arXiv research

Paper proves existence of a CMC hypertorus in 4D sphere using numerical methods.

Authors construct hypertori with constant negative mean curvature in a sphere.

Minimal hypertori found in 4D sphere, solving Bernstein conjecture.

In this paper, we introduce a non linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like $g_{ij}=(1+\frac{1}{l})^{2}δ_{ij}+O(l^{-2})$. The existence of uns…

New CMC surfaces with dihedral symmetry constructed from Darboux transforms.

No proper biharmonic CMC compact hypersurface in a specific warped product space.

We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loo…

Constructs cmc doublings of minimal surfaces via min-max theory.

We translate a classification scheme for periodic CMC surfaces developed by J. Dorfmeister and the author to discrete CMC surfaces in the sense of A. Bobenko and U. Pinkall. The scheme uses the dressing action on discrete CMC surfaces to arrive at a classification for periodic discrete CMC surfaces.

Compactness proven for CMC surfaces with bounded topology and boundary length.

New cosmological spacetimes without CMC Cauchy surfaces found.

Study singularities of CMC 1 surfaces in de Sitter space.

Estimates bandwidth for CMC initial data sets.

The paper explores CMC hypersurfaces in spheres, verifying Yau's conjecture.

Authors review CMC spacelike hypersurfaces and new existence results.

In this paper we numerically construct CMC deformations of the Lawson minimal surfaces $ξ_{g,1}$ using a spectral curve and a DPW approach to CMC surfaces in spaceforms.

New CMC existence result for expanding cosmological spacetimes.

The study constructs free boundary CMC annuli in spherical and hyperbolic balls.

Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.

It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.

The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn…

We propose an extension of the differential system for constant mean curvature (CMC) surfaces in a three dimensional space form to an associated hierarchy of evolution equations by the higher-order commuting symmetries. The infinite sequence of higher-order conservation laws of CMC surfaces admit the corresponding exte…

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…

The paper constructs all cmc hypersurfaces with two principal curvatures.

The abstract proves the existence of CMC-1 surfaces with any complex structure in hyperbolic space.

This paper establishes an interesting connection between the family of CMC surfaces of revolution in $\mathbb E_1^3$ and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the $\mathbb E_1^3$, only spacelike cylinders and stand…

The paper proves properties of triharmonic CMC hypersurfaces with specific curvature conditions.

We use bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.

CMC surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h\in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in $\sn^5$ with $|H|=h$, while a necessar…

(CMC) 1-immersions in hyperbolic 3-manifolds often develop singularities.

Improved gap for mean curvature of biharmonic hypersurfaces in spheres.

Paper proves rigidity of CMC surfaces in curved 3-manifolds.

The paper proves that for a given metric, there exists another metric where the number of constant mean curvature hypersurfaces increases.

We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.

The paper confirms a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.

The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.

In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the $3$-sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. For rational times we obtain closed (possibly bran…

The paper proves properties of triharmonic CMC hypersurfaces with limited curvature types.

This paper continues the investigation of constant mean curvature (CMC) time functions in maximal globally hyperbolic spatially compact spacetimes of constant sectional curvature, which was started in math.DG/0604486. In that paper, the case of flat spacetimes was considered, and in the present paper, the remaining cas…

Many results in mathematical relativity, including results for both the initial data problem and for the evolution problem, rely on the existence of a constant mean curvature (CMC) Cauchy surface in the underlying spacetime. However, it is known that some spacetimes have no CMC Cauchy surfaces (slices). This is an obst…

Study links Hopf differentials to curvature line flows on time-like CMC surfaces.

CMC-1 surfaces found on compact Riemann surfaces with Cantor sets.

We first summarize the characterization of smooth spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in the Schwarzschild spacetime and Kruskal extension. Then use the characterization to prove special SS-CMC foliation property, and verify part of the conjecture by Malec and Ó Murchadha in t…

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $Σ$ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then $Σ$ is either …

In this paper, we give a Simons' type formula for the cmc surfaces in homogeneous $3$-manifolds $E(κ,τ)$, $τ\neq0$. As an application, we give a rigidity result in the case of $κ> 4τ^2$ for the cmc surfaces under a pinching assumption of the second fundamental form.

We investigate CMC-surfaces with periodic metric in a dressing orbit of the cylinder. It is shown, that such surfaces are always of finite type. Using the periodicity conditions for the extended frame of a CMC-surface, we develop an alternative approach to the classification of CMC-tori given by Pinkall and Sterling.

The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.

New closed non-CMC biconservative surfaces found in round 3-sphere.