Legendrian Immersions Research Articles

Almost monotonicity formula for H‐minimal Legendrian surfaces in the Heisenberg group

AbstractWe prove an almost monotonicity formula for H‐minimal Legendrian Surfaces (also called contact stationary Legendrian immersions or Hamiltonian stationary immersions) in the Heisenberg Group . From this formula we deduce a Bernstein‐Liouville type theorem for H‐minimal Legendrian Surfaces. We also present some possible range of applications of this formula.

Mergelyan approximation theorem for holomorphic Legendrian curves

In this paper, we prove a Mergelyan type approximation theorem for immersed holomorphic Legendrian curves in an arbitrary complex contact manifold $(X,\xi)$. Explicitly, we show that if $S$ is a compact admissible set in a Riemann surface $M$ and $f:S\to X$ is a $\xi$-Legendrian immersion of class $\mathscr{C}^{r+2}(S,X)$ for some $r\ge 2$ which is holomorphic in the interior of $S$, then $f$ can be approximated in the $\mathscr{C}^r(S,X)$ topology by holomorphic Legendrian embeddings from open neighbourhoods of $S$ into $X$. This result has numerous applications, some of which are indicated in the paper. In particular, by using Bryant's correspondence for the Penrose twistor map $\mathbb{CP}^3\to S^4$ we show that a Mergelyan approximation theorem and the Calabi-Yau property hold for superminimal surfaces in the $4$-sphere $S^4$.

The Calabi–Yau problem for minimal surfaces with Cantor ends

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in \mathbb{R}^3 with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least 2, for holomorphic null immersions into \mathbb{C}^n with n \geq 3 , for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any selfdual or anti-self-dual Einstein four-manifold.

Holomorphic Legendrian curves in ℂℙ3 and superminimal surfaces in 𝕊4

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into $\mathbb{CP}^3$ as a complete holomorphic Legendrian curve. Under the twistor projection $\pi:\mathbb{CP}^3\to \mathbb S^4$ onto the $4$-sphere, immersed holomorphic Legendrian curves $M\to \mathbb{CP}^3$ are in bijective correspondence with superminimal immersions $M\to\mathbb S^4$ of positive spin according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in $\mathbb S^4$. In particular, superminimal immersions into $\mathbb S^4$ satisfy the Runge approximation theorem and the Calabi-Yau property.

Loose Legendrian and pseudo-Legendrian knots in $3$-manifolds

We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a $3$-manifold $M$ that are transverse to a nowhere-zero vector field $V$ up to the corresponding isotopy relation. Such knots are called $V$-transverse. A framed isotopy class is simple if any two $V$-transverse knots in that class which are homotopic through $V$-transverse immersions are $V$-transverse isotopic. We show that all knot types in $M$ are simple if any one of the following three conditions hold: $1.$ $M$ is closed, irreducible and atoroidal; or $2.$ the Euler class of the $2$-bundle $V^{\perp}$ orthogonal to $V$ is a torsion class, or $3.$ if $V$ is a coorienting vector field of a tight contact structure. Finally, we construct examples of pairs of homotopic knot types such that one is simple and one is not. As a consequence of the $h$-principle for Legendrian immersions, we also construct knot types which are not Legendrian simple.

Darboux Charts Around Holomorphic Legendrian Curves and Applications

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,\xi)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds, and we find a holomorphic Darboux chart around any contractible isotropic Stein submanifolds in an arbitrary complex contact manifold.

On Willmore Legendrian surfaces in $$\mathbb {S}^5$$ S 5 and the contact stationary Legendrian Willmore surfaces

In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo ( arXiv:1211.4227v6 ) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in $$\mathbb {S}^5$$ , and then we use this relation to prove a classification result for Willmore Legendrian spheres in $$\mathbb {S}^5$$ . We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in $$\mathbb {S}^5$$ belongs to [0, 2], then it must be 0 and L is totally geodesic or 2 and L is a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let $$\Sigma $$ be a closed surface and $$(M,\alpha ,g_\alpha ,J)$$ a 5-dimensional Sasakian manifold with a contact form $$\alpha $$ , an associated metric $$g_\alpha $$ and an almost complex structure J. Assume that $$f:\Sigma \mapsto M$$ is a Legendrian immersion. Then f is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if $$(M,\alpha ,g_\alpha ,J)$$ is a Sasakian Einstein manifold, in particular $$\mathbb {S}^5$$ .

