Constant Mean Curvature Research Articles

Volume functionals on pseudoconvex hypersurfaces

The focus of this paper is on a volume form defined on a pseudoconvex hypersurface [Formula: see text] in a complex Calabi–Yau manifold (that is, a complex [Formula: see text]-manifold with a nowhere-vanishing holomorphic [Formula: see text]-form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in [Formula: see text]. We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces [Formula: see text] bounding compact domains [Formula: see text]. That is, we study critical points of the volume functional [Formula: see text] where the ordinary volume [Formula: see text] is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere [Formula: see text] C[Formula: see text] does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in “most” directions but non-negative in directions corresponding to deformations of [Formula: see text] by holomorphic diffeomorphisms. We are led to conjecture a “minimax” characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case [Formula: see text] and the embedding problem studied in our previous paper [S. Donaldson and F. Lehmann, Closed 3-forms in five dimensions and embedding problems, preprint (2022), arXiv:2210.16208]. There are some special features in this case. The volume functional can be defined without reference to the embedding in [Formula: see text] using only a closed “pseudoconvex” real [Formula: see text]-form on [Formula: see text]. In Sec. 4, we review this and develop some of the theory from the point of the symplectic structure on exact [Formula: see text]-forms on [Formula: see text] and the moment map for the action of the diffeomorphisms of [Formula: see text].

Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime

Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime

Twisted-plywood-like tissue formation in vitro. Does curvature do the twist?

Little is known about the contribution of 3D surface geometry to the development of multilayered tissues containing fibrous extracellular matrix components, such as those found in bone. In this study, we elucidate the role of curvature in the formation of chiral, twisted-plywood-like structures. Tissues consisting of murine preosteoblast cells (MC3T3-E1) were grown on 3D scaffolds with constant-mean curvature and negative Gaussian curvature for up to 32 days. Using 3D fluorescence microscopy, the influence of surface curvature on actin stress-fiber alignment and chirality was investigated. To gain mechanistic insights, we did experiments with MC3T3-E1 cells deficient in nuclear A-type lamins or treated with drugs targeting cytoskeleton proteins. We find that wild-type cells form a thick tissue with fibers predominantly aligned along directions of negative curvature, but exhibiting a twist in orientation with respect to older tissues. Fiber orientation is conserved below the tissue surface, thus creating a twisted-plywood-like material. We further show that this alignment pattern strongly depends on the structural components of the cells (A-type lamins, actin, and myosin), showing a role of mechanosensing on tissue organization. Our data indicate the importance of substrate curvature in the formation of 3D tissues and provide insights into the emergence of chirality.

Umbilicity of constant mean curvature hypersurfaces into space forms

In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is totally umbilical.

On constant higher order mean curvature hypersurfaces in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$

Abstract We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ . Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros–Rosenberg type theorem in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ : we show that compact connected hypersurfaces of constant r-th mean curvature embedded in H n × [ 0 , ∞ ) ${\mathbb{H}}^{n}{\times}\left[0,\infty \right)$ with boundary in the slice H n × { 0 } ${\mathbb{H}}^{n}{\times}\left\{0\right\}$ are topological disks under suitable assumptions.

A global Morse index theorem and applications to Jacobi fields on CMC surfaces

In this paper, we establish a “global” Morse index theorem. Given a hypersurface [Formula: see text] of constant mean curvature, immersed in [Formula: see text]. Consider a continuous deformation of “generalized” Lipschitz domain [Formula: see text] enlarging in [Formula: see text]. The topological type of [Formula: see text] is permitted to change along [Formula: see text], so that [Formula: see text] has an arbitrary shape which can “reach afar” in [Formula: see text], i.e. cover any preassigned area. The proof of the global Morse index theorem is reduced to the continuity in [Formula: see text] of the Sobolev space [Formula: see text] of variation functions on [Formula: see text], as well as the continuity of eigenvalues of the stability operator. We devise a “detour” strategy by introducing a notion of “set-continuity” of [Formula: see text] in [Formula: see text] to yield the required continuities of [Formula: see text] and of eigenvalues. The global Morse index theorem thus follows and provides a structural theorem of the existence of Jacobi fields on domains in [Formula: see text].

