Surfaces In 3-manifolds Research Articles

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.

Ruled Surfaces in 3-Dimensional Riemannian Manifolds

In this work, ruled surfaces in 3-dimensional Riemannian manifolds are studied. We determine the expressions for the extrinsic and sectional curvatures of a parametrized ruled surface, where the former one is shown to be non-positive. We also quantify the set of ruling vector fields along a given base curve which allows us to define a relevant reference frame that we refer to as Sannia frame. The fundamental theorem of existence and equivalence of Sannia ruled surfaces in terms of a system of invariants is given. The second part of the article tackles the concept of the striction curve, which is proven to be the set of points where the so-called Jacobi evolution function vanishes on a ruled surface. This characterisation of striction curves provides independent proof for their existence and uniqueness in space forms and disproves their existence or uniqueness in some other cases.

An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds

An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds

Gauss-Bonnet Theorems for Surfaces in Sub-Riemannian Three-dimensional Manifolds

Gauss-Bonnet Theorems for Surfaces in Sub-Riemannian Three-dimensional Manifolds

Characterization of manifolds of constant curvature by ruled surfaces

We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the Gaussian and mean curvatures. We also provide model-independent proof for the known fact that extrinsically flat surfaces in space forms are ruled. This proof allows us to identify the necessary and sufficient condition the curvature tensor must satisfy for an extrinsically flat surface in a generic 3d manifold to be ruled. Further, we show that if a 3d manifold has an extrinsically flat surface tangent to any 2d plane and if they are all ruled surfaces, then the manifold is a space form. As an application, we prove that there must exist extrinsically flat surfaces in the Riemannian product of the hyperbolic plane, or sphere, with the reals, and that do not make a constant angle with the real direction.

Variational equations and Killing magnetic trajectories on timelike surfaces in semi-Riemannian manifolds

In this article, Darboux frame variations for timelike surfaces in semi-Riemannian manifolds are discussed. In addition, the Killing equations are examined by using the Darboux frame curvature variations. Then, magnetic trajectories are generated by means of the variational vector fields. Furthermore, parametric representations of all magnetic trajectories on the de Sitter space $\mathbb{S}_{1}^{2}$ are presented. Moreover, various examples of magnetic trajectories are given in order to illustrate the theoretical results.

Homotopy Motions of Surfaces in 3-Manifolds

Abstract We introduce the concept of a homotopy motion of a subset in a manifold and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behavior of simple loops on a Heegaard surface and monodromies of virtual branched covering surface bundles associated with a Heegaard splitting.

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.

Counting essential surfaces in 3-manifolds

We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected) essential surfaces in terms of their Euler characteristic always has a short generating function and hence has quasi-polynomial behavior. This gives remarkably concise formulae for the number of such surfaces, as well as detailed asymptotics. We give algorithms that allow us to compute these generating functions and the underlying surfaces, and apply these to almost 60,000 manifolds, providing a wealth of data about them. We use this data to explore the delicate question of counting only connected essential surfaces and propose some conjectures. Our methods involve normal and almost normal surfaces, especially the work of Tollefson and Oertel, combined with techniques pioneered by Ehrhart for counting lattice points in polyhedra with rational vertices. We also introduce a new way of testing if a normal surface in an ideal triangulation is essential that avoids cutting the manifold open along the surface; rather, we use almost normal surfaces in the original triangulation.

Codazzi surfaces in 4-manifolds

Codazzi surfaces in 4-manifolds

Index bounds for closed minimal surfaces in 3-manifolds with the Killing property

Index bounds for closed minimal surfaces in 3-manifolds with the Killing property

On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds

Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient K̂ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.

Hopf type theorems for surfaces in Riemannian manifolds

Hopf type theorems for surfaces in Riemannian manifolds

Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

Our main result gives an improved bound on the filling areas of curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu’s classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.

Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds

On considère des processus stochastiques sur des surfaces dans des espaces tridimensionnels de contact sous-riemanniens. En utilisant des approximations riemanniennes définies par le champ de Reeb, on obtient un opérateur différentiel du second ordre sur la surface comme limite des opérateurs de Laplace–Beltrami correspondants. Le processus stochastique associé à l’opérateur limite est défini le long de la foliation caractéristique induite sur la surface par la distribution de contact. Nous montrons que pour ce processus stochastique, les points elliptiques sont inaccessibles alors que les points hyperboliques sont accessibles à travers les séparatrices. On discute les résultats sur des exemples et on identifie des surfaces canoniques dans le groupe de Heisenberg, ainsi que dans les groupes SU(2) et SL(2,R) équipés de leurs structures sous-riemanniennes de contact canoniques, jouant le rôle de cas modèles dans ce contexte. Ces techniques nous permettent de plus de dériver une expression de la courbure gaussienne intrinsèque d’une surface générale dans une variété sous-riemannienne de dimension trois.

The Cohomological Genus and Symplectic Genus for 4-Manifolds of Rational or Ruled Types

An important problem in low dimensional topology is to understand the properties of embedded or immersed surfaces in 4-dimensional manifolds. In this article, we estimate the lower genus bound of closed, connected, smoothly embedded, oriented surfaces in a smooth, closed, connected, oriented 4-manifold with the cohomology algebra of a rational or ruled surface. Our genus bound depends only on the cohomology algebra rather than on the geometric structure of the 4-manifold. It provides evidence for the genus minimizing property of rational and ruled surfaces.

Wintgen inequality for surfaces in Hessian manifolds

Wintgen inequality for surfaces in Hessian manifolds

Self-intersections of closed parametrized minimal surfaces in generic Riemannian manifolds

This article shows that for generic choice of Riemannian metric on a compact oriented manifold M of dimension four, the tangent planes at any self-intersection p in M of any prime closed parametrized minimal surface in M are not simultaneously complex for any orthogonal complex structure on M at p. This implies via geometric measure theory that H_2(M;{{mathbb {Z}}}) is generated by homology classes that are represented by oriented imbedded minimal surfaces.

Multibranched Surfaces in 3-Manifolds

This article is a survey of recent works on embeddings of multibranched surfaces into 3-manifolds.

Null-homologous exotic surfaces in 4–manifolds

In this paper we exhibit infinite families of embedded tori in 4-manifolds that are topologically isotopic but smoothly distinct. The interesting thing about these tori is that they are topologically trivial in the sense that each bounds a topologically embedded solid handlebody. This implies that there are stably ribbon surfaces in 4-manifolds that are not ribbon.