CMC Surfaces Research Articles

CMC surfaces of revolution, elliptic curves, Weierstrass-$$\wp $$ functions, and algebraicity

AbstractThis paper establishes an interesting connection between the family of CMC surfaces of revolution in $$\mathbb {E}_1^3$$ 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 $$\mathbb {E}_1^3$$ E 1 3 , only spacelike cylinders and standard hyperboloids are algebraic. We also show that a similar connection exists between CMC surfaces of revolution in $$\mathbb E^3$$ E 3 and elliptic curves. Further, we use this to reestablish the fact that the only CMC algebraic surfaces of revolution in $$\mathbb E^3$$ E 3 are spheres and right circular cylinders.

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.

CMC-1 surfaces via osculating Möbius transformations between circle patterns

Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induce a P S L ( 2 , C ) PSL(2,\mathbb {C}) -valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same discrete conformal structure. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature H ≡ 1 H\equiv 1 in hyperbolic space. We further establish convergence on triangular lattices.

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].

Spacelike CMC surfaces near null infinity of the Schwarzschild spacetime

Spacelike CMC surfaces near null infinity of the Schwarzschild spacetime

Hierarchy structures in finite index CMC surfaces

Abstract Given ε 0 > 0 {{\varepsilon}_{0}>0} , I ∈ ℕ ∪ { 0 } {I\in\mathbb{N}\cup\{0\}} and K 0 , H 0 ≥ 0 {K_{0},H_{0}\geq 0} , let X be a complete Riemannian 3-manifold with injectivity radius Inj ⁡ ( X ) ≥ ε 0 {\operatorname{Inj}(X)\geq{\varepsilon}_{0}} and with the supremum of absolute sectional curvature at most K 0 {K_{0}} , and let M ↬ X {M\looparrowright X} be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] {H\in[0,H_{0}]} with index at most I. For such M ↬ X {M\looparrowright X} , we prove a structure theorem which describes how the interesting ambient geometry of the immersion is organized locally around at most I points of M, where the norm of the second fundamental form takes on large local maximum values.

Geometry of CMC surfaces of finite index

Abstract Given r 0 > 0 {r}_{0}\gt 0 , I ∈ N ∪ { 0 } I\in {\mathbb{N}}\cup \left\{0\right\} , and K 0 , H 0 ≥ 0 {K}_{0},{H}_{0}\ge 0 , let X X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ r 0 \hspace{0.1em}\text{Inj}\hspace{0.1em}\left(X)\ge {r}_{0} and with the supremum of absolute sectional curvature at most K 0 {K}_{0} , and let M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] H\in \left[0,{H}_{0}] and with index at most I I . We will obtain geometric estimates for such an M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X as a consequence of the hierarchy structure theorem. The hierarchy structure theorem (Theorem 2.2) will be applied to understand global properties of M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X , especially results related to the area and diameter of M M . By item E of Theorem 2.2, the area of such a noncompact M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X is infinite. We will improve this area result by proving the following when M M is connected; here, g ( M ) g\left(M) denotes the genus of the orientable cover of M M : (1) There exists C 1 = C 1 ( I , r 0 , K 0 , H 0 ) > 0 {C}_{1}={C}_{1}\left(I,{r}_{0},{K}_{0},{H}_{0})\gt 0 , such that Area ( M ) ≥ C 1 ( g ( M ) + 1 ) {\rm{Area}}\left(M)\ge {C}_{1}\left(g\left(M)+1) . (2) There exist C > 0 C\gt 0 , G ( I ) ∈ N G\left(I)\in {\mathbb{N}} independent of r 0 , K 0 , H 0 {r}_{0},{K}_{0},{H}_{0} , and also C C independent of I I such that if g ( M ) ≥ G ( I ) g\left(M)\ge G\left(I) , then Area ( M ) ≥ C ( max 1 , 1 r 0 , K 0 , H 0 ) 2 ( g ( M ) + 1 ) {\rm{Area}}\left(M)\ge \frac{C}{{\left(\max \left\{1,\frac{1}{{r}_{0}},\sqrt{{K}_{0}},{H}_{0}\right\}\right)}^{2}}\left(g\left(M)+1) . (3) If the scalar curvature ρ \rho of X X satisfies 3 H 2 + 1 2 ρ ≥ c 3{H}^{2}+\frac{1}{2}\rho \ge c in X X for some c > 0 c\gt 0 , then there exist A , D > 0 A,D\gt 0 depending on c , I , r 0 , K 0 , H 0 c,I,{r}_{0},{K}_{0},{H}_{0} such that Area ( M ) ≤ A {\rm{Area}}\left(M)\le A and Diameter ( M ) ≤ D {\rm{Diameter}}\left(M)\le D . Hence, M M is compact and, by item 1, g ( M ) ≤ A / C 1 − 1 g\left(M)\le A\hspace{0.1em}\text{/}\hspace{0.1em}{C}_{1}-1 .

