Constant Mean Curvature Research Articles

The Equality Case in the Substatic Heintze–Karcher Inequality

We provide a rigidity statement for the equality case of the Heintze–Karcher inequality in substatic manifolds. We apply such a result in the warped product setting to fully remove assumption (H4) in the celebrated Brendle’s characterization of constant mean curvature hypersurfaces in warped products.

Existence of free boundary disks with constant mean curvature in [formula omitted]

Given a surface Σ in R3 diffeomorphic to S2, Struwe [38] proved that for almost every H below the mean curvature of the smallest sphere enclosing Σ, there exists a branched immersed disk which has constant mean curvature H and boundary meeting Σ orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on Σ. Specifically, when Σ itself is convex and has mean curvature bounded below by H0, we obtain existence for all H∈(0,H0). Instead of the heat flow in [38], we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou [7], a key ingredient for extending existence across the measure zero set of H's is a Morse index upper bound.

Foliations of asymptotically flat manifolds by stable constant mean curvature spheres

Foliations of asymptotically flat manifolds by stable constant mean curvature spheres

Min-max theory for capillary surfaces

Abstract We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature 𝑐, and with smooth boundary contacting at any given constant angle 𝜃. Moreover, if 𝑐 is nonzero and 𝜃 is not π 2 \frac{\pi}{2} , then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions.

Stability of Capillary Hypersurfaces with Constant Higher Order Mean Curvature

Stability of Capillary Hypersurfaces with Constant Higher Order Mean Curvature

Local rigidity of constant mean curvature hypersurfaces in space forms

In this paper, we study the local rigidity of constant mean curvature (CMC) hypersurfaces. Let x:Mn→Mn+1(c),n≥4, be a piece of immersed constant mean curvature hypersurface in the (n+1)-dimensional space form Mn+1(c). We prove that if the scalar curvature R is constant and the number g of the distinct principal curvatures satisfies g≤3, then Mn is an isoparametric hypersurface. Further, if Mn is a minimal hypersurface, then Mn is a totally geodesic hypersurface for c≤0, and Mn is either a Cartan minimal hypersurface, a Clifford minimal hypersurface, or a totally geodesic hypersurface for c>0, which solves the high dimensional version of Bryant Conjecture.

On effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes

We study the effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes, both in the spacelike and the timelike case, respectively. In the spacelike case, we study totally umbilic warped product graphs and give a full characterization of embedded surfaces with constant spacetime mean curvature using an Alexandrov Theorem by Brendle and Borghini–Fogagnolo–Pinamonti. In the timelike case, we achieve a characterization of photon surfaces with constant umbilicity factor similar to a result by Cederbaum–Galloway.

Gradient conformal stationarity and the CMC condition in LRS spacetimes

Abstract We study the existence of gradient conformal Killing vectors (CKVs) in the class of locally rotationally symmetric (LRS) spacetimes which generalizes spherically symmetric spacetimes, and investigate some implications for the evolutionary character of marginally outer trapped surfaces. We first study existence of gradient CKVs via the obtention of a relationship between the Ricci curvature and the gradient of the divergence of the CKV. This provides an alternative set of equations, for which the integrability condition is obtained, to analyze the existence of gradient CKVs. A uniqueness result is obtained in the case of perfect fluids, where it is demonstrated that the Robertson-Walker solution is the unique perfect fluid solution with a nonvanishing pressure, admitting a timelike gradient CKV. The constant mean curvature condition for LRS spacetimes is also obtained, characterized by three distinct conditions which are specified by a set of three scalars. Linear combinations of these scalars, whose vanishing define the constant mean curvature condition, turn out to be related to the evolutions of null expansions of 2-spheres along their null normal directions. As such, some implications for the existence of black holes and the character of the associated horizons are obtained. It is further shown that dynamical black holes of increasing area, with a non-vanishing heat flux across the horizon, will be in equilibrium, with respect to the frame of the conformal observers.

On the stability of constant higher order mean curvature hypersurfaces in a Riemannian manifold

AbstractWe propose a notion of stability for constant ‐mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem.

Kα-translators of Weingarten type in Gauss curvature flow

In this paper, we consider Kα-translating solitons in R3 and give some classifications of Kα-translating solitons fulfilling some extrinsic curvature assumptions. We prove that Kα-translating solitons with constant mean curvature are either planes or circular cylinders. We also prove that planes and circular cylinders are the only Kα-translating solitons with constant squared norm of the second fundamental form.

