arXiv research

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

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

The study finds surfaces with constant anisotropic mean curvature foliated by circles in Euclidean space.

Let (Σ, ω) be a compact Riemann surface with constant curvature c. In this work, we proved that the mean curvature flow of a given Hamiltonian diffeomorphism on Σ provides a smooth path in Ham(Σ), the group of all Hamiltonian diffeomorphisms of Σ. This result gives a proof, in the case of graph of Hamiltonian diffeomor…

The study classifies and constructs examples of surfaces with specific curvature and boundary conditions.

Constructs surfaces with constant mean curvature from Bessel equation.

Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.

In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. By using a local test func…

We illustrate the flow or wave character of the metrics and curvatures of evolving manifolds, introducing the Riemann flow and the Riemann wave via the bialternate product Riemannian metric. This kind of evolutions are new and very natural to understand certain flow or wave phenomena in the nature as well as the geomet…

Local fractional derivatives affect Riemann curvature tensor to zero.

We study the canonical metric on a compact Riemann surface of genus at least two. While it is known that the canonical metric is of nonpositive curvature, we show that its Gaussian curvatures are not bounded away from zero nor negative infinity when the surface is close to the compactification divisor of Riemann's modu…

The Weil-Petersson metric's curvature vanishes for surfaces with short geodesics.

Study classifies zero mean curvature surfaces with planar curvature lines.

In this article we give a complete description of the evolution of an area decreasing map $f:M\to N$ induced by its mean curvature in the situation where $M$ and $N$ are complete Riemann surfaces with bounded geometry, $M$ being compact, for which their sectional curvatures $σ_M$, $σ_N$ satisfy $\minσ_M\ge\supσ_N$.

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.

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

We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the total mass tends to zero.

We prove the mean curvature flow of the graph of a symplectomorphism between Riemann surfaces converges smoothly as time approaches infinity.

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 …

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

New symmetries found in Riemann-Cartan geometries.

This paper constructs a family of constant mean curvature immersions of the thrice-punctured Riemann sphere into Euclidean 3-space with asymptotically Delaunay ends via loop group methods.

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

We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the…

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.

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to t…

We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph…

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.

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. They are Moebius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann mani…

Defines new invariants for Riemann-Finsler manifolds, generalizing Preissman's theorem.

We first prove a general gluing theorem which creates new nondegenerate constant mean curvature surfaces by attaching half Delaunay surfaces with small necksize to arbitrary points of any nondegenerate CMC surface. The proof uses the method of Cauchy data matching from \cite{MP}, cf. also \cite{MPP}. In the second part…

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

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…

Geometric quantization results for Riemann surfaces with semi-positive line bundles.

Classifies zero mean curvature surfaces in Lorentz-Minkowski space.

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…

New approach proves existence of gravitating vortices on Riemann surfaces.

New, algebraic surfaces found in curved spaces.

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

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