Compactness proven for CMC surfaces with bounded topology and boundary length.
problem Proving compactness of CMC surfaces with specific constraints.
method Graphical Ck compactness proof for surfaces with bounded topology, area, and boundary length. result Space of free boundary CMC surfaces is compact in the Ck graphical sense away from a finite set of points. The study constructs free boundary CMC annuli in spherical and hyperbolic balls.
problem Finding free boundary CMC annuli in spherical and hyperbolic balls.
method Constructing free boundary CMC annuli with constant mean curvature H in geodesic balls of S^3 and H^3.
result Embedded free boundary CMC annuli exist for certain mean curvatures in both spaces.
The paper classifies CMC free boundary hypersurfaces in rotational domains.
problem Existence and uniqueness of free boundary constant mean curvature hypersurfaces in rotational domains.
method Classification and construction of CMC free boundary hypersurfaces under specific conditions.
result Classification of CMC free boundary hypersurfaces as topological disks or annuli.
Study proves all free boundary CMC annuli are of finite type.
problem Free boundary constant mean curvature annuli in the unit ball.
method Adapted Sklyanin's K-matrix formalism to sinh-Gordon equation.
result All free boundary CMC annuli are of finite type.
Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.
problem Understanding topological constraints for stable free boundary CMC surfaces in negatively curved settings.
method Established intrinsic area-length-topology inequalities via a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator.
result Explicit topological restrictions for stable free boundary CMC surfaces, showing low genus and few boundary components.
Study applies Penrose and mass theorems to 3-manifolds with nonnegative scalar curvature and CMC boundary.
problem Derive inequalities on compact 3-manifolds with nonnegative scalar curvature and CMC boundary. method Apply Riemannian Penrose inequality and positive mass theorem; construct a collar extension.
result Prove inequalities on the boundary of compact 3-manifolds with nonnegative scalar curvature and CMC boundary. The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.
problem Stability of CMC hypersurfaces with free boundaries.
method Analysis and numerical computations.
result Equilibrium hypersurfaces are stable without self-intersection in all dimensions.
We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean curvature and the ambient metric are allowed to vary simultaneously, provided that the infinitesimal ambient symmetries change smoothly. We…
Study proves behaviors of CMC surfaces near future null-infinity in Schwarzschild spacetime.
problem Analyzing spacelike CMC surfaces near future null-infinity in Schwarzschild spacetime.
method Proves asymptotic hyperbolicity, derives boundary data expressions, and shows compatibility conditions.
result Compatibility conditions and asymptotic behaviors of spacelike CMC surfaces near future null-infinity.
The paper bounds the index of CMC surfaces with capillary boundary.
problem Bounding the index of CMC surfaces with capillary boundary.
method Comparison of second variations of area and energy, derived second variation formulae.
result The index is bounded linearly by genus, boundary components, and contact angle.
Study finds conditions for free boundary CMC surfaces in conformally Euclidean 3-balls.
problem Conditions for existence of free boundary CMC surfaces in conformally Euclidean 3-balls.
method Analyzes pinching conditions on the traceless second fundamental tensor involving support function, positional conformal vector field, and potential function.
result Either a disk or an annulus rotationally symmetric surface is found under specific conditions.
Proves existence and uniqueness of CMC solutions in product manifolds.
problem Existence and uniqueness of solutions to CMC equation with Neumann boundary data.
method Analyzes product manifold MnimesR with specific curvature conditions. result Proves existence and uniqueness of solutions.
Study on stable CMC surfaces in slabs with boundary conditions.
problem Characterizing and proving properties of stable CMC surfaces in slabs.
method Analyzing stable constant mean curvature (CMC) hypersurfaces in product spaces with free boundary conditions.
result No stable CMC surface connects boundary components of a slab with width greater than a certain limit.
Research shows conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.
problem Conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.
method Analyzes the geometry of the domain's boundary to determine foliation conditions.
result Conditional foliation is possible but not guaranteed, depending on the domain's geometry.
We prove the non-vanishing of the CMC flux of the boundaries of certain Riemannian manifolds with constant mean curvature.