Examples of cosmological spacetimes without CMC Cauchy surfaces

CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity as finding solutions to the vacuum Einstein constraint equations is made much simpler by assuming CMC initial data. However, Bartnik (Commun Math Phys 117(4):615–624, 1988) constructed a cosmological spacetime without a CMC Cauchy surface whose spatial topology is the connected sum of two three-dimensional tori. Similarly, Chruściel et al. (Commun Math Phys 257(1):29–42, 2005) constructed a vacuum cosmological spacetime without CMC Cauchy surfaces whose spatial topology is also the connected sum of two tori. In this article, we enlarge the known number of spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by generalizing Bartnik’s construction. Specifically, we show that there are cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies are the connected sum of any compact Euclidean or hyperbolic three-manifold with any another compact Euclidean or hyperbolic three-manifold. Analogous examples in higher spacetime dimensions are also possible. We work with the Tolman–Bondi class of metrics and prove gluing results for variable marginal conditions, which allows for smooth gluing of Schwarzschild to FLRW models.

Visualization of Isometric Deformations of Helicoidal CMC Surfaces

Helicoidal surfaces of constant mean curvature were fully described by do Carmo and Dajczer. However, the obtained parameterizations are given in terms of somewhat complicated integrals, and as a consequence, not many examples of such surfaces are visualized. In this paper, by using these methods in some particular cases, we provide several interesting visualizations involving these surfaces, mostly as an isometric deformation of a rotational surface. We also give interpretations of some older results involving helicoidal surfaces, motivated by the work carried out by Malkowsky and Veličković. All of the graphics in this paper were created in Wolfram Mathematica.

Sharp pinching theorems for complete submanifolds in the sphere

Abstract For every complete and minimally immersed submanifold f : M n → S n + p f\colon M^{n}\to\mathbb{S}^{n+p} whose second fundamental form satisfies | A | 2 ≤ n ⁢ p / ( 2 ⁢ p − 1 ) \lvert A\rvert^{2}\leq np/(2p-1) , we prove that it is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in S 4 \mathbb{S}^{4} , thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete M n M^{n} . We also obtain the corresponding result for complete hypersurfaces with non-vanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension n ≤ 6 n\leq 6 , a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work by Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.

Simulating Capillary Folding of Thin Elastic Sheets with Pinned Contact Lines

We consider the capillary folding of thin elastic sheets with pinned contact lines in three dimensions. The folding occurs due to the interaction between the elastic sheet and a droplet deposited on top of it. Firstly, we derive the equilibrium equations by minimizing the total energy of the system. This energy comprises the interfacial energies and the elastic energy, which is described by the nonlinear Koiter's model. Then we develop a time-splitting numerical scheme to solve the gradient flow dynamics. Following this dynamics, the system evolves towards the equilibrium state. At each time step, we first determine the droplet surface with a constant mean curvature by minimizing a squared area functional, subject to the boundary condition at the pinned contact line. The elastic sheet, discretized by the C1-conforming subdivision element method, is then evolved by taking into account the fluid pressure and the capillary force. We propose a novel remeshing strategy to avoid the need for interpolating the capillary force on the sheet boundary and to prevent simulations from breakdown due to distorted meshes. We perform numerical tests to demonstrate the accuracy of the numerical method, as well as its ability to predict folded structures for various types of sheets.

Ricci Solitons on Riemannian Hypersurfaces Arising from Closed Conformal Vector Fields in Riemannian and Lorentzian Manifolds

This paper investigates Ricci solitons on Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds. We provide conditions under which a Riemannian hypersurface, exhibiting specific properties related to a closed conformal vector field of the ambiant manifold, forms a Ricci soliton structure. The characterization involves a delicate balance between geometric quantities and the behavior of the conformal vector field, particularly its tangential component. We extend the analysis to ambient manifolds with constant sectional curvature and establish that, under a simple condition, the hypersurface becomes totally umbilical, implying constant mean curvature and sectional curvature. For compact hypersurfaces, we further characterize the nature of the Ricci soliton.

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

Screw motion surfaces of constant mean curvature in homogeneous 3-manifolds

We study the geometry of non-minimal surfaces of supercritical constant mean curvature invariant under screw motions in the homogeneous 3-manifolds E(κ,τ) including the space-forms of non-negative curvature. We give a complete classification, thereby unifying and extending various previous results. We give the first classification for the Berger sphere case, and we exhibit a new family of screw motion CMC surfaces, called tubes.

Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface [Formula: see text] is the complex structure of a complete surface of constant mean curvature [Formula: see text] ([Formula: see text]) in the three-dimensional hyperbolic space [Formula: see text]. We go further and establish a jet interpolation theorem for complete conformal [Formula: see text] immersions [Formula: see text]. As a consequence, we show the existence of complete densely immersed [Formula: see text] surfaces in [Formula: see text] with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in [Formula: see text] which is also established in this paper.

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