arXiv research

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

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…

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…

Paper introduces v-CMC linking causality and utility.

New CMC existence result for expanding cosmological spacetimes.

The paper classifies CMC free boundary hypersurfaces in rotational domains.

Study proves behaviors of CMC surfaces near future null-infinity in Schwarzschild spacetime.

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.

In this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3-manifolds. We require the metric have the form g_{ij}=δ_{ij}+h_{ij} with h_{ij}=O_{4}(r^{-1}) and R=O(r^{-3-τ}),τ>0. We do not require the metric to be close to Schwarzschild metric in any sense or to satisf…

Study finds conditions for free boundary CMC surfaces in conformally Euclidean 3-balls.

We give necessary conditions on complete embedded \cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are two-dimensional varieties in the moduli spaces of general \cmc surfaces. We characterize fundamental domains of our \cmc surfaces by associated great circle polyg…

The paper extends an Alexandrov theorem to Minkowski spacetime.

Study proves all free boundary CMC annuli are of finite type.

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

New cosmological spacetimes without CMC Cauchy surfaces found.

We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature (CMC) 1 with singularities and with elliptic ends in de Sitter 3-space. An immersed end of a CMC 1 surface is an ``elliptic end'' if the monodromy representation at the end is diagonalizable with eigenvalues in the unit cir…

DPW method reconstructs minimal and symmetric CMC surfaces in 3-sphere.

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

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 techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by glu…

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 develop a conservation law for constant mean curvature (CMC) surfaces introduced by Korevaar, Kusner and Solomon, and provide a converse, so as to characterize CMC surfaces by a conservation law. We work with `twizzler' construction, which applies a screw-motion to some base curve. We show that, excluding cylinders,…

Research shows conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.

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

Study finds non-CMC biconservative hypersurfaces in spheres, proving their existence but not embeddability.

We consider proper-biharmonic flat tori with constant mean curvature (CMC) in spheres and find necessary and sufficient conditions for certain rectangular tori and square tori to admit full CMC proper-biharmonic immersions in $\mathbb{S}^n$, as well as the explicit expressions of some of these immersions.

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…

Given $H\in [0,\infty),$ some sufficient conditions for existence of CMC $H$ graphs with boundary in two parallel planes of $\mathbb{H}^2\times\mathbb{R}$ are presented. Height estimates for outwards-oriented CMC surfaces (horo)cyllindrically bounded are also exhibited.

Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate to any conformally immersed constrained Willmore torus f a compact Riemann surface Σ, such that f can be reconstructed in terms of algebraic data on Σ. Particularly interesting exam…

Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.

The paper bounds eigenvalues of the Jacobi operator and derives rigidity results for CMC hypersurfaces.

We derive necessary conditions on the parameters of the ends of a CMC-1 trinoid in hyperbolic 3-space $H^{3}$ with symmetry plane by passing to its conjugate minimal surface. Together with Daniel's results, this yields a classification of generic symmetric trinoids. We also discuss the relation to other classification …

We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surface…

The Clifford tori in the 3-sphere are a one-parameter family of flat, two-dimensional, constant mean curvature (CMC) surfaces. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small ca…

The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in $S ^ 3$ result when these spectral curves satisfy periodicity conditions. We prove that the spectra…

The paper proves surfaces close to spheres under specific conditions.

We compute a Simons' type formula for the stress-energy tensor of biharmonic maps from surfaces. Specializing to Riemannian immersions, we prove several rigidity results for biharmonic CMC surfaces, putting in evidence the influence of the Gaussian curvature on pseudo-umbilicity. Finally, the condition of biharmonicity…

It is shown by several authors going back to Huisken-Yau that asymptotically Schwarzschildean time-slices possess a unique foliation by stable constant mean curvature (CMC) spheres defining the so-called CMC center of mass. We analyze how the leaves of this foliation evolve in time under the Einstein equations. More pr…

Proves existence of solution to Lichnerowicz equation on non-CMC manifolds.

New CMC surfaces with dihedral symmetry constructed from Darboux transforms.

Paper proves no specific CMC hypersurfaces in hyperbolic space.

New method improves curvature estimates for stable surfaces.

The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces which are products of lower-dimensional spheres called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tor…

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…

In this note we discuss graphs over a domain $Ω\subset N^2$ in the product manifold $N^2\times \mathbb{R}$. Here $N^2$ is a complete Riemannian surface and $Ω$ has peice-wise smooth boundary. Let $γ\subset\partialΩ$ be a smooth connected arc and $Σ$ be a complete graph in $N^2\times \mathbb{R}$ over $Ω$. We show that i…

Study on biconservative surfaces in 4D hyperbolic space, providing extrinsic descriptions.

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

Study CMC hypersurfaces in $\mathbb{H}^2 imes\mathbb{H}^2$ with a specific symmetry.