The paper proves gap results for constant mean curvature surfaces in Euclidean and hyperbolic spaces.
problem Understanding the properties of constant mean curvature surfaces.
method Proving natural inequalities and using them to show that CMC surfaces in Euclidean and hyperbolic spaces are either spheres, cylinders, or disks.
result Gap theorems for CMC surfaces in Euclidean and hyperbolic spaces complete the picture.
We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…
Given H∈[0,∞), some sufficient conditions for existence of CMC H graphs with boundary in two parallel planes of H2×R are presented. Height estimates for outwards-oriented CMC surfaces (horo)cyllindrically bounded are also exhibited.
Study bounds the index of CMC surfaces in 3-manifolds.
problem Bounding the index of CMC surfaces in 3-manifolds.
method Proved index estimates for CMC surfaces in certain 3-dimensional submanifolds.
result Index of CMC surfaces is bounded below by a linear function of its genus when mean curvature is large enough.
This thesis constructs cmc 1/2 surfaces from catenoids, proving convergence and solving boundary value problems.
problem Creating cmc 1/2 surfaces with positive genus in H2imesR. method Analytic gluing construction, solving mean curvature equation via perturbative methods and linear analysis.
result Construction of cmc 1/2 annuli asymptotic to horizontal catenoids, proving convergence to horocylinders.
The paper bounds eigenvalues of the Jacobi operator and derives rigidity results for CMC hypersurfaces.
problem Bounding the first eigenvalue of the Jacobi operator for CMC hypersurfaces.
method Geometric upper bounds for eigenvalues and rigidity results.
result New rigidity results for the area and length of CMC hypersurfaces.
Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.
problem Analyzing the Morse index and eigenvalues of free-boundary CMC hypersurfaces in the upper hemisphere.
method Proved results using the norm squared of the second fundamental form and eigenvalue estimates.
result Proved bounds on Morse index and eigenvalues for free-boundary CMC hypersurfaces.
The study examines graphs over domains in product manifolds, revealing properties of geodesics and constant curvature.
problem Properties of graphs over domains in product manifolds.
method Analyzes minimal, translating, and CMC graphs over domains with piecewise smooth boundaries.
result Geodesic arcs in the boundary of domains for minimal and translating graphs, and constant curvature for CMC graphs.
Proves rigidity of boundaries with constant mean curvature in warped product manifolds.
problem Rigidity and compactness of boundaries with constant mean curvature in warped product manifolds.
method Distributional CMC-rigidity proof for rectifiable boundaries.
result Characterizes limits of boundaries with converging mean curvatures.
New CMC existence result for expanding cosmological spacetimes.
problem Establishing a new constant mean curvature (CMC) existence result for cosmological spacetimes.
method Construction of barriers in the support sense and asymptotic limit of mean curvature flow.
result The existence of a CMC Cauchy surface in expanding cosmological spacetimes.
In this paper we generalize the Local Removable Singularity Theorem in [16] for minimal laminations to the case of weak H-laminations (with H∈R constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mea…
Study finds non-CMC biconservative hypersurfaces in spheres, proving their existence but not embeddability.
problem Characterizing and proving the existence of non-CMC biconservative hypersurfaces in spheres.
method Analyzing p-elastic curves of profile curves of biconservative rotational hypersurfaces in space forms. result Existence of a discrete biparametric family of non-CMC closed biconservative hypersurfaces in Sn(ρ), none of which can be embedded. New closed non-CMC biconservative surfaces found in round 3-sphere.
problem Existence of closed biconservative surfaces in space forms.
method Characterization of profile curves and proof of existence using curvature energy.
result Existence of a discrete family of closed, non-CMC biconservative surfaces in S3(ρ). We prove the existence and uniqueness of the Dirichlet problem for spacelike, spherically symmetric, constant mean curvature equation with symmetric boundary data in the extended Schwarzschild spacetime. As an application, we completely solve the CMC foliation conjecture which is posted by Malec and O Murchadha in 2003…
The paper extends Liebmann's Theorem to convex hypersurfaces with boundary.
problem Proving properties of convex hypersurfaces with boundary in Euclidean space.
method Analyzing locally convex, embedded, compact, connected CMC hypersurfaces bounded by a closed strictly convex submanifold.
result Spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a (n−1)−sphere. In this note, we observe that if B is a ball in a Euclidean space with dimension n, n≥3, then a stable CMC hypersurface Σ with free boundary in B satisfies \[ nA\leq L\leq nA\left( \frac{1+\sqrt{1+4(n+1)H^2}}{2} \right)\,, \] where L, A and H denote the length of ∂Σ, the area of Σ and the…
We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary,…
The paper proves bounds on mean curvature for CMC foliations with Ricci curvature constraints.
problem Bounding mean curvature for CMC foliations with specific Ricci curvature conditions.
method Analyzing foliations on compact Riemannian manifolds with Ricci curvature constraints.
result Proves that for a foliation by CMC hypersurfaces with specific Ricci curvature constraints, the mean curvature is constant and all leaves are totally umbilical.
New discrete cmc surfaces defined from sphere packings and combinatorics.
problem Creating constant mean curvature surfaces from discrete data.
method Discrete cmc surfaces defined via sphere packings and combinatorial patterns.
result Construction of discrete cmc surfaces from orthogonal ring patterns.
Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.
problem Proving upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.
method Analyzing a weighted eigenvalue problem and using a Lorentz-Sobolev inequality to study eigenfunctions and index/nullity in neck regions.
result Upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces proved.
Two Bartnik mass definitions are shown to be equivalent under convexity conditions.
problem Equivalence of two Bartnik mass definitions for compact Riemannian manifolds.
method Gluing Bartnik extensions and convexity conditions.
result Equivalence of Bartnik mass definitions under convexity conditions.
Let R+n+1 \ be the half-space model of the hyperbolic space Hn+1. It is proved that if Γ⊂{xn+1=0}⊂∂∞Hn+1 is a bounded C0 Euclidean graph over {x1=0, xn+1=0} then, given $\left\vert H\right\vert <…
Smooth approximations near singularities of constant mean curvature surfaces are found.
problem Finding smooth approximations for constant mean curvature surfaces near singular points.
method Proving the existence of sequences of smooth CMC hypersurfaces converging to a given one in a ball centered at the singularity.
result Smooth approximations exist in a ball centered at the singularity of a CMC hypersurface.
For all N≥9, we find smooth entire epigraphs in RN, namely smooth domains of the form Ω:={x∈RN / xN>F(x1,…,xN−1)}, which are not half-spaces and in which a problem of the form Δu+f(u)=0 in Ω has a positive, bounded solution with 0 Dirichlet boundary data and constant Neum…
New examples show non-rotational annuli in a ball, solving a uniqueness problem.
problem Uniqueness of annular solutions in a ball.
method Constructing a family of compact embedded CMC annuli with free boundary in the unit ball.
result Non-rotational annuli found, providing a counterexample to Nitsche and Wente's uniqueness problem.
We establish existence and uniqueness of compact graphs of constant mean curvature in MxR over bounded multiply connected domains of Mx{0} with boundary lying in two parallel horizontal slices of MxR
A version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol3 is proved. Moreover, we construct examples of admissible domains where the results may be applied.
We consider a variational problem for submanifolds Q ⊂ M with nonempty boundary ∂Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutio…
The paper proves existence of solutions for Einstein-type elliptic systems on AE manifolds.
problem Analyzing semi-linear systems of partial differential equations motivated by the conformal formulation of Einstein constraint equations.
method Proving existence theorems under suitable conditions, including smallness assumptions on free parameters.
result Existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations.
In this paper we study sets in the n-dimensional Heisenberg group $\hhn$ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in $\hhn$. We define a notion of mean curvature for hypersurfaces and we show that the boundary of a…
We will discuss some sharp estimates for CMC graphs in a Riemannian 3-manifold MxR whose boundary is contained in a slice. We will start by giving sharp lower bounds for the geodesic curvature of the boundary and improve these bounds when assuming additional restrictions on the maximum height that such a surface reache…
In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean curvature surface homologous to a subset of the interior boundary components. This boun…
Proves rigidity of stable free boundary hypersurfaces in 5-manifolds.
problem Stability and rigidity of free boundary hypersurfaces in 5-manifolds.
method Combining k-tri-Ricci curvature and 3-intermediate Ricci curvature. result Improves rigidity result to 5-dimensions and extends to free boundary case.