Constant Mean Curvature Hypersurfaces in Anti-de Sitter Space

Abstract We study entire spacelike constant mean curvature hypersurfaces in Anti-de Sitter space of any dimension. First, we give a classification result with respect to their asymptotic boundary, namely we show that every admissible sphere $\Lambda $ is the boundary of a unique such hypersurface, for any given value $H$ of the mean curvature. We also demonstrate that, as $H$ varies in $\mathbb {R}$, these hypersurfaces analytically foliate the invisible domain of $\Lambda $. Finally, we extend Cheng-Yau Theorem to the Anti-de Sitter space, which establishes the completeness of any entire constant mean curvature hypersurface.

A practical algorithm for the design of multiple-sized porous scaffolds with triply periodic structures

In this study, we present a practical volume-merging method for generating multiple-sized porous structures that exhibit geometries with triply periodic minimal surface (TPMS) lattice structures. The proposed method consists of three stages: (1) designing the physical models with a signed distance field, (2) performing a merging operation for the porous scaffolds, and (3) assembling different units into a composite structure. The significant advantages of the proposed algorithm can be summarized as follows: Our method is independent of the model shape; the designed structures maintain a smooth surface with a constant mean curvature, and the mathematical computational complexity is low. We can join two different-sized triply periodic minimal surface lattices in the radial direction, where the transition region is obtained by smooth interpolation between the two lattice structures with different cell sizes or types. However, constructing large-sized models is only conceptually possible due to computational cost and memory storage constraints. To overcome these limitations, we present a practical method that can efficiently assemble large-scaled models at a low computational cost. The proposed method is based on a Boolean union operation of basic units of TPMS. Thus, it is simple to generate large-scale three-dimensional multiple-sized porous volumes based on our proposed method, which can be applied to many applications in mechanical and electrical engineering. The produced multi-scale compound scaffolds have smooth surfaces without fractures, making them suitable for straightforward application in additive manufacturing. Several numerical tests are conducted to validate the efficiency of the proposed algorithm.

Biconservative surfaces with constant mean curvature in lorentzian space forms

Biconservative surfaces with constant mean curvature in lorentzian space forms

Annular type surfaces with fixed boundary and with prescribed, almost constant mean curvature

We prove existence and nonexistence results for annular type parametric surfaces with prescribed, almost constant mean curvature, characterized as normal graphs of compact portions of unduloids or nodoids in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{3}$$\\end{document}, and whose boundary consists of two coaxial circles of the same radius.

Geometry of constant mean curvature surfaces in $\mathbb{R}^{3}$

The crowning achievement of this paper is the proof that round spheres are the only complete, simply-connected surfaces embedded in \mathbb{R}^{3} with nonzero constant mean curvature. Fundamental to this proof are new results including the existence of intrinsic curvature and radius estimates for compact disks embedded in \mathbb{R}^{3} with nonzero constant mean curvature. We also prove curvature estimates for compact annuli embedded in \mathbb{R}^{3} with nonzero constant mean curvature and apply them to obtain deep results on the global geometry of complete surfaces of finite topology embedded in \mathbb{R}^{3} with constant mean curvature.

Stability and Deformation Criteria in Free Boundary CMC Immersions

Let ∑n and M n+1 be smooth manifolds with smooth boundary. Given a free boundary constant mean curvature (CMC) immersion φ: ∑ → M, we found results related to the existence and uniqueness of a deformation family of φ, {φt}t ∈I , composed by free boundary CMC immersions. In addition, we give to some criteria of stability and unstability for this type of deformations. These results are obtained from properties of the eigenvalues and eigenfunctions of the Jacobi operator Jφ associated to φ and establishing conditions for this operator such as Dim(Ker(Jφ)) = 0, or if Dim(Ker(Jφ)) = 1 and, for f ∈ Ker(Jφ); f ≠ 0, ∫∑ volφ*(g) ≠ 0. The deformation family is unique up to diffeomorphisms.