Rigidity of capillary surfaces in compact 3-manifolds with strictly convex boundary

AbstractIn this paper, we obtain one sharp estimate for the length $L(\partial\Sigma)$ of the boundary $\partial\Sigma$ of a capillary minimal surface Σ2 in M3, where M is a compact three-manifolds with strictly convex boundary, assuming Σ has index one. The estimate is in term of the genus of Σ, the number of connected components of $\partial\Sigma$ and the constant contact angle θ. Making an extra assumption on the geometry of M along $\partial M$, we characterize the global geometry of M, which is saturated only by the Euclidean three-balls. For capillary stable CMC surfaces, we also obtain similar results.

Mean curvature flow for generating discrete surfaces with piecewise constant mean curvatures

Piecewise constant mean curvature (P-CMC) surfaces are generated using the mean curvature flow (MCF). As an extension of the known fact that a CMC surface is the stationary point of an energy functional, a P-CMC surface can be obtained as the stationary point of an energy functional of multiple patch surfaces and auxiliary surfaces between them. A new formulation is presented for the MCF as the negative gradient flow of the energy functional for multiple patch continuous surfaces, which are further discretized so as to determine the change in the vertex positions of triangular meshes on the surface as well as along the internal boundaries between patches. Numerical examples show that multiple patch surfaces approximately reach the specified mean curvatures through the proposed method, which can diversify the options for the shape design using CMC surfaces.

Discrete constant mean curvature surfaces on general graphs

The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC in short) surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we show that the discrete isothermic CMC surfaces can be obtained by the discrete holomorphic data (a solution of the additive rational Toda system) via the discrete generalized Weierstrass type representation.

Experimental Investigation Into the Effect of a Ceramic Matrix Composite Surface on Film Cooling

Abstract Ceramic matrix composite (CMC) components enable high turbine entry temperatures, which can lead to improved efficiencies in gas turbines. Implementing film cooling over CMC components, similar to how it is employed for conventional metal components, can extend part life and push operating temperatures beyond the temperature capabilities of CMCs alone. However, CMCs have a unique surface topology that can influence film cooling performance. Often this topology takes the form of an irregular wavy pattern due to the weave of the fibers that make up the strengthening component of the composite. In this study, shaped 7–7–7 film cooling holes are embedded in a five-harness-satin weave pattern representative of a CMC, at two orientations of the pattern. Detailed adiabatic film effectiveness measurements are obtained in a wind tunnel using an infrared camera while near-wall flowfield measurements are obtained with a high-speed particle image velocimetry system. A range of blowing ratios between one and three are investigated at a density ratio of 1.5 and freestream turbulence intensities of 0.5% and 13%. Across the majority of the tested conditions, the CMC surfaces result in lower film cooling performance than a smooth surface. At a freestream turbulence intensity of 0.5%, the adiabatic film effectiveness is moderately insensitive to the blowing ratio for both weave orientations. The boundary layer over the CMC surfaces increases the mixing between the coolant and the mainstream through a combination of increased turbulence, reduced near-wall velocities, and a thicker boundary layer.

Mean convex properly embedded $[\varphi,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$

We establish curvature estimates and a convexity result for mean convex properly embedded [\varphi,\vec{e}_{3}] -minimal surfaces in \mathbb{R}^3 , i.e., \varphi -minimal surfaces when \varphi depends only on the third coordinate of \mathbb{R}^3 . Led by the works on curvature estimates for surfaces in 3-manifolds, due to White for minimal surfaces, to Rosenberg, Souam and Toubiana for stable CMC surfaces, and to Spruck and Xiao for stable translating solitons in \mathbb{R}^3 , we use a compactness argument to provide curvature estimates for a family of mean convex [\varphi,\vec{e}_{3}] -minimal surfaces in \mathbb{R}^{3} . We apply this result to generalize the convexity property of Spruck and Xiao for translating solitons. More precisely, we characterize the convexity of a properly embedded [\varphi,\vec{e}_{3}] -minimal surface in \mathbb{R}^{3} with non-positive mean curvature when the growth at infinity of \varphi is at most quadratic.