The Oka principle for holomorphic Legendrian curves in $$\mathbb {C}^{2n+1}$$ C 2 n + 1

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ of continuous maps from $M$ to the sphere $\mathbb S^{4n-1}$. If $M$ has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of $\mathscr L(M,\mathbb C^{2n+1})$ in terms of the homotopy groups of $\mathbb S^{4n-1}$. It follows that $\mathscr L(M,\mathbb C^{2n+1})$ is $(4n-3)$-connected.

Lagrangian Homothetic Solitons for the Inverse Mean Curvature Flow

We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus on the Lagrangian case, obtaining numerous examples and uniqueness results for some products of circles and spheres that generalize the Clifford torus to arbitrary dimension. We also characterize the pseudoumbilical ones in terms of soliton curves for the inverse curve shortening flow and minimal Legendrian immersions in odd-dimensional spheres. As a consequence, we classify the rotationally invariant Lagrangian homothetic solitons for the inverse mean curvature flow.

Minimal Lagrangian submanifolds via the geodesic Gauss map

For an oriented isometric immersed submanifold of the n-sphere, the spherical Gauss map is the Legendrian immersion of its unit normal bundle into the unit sphere subbundle of the tangent bundle of the sphere, and the geodesic Gauss map projects this into the manifold of oriented geodesics (the Grassmannian of oriented 2-planes in Euclidean n+1-space), giving a Lagrangian immersion of the unit normal bundle into a Kaehler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of the immersion, and show that when it has conformal shape form this depends only on its mean curvature. In particular we deduce that the geodesic Gauss map of every minimal surface in the n-sphere is minimal Lagrangian. We also give simple proofs that: deformations of the immersion always correspond to Hamiltonian deformations of its geodesic Gauss map; the mean curvature vector of the geodesic Gauss map is always a Hamiltonian vector field. This extends work of Palmer on the case when the immersion is a hypersurface.

The Gauss map of a hypersurface in Euclidean sphere and the spherical Legendrian duality

We investigate the Gauss map of a hypersurface in Euclidean n-sphere as an application of the theory of Legendrian singularities. We can interpret the image of the Gauss map as the wavefront set of a Legendrian immersion into a certain contact manifold. We interpret the geometric meaning of the singularities of the Gauss map from this point of view.

Mapped null hypersurfaces and Legendrian maps

For an ( m + 1 ) -dimensional space–time ( X m + 1 , g ) , define a mapped null hypersurface to be a smooth map ν : N m → X m + 1 (that is not necessarily an immersion) such that there exists a smooth field of null lines along ν that are both tangent and g -orthogonal to ν . We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle S T ∗ M of an immersed spacelike hypersurface μ : M m → X m + 1 . We show that a Legendrian map λ ˜ : L m − 1 → ( S T ∗ M ) 2 m − 1 defines a mapped null hypersurface in X . On the other hand, the intersection of a mapped null hypersurface ν : N m → X m + 1 with an immersed spacelike hypersurface μ ′ : M ′ m → X m + 1 defines a Legendrian map to the spherical cotangent bundle S T ∗ M ′ . This map is a Legendrian immersion if ν came from a Legendrian immersion to S T ∗ M for some immersed spacelike hypersurface μ : M m → X m + 1 .

Hamiltonian-minimal Lagrangian submanifolds in complex space forms

Using Legendrian immersions and, in particular, Legendre curves in odd-dimensional spheres and anti-de Sitter spaces, we construct new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one-parameter families of embeddings of quotients of certain product manifolds. We also give new examples of minimal Lagrangian submanifolds in complex projective and hyperbolic spaces. Making use of all these constructions, we get Hamiltonian-minimal and special Lagrangian cones in complex Euclidean space as well.

