arXiv research

New CMC surfaces with dihedral symmetry constructed from Darboux transforms.

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.

Study singularities of CMC 1 surfaces in de Sitter space.

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

New cosmological spacetimes without CMC Cauchy surfaces found.

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

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.

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

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

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…

New CMC existence result for expanding cosmological spacetimes.

Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.

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

Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular …

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 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 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…

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

We investigate the dressing action on surfaces of constant mean curvature (CMC surfaces) in Euclidean space. In particluar, we show that for CMC surfaces with umbilics the isotropy group under dressing is always trivial. This result is applied to the investigation of CMC surfaces whose metric is invariant under a group…

Study bounds CMC surface index in 3-manifolds using energy.

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…

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

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…

Researchers classify cmc surfaces using Jacobi elliptic functions.

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,…

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.

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute cur…

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

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…

The study provides optimal estimates for surfaces close to constant mean curvature.

We start the investigation of immersions $Ψ$ of a simply connected domain $D$ into three dimensional Euclidean space $R^3$, which have constant mean curvature (CMC-immersions), and allow for a group of automorphisms of $D$ which leave the image $Ψ(D)$ invariant. On one hand, this leads to a detailed description of symm…

We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via ano…

We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with thre…

New bounds on genus and area for CMC surfaces in 3-manifolds.

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

We prove that globally subanalytic nonsingular CMC surfaces of $\mathbb{R}^3$ are only planes, round spheres or right circular cylinders

Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are presented: the Umehara-Yamada perturbation, the classical and Bryant's Weierstrass type representations, and the duality…

In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; the…

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

New vacuum spacetimes without CMC Cauchy surfaces found.

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…

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…

Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of higher genus the theory is far more involved due to the non abelian nature of their…

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

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

Formula calculates volume of CMC surfaces with translational periods, disproving isoperimetric conjecture.

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:Σ\to \R^3$ are nondegenerate; that is, the Jacobi operator $Δ_f + |A_f|^2$ has no $L^2$ kernel. In fact, if $Σ$ has genus zero and $f(Σ)$ is contained in a half-space, then we find an explicit upper bound for the dimension of the $L^2$ j…