The minimal surface equation $Q$ in the second order contact bundle of $R^3$, modulo translations, is provided with a complex structure and a canonical vector-valued holomorphic differential form $Omega$ on $Q\0$. The minimal surfaces $M$ in $R^3$ correspond to the complex analytic curves $C$ in $Q$, where the derivative of the Gauss map sends $M$ to $C$, and $M$ is equal to the real part of the integral of $Ω$ over $C$. The complete minimal surfaces of finite topological type and with flat points at infinity correspond to the algebraic curves in $Q$.
We provide a congruence theorem for minimal surfaces in $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in $S^5$ with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we will give a characterization of flat minimal surfaces in $S^5$ with constant contact angle.
We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.
We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $ϕ$. For the normal case, we prove that a $ϕ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $ϕ$-invariant submanifold $N$ everywhere transverse to both the Reeb vector fields but not orthogonal to them, we prove that it is minimal if and only if the angle between the tangential component $ξ$ (with respect to $N$) of a Reeb vector field and the Reeb vector field itself is constant along the integral curves…
In this paper we introduce the notion of contact angle. We deduce formulas for Laplacian and Gaussian curvature of a minimal surface in $S^{2n+1}$ and give a characterization of the generalized Clifford Torus as the only non-legendrian minimal surface in $S^5$ with constant Contact and Kaehler angles.
We provide a characterization of the Clifford Torus in S3 via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S3 with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed.
In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian minimal surfaces can be defined as the critical points of the '`horizontal'' area functional. We derive an intrinsic equation for minimal surfaces associated to a generic Sub-Riemannian structure of co-rank 1 in terms of the canonical volume form and the ``horizontal'' normal. The presented construction permits to describe the Sub-Riemannian minimal surface…
Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.
Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the firs…
In the present paper we consider the problem of local equivalence of second order ODEs which are cubic in second derivative under the action of the pseudogroup of contact transformations. We show how it may be reduced to the equivalence problem of 2-webs in $\mathbb{R}^3$ under the action of finite-dimensional group, admitted by the associated second order equation.
In the present paper we establish the necessary and sufficient conditions for two ordinary differential equations of the form $y"{}^2+A(x,y,y') y"+B(x,y,y')=0$ to be equivalent under the action of the pseudogroup of contact transformations. These conditions are formulated in terms of integrals of some one-dimensional distributions.
In this paper we introduce the notion of contact angle for an immersed surface in three dimensional sphere. We deduce formulas for the Laplacian and for the Gaussian curvature, and we classify minimal surfaces in $S^3$ with constant contact angle. Also, we give an example of a minimal surface in $S^3$ with non constant contact angle.
In \cite{Luo}, the present author proved that if $L$ is a contact stationary Legendrian surface in $\mathbb{S}^5$ with the canonical Sasakian structure and the square length of its second fundamental form belongs to $[0,2]$. Then we have that $L$ is either totally umbilical or is a flat minimal Legendrian torus. In this addendum we further prove that if $L$ is a totally umbilical contact stationary Legendrian surface in $\mathbb{S}^5$ , then $L$ is totally geodesic.
We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467
A contact stationary Legendrian submanifold of $S^{2n+1}$ is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact stationary Legendrian submanifold (actually minimal and Legendrian) is the real, equatorial $n$-sphere $S_0$. This paper develops a method for constructing contact stationary (but not minimal) Legendrian submanifolds of $S^{2n+1}$ by gluing together configurations of sufficiently many $U(n+1)$-rotated copies of $S_0$ at isolated points of suitably transverse intersection. The resulting submanifolds are very symmetric; are geometricall…
A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${\mathcal O}_{{\mathbb P}^1}(1)^{\oplus p} \oplus {\mathcal O}_{{\mathbb P}^1}^{\oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C,$ which is the germ of submanifolds ${\mathcal C}^C_x \subset {\mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small…
In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphere $S^5$ with constant Contact angle and with a parallel normal vector field must be constant.
The stability and the index of complete one-sided minimal surfaces of certain three-dimensional Riemannian manifolds with positive scalar curvature are studied.
In this paper, we determine all conformal minimal immersions of 2-spheres in complex Grassmann manifold $G(2,N; \mathbb{C})$ with parallel second fundamental form.
We prove the existence of minimal surfaces in a bounded convex subset of $\mathbb R^3$, $\mathcal M$, intersecting the boundary of $\mathcal M$ with a fixed contact angle. The proof is based on a min-max construction in the spirit of Almgren-Pitts for the capillarity functional.
An infinite sequence of commuting nonpolynomial contact symmetries of the two-dimensional minimal surface equation is constructed. Local and nonlocal conservation laws for $n$-dimensional minimal area surface equation are obtained by using the Noether identity.
For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type ${\sf C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomial of degree $\le d$ in the minors of $(u_{ij})$, with coefficients functions of the contact coordinates $x^i$, $u$, $u_i$ (e.g., Monge-Ampère equations have degree 1). For $\mathfrak{g}$ of type ${\sf A}$ or ${\sf G}$ we show that this gives all invariant second-order PDEs. For …
We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. O…
