of zero mean curvature surfaces in Lorentz-Minkowski 3-space.

In this paper we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable c…

Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution equation by $T$-curvature at the boundary with the condition that the $Q$-curvature …

Constructs solutions for gravitational instantons from minimal surfaces.

Infinite-time blow-up in high-dimensional mean curvature flow.

In this paper, we prove that the even solution of the mean field equation $Δu=λ(1-e^u)$ on $S^2$ must be axially symmetric when $4<λ\leq 8$. In particular, zero is the only even solution for $λ=6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.

The study introduces canonical coordinates for Lorentz surfaces and proves a Bonnet-type theorem.

Study classifies zero mean curvature surfaces with planar curvature lines.

Geometric description of Riemann zero mean curvature surfaces in Lorentz-Minkowski space.

Study duality of zero mean curvature surfaces in Heisenberg group.

Classifies surfaces with zero mean curvature in a light cone.

Solves surface problem in 3D light cone.

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in $\mathbb R^{n+1}$ over general domains $Ω$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result …

Paper investigates reflection principles for zero mean curvature surfaces in isotropic 3-space.

Let $(M^n,g),~n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\partial M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of $g$ that is complete, has zero scalar curvature on $M$ and has mean curv…

The paper solves a Nirenberg problem on half spheres, finding multiple blow-ups.

New examples of mixed-type zero-curvature graphs found.

A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the …

The paper examines translating solitons and their relation to Lagrangian mean curvature flows with zero Maslov class.

The paper solves conditions for prescribing scalar and Gauss curvatures on manifolds with zero first eigenvalue.

We give necessary and sufficient local conditions for the simultaneous unitarizability of a set of analytic matrix maps from an analytic 1-manifold into SL_n(C) under conjugation by a single analytic matrix map. We apply this result to the monodromy arising from an integrable partial differential equation to construct …

Constant mean curvature surfaces with n ends created using DPW method.

New framework for zero mean curvature surfaces in isotropic 3-space.

The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.

Unbounded convex domains have zero mean curvature on disconnected boundaries.

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …

The method of contact integrable extensions is used to find new zero-curvature representation for Plebañski's second heavenly equation.

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found …

No regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces are found.

Improves Bernstein theorem for zero mean curvature hypersurfaces in Lorentz-Minkowski space.

The paper proves rigidity results for manifolds with scalar curvature and boundary.

The paper studies CMC foliations and their conformal aspects on Riemannian manifolds.

Study on surfaces in neutral space forms with zero mean curvature.

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bo…

Introduces fractional length and nonlocal curvature for smooth curves.

New, algebraic surfaces found in curved spaces.

This thesis constructs cmc 1/2 surfaces from catenoids, proving convergence and solving boundary value problems.

New game approximates mean curvature flow evolution.

Factorable surfaces, i.e. graphs associated with the product of two functions of one variable, constitute a wide class of surfaces. Such surfaces in the pseudo-Galilean space with zero Gaussian and mean curvature were obtained in [1]. In this study, we provide new classification results relating to the factorable surfa…