Spinorial Representation of Surfaces in Lorentzian Homogeneous Spaces of Dimension $$\varvec{3}$$

We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension 3. We in particular obtain a representation theorem for surfaces in \(\mathbb {L}(\kappa ,\tau )\) spaces. We then recover the Calabi correspondence between minimal surfaces in \(\mathbb {R}^3\) and maximal surfaces in \(\mathbb {R}_1^3\), and obtain a new Lawson type correspondence between CMC surfaces in \(\mathbb {R}_1^3\) and in the 3-dimensional pseudo-hyperbolic space \(\mathbb {H}_1^{3}.\)

Double CMC surfaces in Euclidean and hyperbolic spaces

We establish a rigidity theorem for $2$-manifolds immersed in Euclidean and hyperbolic spaces, which have constant mean curvatures with respect to both induced metrics. Using the relation between the two mean curvatures, we prove that the two immersions a

On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature

In this paper, we study the principal eigenvalue $\mu(\mathscr{F}_k^-,E)$ of the fully nonlinear operator \[ \mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u| \] on a set $E \Subset \mathbb{R}^n$, where $h \in [0,\infty)$ and $\mathcal{P}_k^-(\nabla^2 u)$ is the sum of the smallest $k$ eigenvalues of the Hessian $\nabla^2 u$. We prove a lower estimate for $\mu(\mathscr{F}_k^-,E)$ in terms of a generalized Hausdorff measure $\mathscr{H}_\Psi(E)$, for suitable $\Psi$ depending on $k$, moving some steps in the direction of the conjecturally sharp estimate \[ \mu(\mathscr{F}_k^-,E) \ge C \mathscr{H}^k(E)^{-2/k}. \] The theorem is used to study the spectrum of bounded submanifolds in $\mathbb{R}^n$, improving on our previous work in the direction of a question posed by S.T. Yau. In particular, the result applies to solutions of Plateau's problem for CMC surfaces.

Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals

We describe the construction of CMC surfaces with symmetries in mathbb {S}^{3} and mathbb {R}^{3} using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.

Impact of Ceramic Matrix Composite Topology on Friction Factor and Heat Transfer

AbstractCeramic matrix composites (CMCs) are of interest for hot section components of gas turbine engines due to their low weight and favorable thermal properties. To implement this advanced composite in a gas turbine engine, characterizing the influence of CMC’s surface topology on heat transfer and cooling performance is critical. However, very few published studies have reported the flow and heat transfer effects caused by this unique surface topology. This study is an experimental and computational investigation to evaluate the effect of weave orientations, relevant to CMC surfaces, on the resulting pressure loss and convective heat transfer within an internal channel. The weave pattern was additively manufactured as the walls of a scaled-up coupon containing a single channel. For each of the three weave orientations, bulk pressure losses and convective heat transfer coefficients were measured over a range of Reynolds numbers. Scaling the pressure losses in terms of a friction factor and convective heat transfer coefficients in terms of a Nusselt number showed the importance of choosing the appropriate definition of the hydraulic diameter, which was particularly important for the friction factor. A coupon having one wall with the weave surface increased pressure loss and heat transfer compared to a smooth wall with the largest increases occurring when the CMC weave strands were perpendicular to the flow. Friction factor augmentations were much higher than heat transfer augmentations. When adding the weave to a second channel wall, pressure loss and heat transfer were further increased. Orienting the CMC strands perpendicular to the flow consistently showed the largest augmentations in heat transfer over a smooth channel, but at a much higher pressure loss penalty than that seen with the CMC strands parallel to the flow.

A nonexistence result for CMC surfaces in hyperbolic 3-manifolds

We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.

Stable surfaces and free boundary marginally outer trapped surfaces

We explore various notions of stability for surfaces embedded and immersed in spacetimes and initial data sets. The interest in such surfaces lies in their potential to go beyond the variational techniques which often underlie the study of minimal and CMC surfaces. We prove two versions of Christodoulou–Yau estimate for \({\mathbf {H}}\)-stable surfaces, a Cohn-Vossen type inequality for non-compact stable marginally outer trapped surface (MOTS), and a global theorem on the topology of \({\mathbf {H}}\)-stable surfaces. Moreover, we give a definition of capillary stability for MOTS with boundary. This notion of stability leads to an area upper bound inequality and a local splitting theorem for free boundary stable MOTS. Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. These are straightforward generalizations of Chen–Fraser–Pang and Carlotto–Franz results for free boundary minimal surfaces, respectively.