Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.
problem Understanding Ricci flow on discrete surfaces of revolution.
method Explicit parametrizations and Ricci flow analysis for discrete surfaces of revolution.
result Discrete surfaces of revolution approach constant Gaussian curvature under Ricci flow.
Geodesic boundaries of surfaces with constant mean curvature are studied.
problem Understanding geodesic boundaries of surfaces with constant mean curvature.
method Generalization of results from minimal surfaces to constant mean curvature surfaces.
result Geodesic boundaries of constant mean curvature surfaces are explored.
Planes and spheres are the only stationary surfaces with constant Gauss curvature.
problem Finding surfaces with constant Gauss curvature that are stationary under a specific energy function.
method Proving the uniqueness of stationary surfaces by considering different curvature conditions.
result Planes and spheres are the only stationary surfaces with constant Gauss curvature.
The paper derives height estimates for surfaces with constant curvature in warped product spaces.
problem Estimating heights of surfaces with constant curvature in warped product spaces.
method Use of conformal parameters and geometric applications to derive height estimates.
result Derives height estimates for surfaces with positive extrinsic or mean curvature in RimesfR2. Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in the Heisenberg group.
method Proves surfaces in a neighborhood of non-umbilic points are solutions to a sinh-Gordon equation with a differential constraint.
result Surfaces in Heisenberg group with constant mean curvature described by solutions to sinh-Gordon equation.
Open problems on surfaces with boundary and constant mean curvature.
problem Open problems on compact constant mean curvature surfaces with boundary.
method Survey of current status of open problems.
result Collection of open problems in the theory of surfaces.
The paper classifies surfaces with constant skew curvature in 3-space forms.
problem Classifying surfaces with constant skew curvature in 3-space forms.
method Variational characterization and flow of binormal vector field.
result Classification of rotational surfaces with constant skew curvature.
This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure …
The study classifies constant mean curvature surfaces in curved spaces.
problem Classifying constant mean curvature surfaces in curved spaces.
method Analyzes constant mean curvature isometric immersions into S2imesR and H2imesR. result Provides new classifications of constant mean curvature surfaces in various curved spaces.
Constructs special surfaces in hyperbolic space with constant mean curvature.
problem Creating surfaces with specific geometric properties in hyperbolic space.
method Using the DPW method to construct surfaces with constant mean curvature.
result Constructs surfaces with constant mean curvature in hyperbolic space.
The paper classifies translation surfaces with constant curvature in a specific connection.
problem Classifying translation surfaces with constant curvature in a semi-symmetric non-metric connection.
method Completely classified translation surfaces of constant sectional curvature in a semi-symmetric non-metric connection.
result Translation surfaces of constant curvature are generalized cylinders, similar to the Levi-Civita connection but with additional non-constant curvature cases.
In this paper, we prove a general halfspace theorem for constant mean curvature surfaces. Under certain hypotheses, we prove that, in an ambient space M^3, any constant mean curvature H_0 surface on one side of a constant mean curvature H_0 surface Σ_0 is an equidistant surface to Σ_0. The main hypotheses of the theore…
Flat surfaces in Lie groups with constant curvature are flat.
problem Characterizing surfaces in Lie groups with constant Gaussian curvature.
method Analyzing surfaces as products of curves and using bi-invariant metrics.
result All surfaces of constant curvature in 3D Lie groups are flat.
Study constant mean curvature surfaces with integrable boundary conditions.
problem Understanding surfaces with constant mean curvature under specific boundary conditions.
method Used generalized Weierstrass representation to determine potentials.
result Determined potentials for surfaces satisfying integrable boundary conditions.
Study geodesic curvature of logarithmic spirals on curved surfaces.
problem Understanding geodesic curvature on curved surfaces.
method Computed geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature.
result Asymptotic behavior of geodesic curvature is independent of the ambient surface's curvature.