Triply periodic minimal surfaces based topology optimization for the hydrodynamic and convective heat transfer

Triply periodic minimal surfaces (TPMS), possessing diverse and crucial properties for thermal convection analysis, hold significant promise for topology optimization in convective and hydrodynamic systems. This study aims to develop an innovative topology optimization method based on TPMS for channel shape design. We focus on maximizing heat dissipation and mass flux while maintaining a constant mean curvature. The channel structures undergo free evolution during the optimization process, resulting in efficient and intricate geometries based on the TPMS assumption. To incorporate TPMS properties, we modify the original energy formulation, which comprises kinetic energy, thermal energy, and Ginzburg–Landau energy. The governing system encompasses the phase field model, the Darcy–Stokes model, and the reaction–diffusion heat transfer model. The hydrodynamic response within the phase change structure is computed by solving the transient Darcy–Stokes equation augmented with a temperature-dependent diffusion term. To achieve second-order temporal and spatial accuracy, we employ the Crank–Nicolson method for the time scale and the central difference method for the spatial scale. Through the utilization of the Lagrange multiplier method, we establish the unconditional decrease of the discretized original energy. This implies that the proposed scheme can utilize a large time step. To verify the robustness of our proposed method, we conduct several numerical tests from both qualitative and quantitative perspectives. The results demonstrate the effectiveness of our method in enhancing heat and mass transfer capabilities.

Multispherical shapes of vesicles with intramembrane domains

Phase separation of biomembranes into two fluid phases, a and b, leads to the formation of vesicles with intramembrane a- and b-domains. These vesicles can attain multispherical shapes consisting of several spheres connected by closed membrane necks. Here, we study the morphological complexity of these multispheres using the theory of curvature elasticity. Vesicles with two domains form two-sphere shapes, consisting of one a- and one b-sphere, connected by a closed ab-neck. The necks’ effective mean curvature is used to distinguish positive from negative necks. Two-sphere shapes of two-domain vesicles can attain four different morphologies that are governed by two different stability conditions. The closed ab-necks are compressed by constriction forces which induce neck fission and vesicle division for large line tensions and/or large spontaneous curvatures. Multispherical shapes with one ab-neck and additional aa- and bb-necks involve several stability conditions, which act to reduce the stability regimes of the multispheres. Furthermore, vesicles with more than two domains form multispheres with more than one ab-neck. The multispherical shapes described here represent generalized constant-mean-curvature surfaces with up to four constant mean curvatures. These shapes are accessible to experimental studies using available methods for giant vesicles prepared from ternary lipid mixtures.Graphic abstract

A survey about a Do Carmo’s question on complete stable constant mean curvature hypersurfaces

A survey about a Do Carmo’s question on complete stable constant mean curvature hypersurfaces

POINTWISE 1-TYPE GAUSS MAP OF DEVELOPABLE SMARANDACHE RULED SURFACES IN $\mathbb{E}^{3}$

In this paper, we study developable TN, TB, and NB-Smarandache ruled surface with pointwise 1-type Gauss map. In particular, we obtain every such developable TN-Smarandache ruled surface has constant mean curvature, developable TB Smarandache ruled surface is minimal if and only if the curve is a planar curve or helix, and developable NB-Smarandache ruled surface is always plane. Also, we give some examples.

The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

AbstractWe prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3.

Quasiconformal Gauss maps and the Bernstein problem for Weingarten multigraphs

Abstract: We prove that any complete, uniformly elliptic Weingarten surface in Euclidean $3$-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that planes are the only complete, uniformly elliptic Weingarten multigraphs. We also show that this result holds for a large class of non-uniformly elliptic Weingarten equations. In particular, this solves in the affirmative the Bernstein problem for entire graphs for that class of elliptic equations. To obtain these results, we prove that planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form.

A note on starshaped hypersurfaces with almost constant mean curvature in space forms

A note on starshaped hypersurfaces with almost constant mean curvature in space forms

Constant mean curvature foliation of the Taub–NUT metric

Constant mean curvature foliation of the Taub–NUT metric