Compact Minimal Surfaces Research Articles

Simons-type inequalities for minimal surfaces with constant Kähler angle in a complex hyperquadric

In this paper, we establish Simons-type inequalities and obtain some pinching theorems for compact minimal surfaces with constant Kähler angle immersed into a complex hyperquadric, and we characterize all these minimal surfaces when equality holds in the pinching theorems.

Addendum to the paper: Compact embedded minimal surfaces in the Berger sphere

We construct a two discrete parameter family of compact minimal surfaces embedded in the Berger sphere which may be considered as the analogue of the helicoidal Karcher-Scherk surfaces.

Minimal surfaces in the product of two dimensional real space forms endowed with a neutral metric

We investigate minimal surfaces in products of two-spheres $\mathbf{S}^2_p\times\mathbf{S}^2_p$, with the neutral metric given by $(g,-g)$. Here $\mathbf{S}^2_p\subset \mathbf{R}^{p,3-p}$, and $g$ is the induced metric on the sphere. We compute all totally geodesic surfaces and we give a relation between minimal surfaces and the solutions of the Gordon equations. Finally, in some cases we give a topological classification of compact minimal surfaces.

Fields of Smooth Compact Minimal Surfaces

The minimality of surfaces defined by certain classical embeddings is proved and properties of sections of the surfaces are studied.

The geometry of stable minimal surfaces in metric Lie groups

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depends only on the maximum distance of points of the boundary $\partial M$ to a vertical geodesic of $X$. We also give a generalization of the classical Rado's Theorem in $\mathbb{R}^3$ to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in $X$, and we study the geometry, existence and uniqueness of this type of Plateau problem.

Penrose-type inequalities with a Euclidean background

The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.

Rigidity of compact surfaces in homogeneous 3-manifolds with constant mean curvature

In this paper, we establish a rigidity theorem for compact constant mean curvature surfaces of the Berger sphere in terms of the surfaces' geometric invariants. This extends the previous similar result on compact minimal surfaces of the Berger sphere.

Uniform approximation by complete minimal surfaces of finite total curvature in ℝ³

We prove that any compact minimal surface in R 3 \mathbb {R}^3 can be uniformly approximated by complete minimal surfaces of finite total curvature in R 3 \mathbb {R}^3 . This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on regions of finite conformal type. We deal only with the orientable case.

A spectral curve approach to Lawson symmetric CMC surfaces of genus 2

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construction. The parametrization uses Hitchin's abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1-dimensional torus. This description is used to develop a spectral curve theory for the Lawson surface. This theory applies as well to other CMC and minimal surfaces with the same holomorphic symmetries as the Lawson surface but different Riemann surface structure. Additionally, we study the space of isospectral deformations of compact minimal surface of genus $g\geq2$ and prove that it is generated by simple factor dressing.

On stable compact minimal submanifolds

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface are the complex ones.

Minimal Surfaces in $\mathbb{S}^{2} \times\mathbb{S}^{2}$

A general study of minimal surfaces of the Riemannian product of two spheres \(\mathbb {S}^{2}\times \mathbb {S}^{2}\) is tackled. We establish a local correspondence between (non-complex) minimal surfaces of \(\mathbb {S}^{2} \times \mathbb {S}^{2}\) and a certain pair of minimal surfaces of the sphere \(\mathbb {S}^{3}\). This correspondence also allows us to link minimal surfaces in \(\mathbb{S}^{3}\) and in the Riemannian product \(\mathbb {S}^{2} \times \mathbb {R}\). Some rigidity results for compact minimal surfaces are also obtained.

On minimal surfaces in a class of Finsler 3-spheres

Perturbing the classical metric on the round 3-sphere \(S^3\) by the Killing vector fields tangent to Hopf fibers, one gets a class of Finsler metrics of Randers type with constant flag curvature \(\mathbf{K}=1\), depending on one parameter \(k>1\), called Bao–Shen’s (J Lond Math Soc 66:453–467, 2002) metrics. The corresponding spheres will be called Bao–Shen’s spheres, which are proper candidates of positively curved Finsler space forms. In this paper, we study the minimal surfaces in Bao–Shen’s spheres. We first study submanifolds isometrically immersed in a Randers manifold by the method of Zermelo’s navigation. Then we give a clear formula of the mean curvature of the surface in a Bao–Shen’s sphere by introducing the volume ratio function to show its relation with the mean curvature of the surface in round 3-sphere. As an application, we find an interesting family of minimal surfaces with respect to Busemann–Hausdorff volume form in Bao–Shen’s sphere called helicoids. This family contains the compact minimal surfaces \(\tau _{m,n}\) in round 3-sphere constructed by Lawson (Ann Math 92(3):335–374, 1970), including great 2-spheres, Clifford torus, Klein bottles, etc. Moreover, two rigidity results are given.

Higher genus minimal surfaces in S3 and stable bundles

We consider compact minimal surfaces of genus 2 which are homotopic to an embedding. We prove that such surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic connections are at the Weierstrass points of the Riemann surface and are at most quadratic. For the existence proof of the DPW potential, we give a characterization of stable extensions of spin bundles S by its dual in terms of an associated element of . We also show that the family of holomorphic structures associated to a minimal surface of genus in S 3 is generically stable.

Relative isoperimetric inequalities for minimal submanifolds outside a convex set

AbstractLet C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ∩∂C and ∂Σ ∼ ∂C is radially connected from a point p ∈ ∂Σ∩∂C. We introduce a modified volume Mp(Σ) of Σ and obtain a sharp isoperimetric inequality where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.

An isoperimetric constant associated to horizons in [formula omitted] blown up at two points

Let g be a metric on S 3 with positive Yamabe constant. When blowing up g at two points, a scalar flat manifold with two asymptotically flat ends is produced and this manifold will have compact minimal surfaces. We introduce the Θ -invariant for g which is an isoperimetric constant for the cylindrical domain inside the outermost minimal surface of the blown-up metric. Further we find relations between Θ and the Yamabe constant and the existence of horizons in the blown-up metric on R 3 .

Minimal surfaces in circle bundles over Riemann surfaces

For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $\Sigma$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the complement of a finite number of points in $\Sigma$.

A Payne–Rayner Type Inequality for the Robin Problem on Arbitrary Minimal Surfaces in $${\mathbb{R}^N}$$

We prove a Payne–Rayner type inequality for the first eigenfunction of the Laplacian with Robin boundary condition on any compact minimal surface with boundary in $${\mathbb{R}^N}$$ . We emphasize that no topological condition is necessary on the boundary.

On minimal surfaces bounded by two convex curves in parallel planes

We prove that a compact minimal surface bounded by two closed convex curves in parallel planes close enough to each other must be topologically an annulus.

Isoperimetric bounds for the first eigenvalue of the Laplacian

In this paper, generalizing an earlier result by Payne–Rayner, we prove an isoperimetric lower bound for the first eigenvalue of the Laplacian in the fixed membrane problem on a compact minimal surface in a Euclidean space Rn with weakly connected boundary. We also prove an isoperimetric upper bound for the first eigenvalue of the Laplacian of an embedded closed hypersurface in Rn.

Minimal surfaces, Hopf differentials and the Ricci condition

We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials, which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture for exceptional minimal surfaces in odd dimensional spheres or in S 4m .