Classification of constant curvature surfaces in Berger spheres.
problem Identifying complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres.
method Complete classification through detailed analysis of Clifford tori and spheres.
result Rotationally invariant spheres with constant Gauss curvature are the only topological spheres in Berger spheres for K>KP. The study uses isothermic coordinates to analyze space-like surfaces with constant curvature.
problem Global properties of space-like surfaces with constant mean curvature in Lorentz-Minkowski space.
method Isothermic coordinate systems
result Global properties of space-like surfaces with constant mean curvature explored.
Constructs surfaces with specific topologies and curvatures.
problem Creating surfaces with prescribed genus and ends.
method Using a family of constant mean curvature surfaces constructed in \cite{Kleene}, resolving tangency points over catenoidal necks.
result Complete embedded surfaces with freely prescribed genus and ends.
Study on discrete surfaces with constant principal curvature for nanocarbon applications.
problem Understanding discrete geometry properties of nanocarbon materials.
method Developed discrete surface theory on 3-ary oriented trees, defined discrete principal directions, constructed examples of discrete CPC surfaces.
result Construction of discrete constant principal curvature surfaces, including discrete CPC tori.
In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.
The study proves the existence of certain surface types in 3D spaces.
problem Existence of constant mean curvature surfaces in specific 3D spaces.
method Analysis of homology classes of closed 3-manifolds.
result Existence of constant mean curvature surfaces in specified 3D spaces.
Paper classifies special slant surfaces with varying curvature.
problem Classifying surfaces with non-constant mean curvature.
method Analyzing special slant surfaces in complex space forms.
result Complete classification of surfaces with non-constant mean curvature.
We study parabolic linear Weingarten surfaces in hyperbolic space $\rlopezh^3$. In particular, we classify two family of parabolic surfaces: surfaces with constant Gaussian curvature and surfaces that satisfy the relation aκ1+bκ2=c, where κi are the principal curvatures, and a,b and c are constant.
The paper proves constant mean curvature surfaces in specific manifold types.
problem Existence of surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
method Combines min-max theory with inverse mean curvature flow.
result Existence of compact surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
Paper proves a theorem about constant mean curvature surfaces in isotropic 3-space.
problem Understanding constant mean curvature surfaces in isotropic 3-space.
method Value distribution theorem of Gaussian curvature applied to CMC surfaces.
result Implication of a Bernstein-type theorem for CMC surfaces in isotropic 3-space.
Simply connected surfaces with large constant mean curvature and free boundaries concentrate at critical points of the boundary's mean curvature.
problem Surfaces with large constant mean curvature and free boundaries.
method Proving concentration at critical points of the boundary's mean curvature.
result Simply connected H-surfaces concentrate at critical points of the boundary's mean curvature.
Constructs surfaces with constant mean curvature from Bessel equation.
problem Creating surfaces with constant mean curvature.
method From the Bessel equation, constructs immersions of the twice-punctured Riemann sphere into R^3.
result Family of constant mean curvature surfaces constructed.
We explain how the current knowledge on the set of complete noncompact constant mean curvature surfaces can be exploited to produce new examples of compact constant mean curvature surfaces of genus greater than or equal to 3.
A surface in homogenous space Sol is said to be an invariant surface if it is invariant under some of the two 1-parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classi…
The paper finds new constant p-mean curvature surfaces in the Heisenberg group.
problem Discovering new examples of constant p-mean curvature surfaces. method Utilizing the theory and approach for constructing such surfaces.
result Complete description of rotationally invariant surfaces of constant p-mean curvature. Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in sub-Lorentzian Heisenberg group.
method First-variation formula derivation and isoperimetric candidates classification.
result Characterization and conjecture of isoperimetric maximizers.
The paper finds formulas for special surface shapes in 3D space.
problem Creating formulas for constant mean curvature surfaces.
method Weierstrass representations for discrete surfaces in isotropic space.
result Constructs examples of surfaces with discrete parametrizations.
New method to find surfaces in null cones with constant curvature near black hole indicators.
problem Finding surfaces in null cones with constant curvature near black hole indicators.
method Flow techniques to foliate null cones by surfaces of constant spacetime mean curvature.
result A neighbourhood of stable MOTS in a null cone can be foliated by hypersurfaces of constant spacetime mean curvature.