Singularities of flat fronts in hyperbolic space

It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.

Isomorphism of the Groups of Vassiliev Invariants of Legendrian and Pseudo-Legendrian Knots

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of C-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples of contact 3-manifolds where Vassiliev invariants of Legendrian and of framed knots are different. Moreover in these examples Vassiliev invariants of Legendrian knots distinguish Legendrian knots that are isotopic as framed knots and homotopic as Legendrian immersions. This raised the question what information about Legendrian knots can be captured using Vassiliev invariants. Here we answer this question by showing that for any contact 3-manifold with a cooriented contact structure the groups of Vassiliev invariants of Legendrian knots and of knots that are nowhere tangent to a vector field that coorients the contact structure are canonically isomorphic.

Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds

We show that for a large class of contact three-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's J +-type invariants of wave fronts on a surface F is isomorphic to the group of Vassiliev invariants of framed knots in the spherical cotangent bundle ST ∗F of F. On the other hand, we construct the first examples of contact manifolds for which Vassiliev invariants of Legendrian knots can distinguish Legendrian knots that realize isotopic framed knots and are homotopic as Legendrian immersions.

Arnold-type invariants of wave fronts on surfaces

Recently, Arnold's St and J ± invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface F. All invariants satisfying the axioms which naturally generalize the axioms used by Arnold are explicitly described. We also give an explicit formula for the finest order one J +-type invariant of fronts on an orientable surface F≠ S 2. We obtain necessary and sufficient conditions for an invariant of nongeneric fronts with one nongeneric singular point to be the Vassiliev-type derivative of an invariant of generic fronts. As a byproduct, we calculate all homotopy groups of the space of Legendrian immersions of S 1 into the spherical cotangent bundle of a surface.

Multivalued solutions to the eikonal equation in stratified media

In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geometric solutions, using the notion of Legendrian immersions with an initial point source or an initial smooth front. Then, we study the singularities of the solutions in the case of a smooth or piecewise Hamiltonian in a boundaryless stratified medium. Finally, we study the singularities of the solutions in a domain with a boundary that describes the propagating field in a waveguide.

The Euler Characteristic of a Generic Wavefront in a 3-Manifold

We give a relation between Euler characteristics of a generic closed Legendrian surface and its wavefront. 0. INTRODUCTION In this note we shall compute the Euler characteristic of a generic wave front in a 3-manifold. Let N be a (2n + l)-dimensional smooth manifold and K be a contact structure on N (i.e., K is a nondegenerate tangent hyperplane field on N ). An immersion i: L N is said to be Legendrian if dimL = n and di,(TxL) c Kx for any x E L. We say that a smooth fibre bundle t: E -4 M is Legendrian if its total space E is furnished with a contact structure and its fibres are Legendrian submanifolds. For a Legendrian immersion i: L -E, 7t o i : L -* M is called a Legendrian map and the image of the Legendrian map Xt o i is called the wavefront of i. It is denoted by W(i). From now on, we only consider the case of n = 2. Then it is known that a generic wavefront has (semicubic) cuspidal edges (A2), swallowtails (A3), and points of transversal self-intersection (A1A1, A1A2, AIAIAI) as singularities [1] (see Figure 1 on the next page). We shall refer to the AI AI A I-type point as a triple point of i. If L is a closed surface, then the number of swallowtails and triple points are finite. Our main result is the following: Theorem. Let i : L -? E be a generic Legendrian immersion of a closed surface. Then we have X(W(i)) = X(L) + T(i) + s() 2' where x (X) is the Euler characteristic of X, T(i) is the number of triple points on W(i), and S(i) is the number of swallowtails. We remark that the corresponding result for a generic wavefront in a 2manifold is easily verified and that X(W(i)) = -d(i), where d(i) denotes the number of double points on W(i). Received by the editors December 23, 1991. 1991 Mathematics Subject Classification. Primary 58C27.