It is shown that the equation which describes constant mean curvature surface via the generalized Weierstrass-Enneper inducing has Hamiltonian form. Its simplest finite-dimensional reduction has two degrees of freedom, integrable and its trajectories correspond to well-known Delaunay and do Carmo-Dajzcer surfaces (i.e.…
New discrete models for constant mean curvature surfaces and tori.
problem Creating discrete models for constant mean curvature surfaces and tori.
method Integrable theory of discrete polarised curves and Darboux transforms.
result Closed-form discrete parametrisations of discrete isothermic cylinders, discrete constant mean curvature cylinders, and discrete isothermic tori.
Study non-minimal surfaces in homogeneous 3-manifolds with constant mean curvature.
problem Classify surfaces of constant mean curvature in homogeneous 3-manifolds.
method Investigate screw motions and classify surfaces in E(κ,τ) including space-forms. result Complete classification of non-minimal surfaces of supercritical constant mean curvature.
In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space ł3 and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in ł3 of constant mean c…
We classify homothetical surfaces with constant mean curvature in hyperbolic space.
problem Classifying surfaces with constant mean curvature in hyperbolic space.
method Using the upper half-space model, we define surfaces by z=φ(x)ψ(y) and prove they are parabolic. result All homothetical surfaces with constant mean curvature in hyperbolic space are parabolic.
Constructs metrics on Riemann surfaces with singularities.
problem Creating constant curvature metrics on surfaces with specific singularities.
method Using meromorphic 1-forms and ODEs to construct conformal metrics.
result Classified constant curvature metrics on S2 with two conical singularities. The abstract finds conditions for creating curves of constant curvature.
problem Finding conditions for closed embedded curves of constant curvature.
method Using Almgren-Pitts min-max method for geodesic curvature.
result Closed embedded curves of any prescribed constant curvature found in S2. New flow generates surfaces with constant curvature.
problem Creating surfaces with specific curvature properties.
method Framed curvature flow, analyzing trajectory surfaces.
result Trajectory surfaces of constant mean or Gaussian curvature.
In this work, we study spacelike surfaces in Minkowski space E13 foliated by pieces of circles and that satisfy a linear Weingarten condition of type aH+bK=c, where a,b and c are constant and H and K denote the mean curvature and the Gauss curvature respectively. We show that such surfaces must be surfa…
The study pinches conditions for constant mean curvature surfaces in convex 3-manifolds.
problem Understanding the topology and geometry of constant mean curvature surfaces with free boundaries in convex 3-manifolds.
method Provided pinching conditions on the traceless second fundamental form to guarantee surface topology.
result The surface is either a disk, annulus, spherical cap, or Delaunay surface under certain conditions.
The paper establishes curvature inequalities and rigidity results for CMC and STCMC surfaces in Riemannian and Lorentzian geometry.
problem Curvature inequalities and rigidity for surfaces with constant mean curvature and spacetime constant mean curvature.
method Analysis of curvature inequalities and rigidity results for surfaces in both Riemannian and Lorentzian settings, using stability conditions and extrinsic curvature sign conditions.
result Sharp inequality for spacetime constant mean curvature surfaces and rigidity for the equality case.
The paper characterizes surfaces in Heisenberg group with constant p-mean curvature.
problem Characterizing surfaces with constant p-mean curvature in the Heisenberg group. method Using the fundamental theorem of surfaces in H1, the existence of constant p-mean curvature surfaces is linked to solutions of a nonlinear ODE. result Complete set of solutions to the ODE (1.2) or (1.5) divides constant p-mean curvature surfaces into several classes. Classifies surfaces with constant Gaussian curvature in Euclidean 3-space.
problem Classifying surfaces with constant Gaussian curvature in Euclidean 3-space.
method Analyzing surfaces as implicit equations and proving properties based on Gaussian curvature.
result Surfaces with constant Gaussian curvature are either surfaces of revolution, cylindrical surfaces, conical surfaces, or have specific forms.