Related papers: Branch points of area-minimizing projective planes

Surfaces admitting flows all whose orbits are dense are called minimal. Minimal orientable surfaces were characterized by J.C. Beni\`{e}re in 1998, leaving open the nonorientable case. This paper fills this gap providing a characterization…

We study the metric of minimal area on a punctured Riemann surface under the condition that all nontrivial homotopy closed curves be longer than or equal to $2\pi$. By constructing deformations of admissible metrics we establish necessary…

We define a laminar branched surface to be a branched surface satisfying the following conditions: (1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after…

In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a…

In (the surface of) a convex polytope P^3 in R^4, an area-minimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an area-minimizing G-invariant…

We classify branched immersed disks in space forms with non-zero parallel mean curvature vector and non-orthogonal constant contact angle along the boundary in 4-dimensional space form. For higher codimensional case, we prove a codimension…

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at…

This article proves that if M is a smooth manifold of dimension at least four, then for generic choice of metric on M, all prime parametrized minimal surfaces in M are free of branch points and lie on nondegenerate critical submanifolds for…

In this paper, we prove that any surface corresponding to linear second-order ODEs as a submanifold is minimal in all classes of third-order ODEs $y'''=f(x, y, p, q)$ as a Riemannian manifold where $y'=p$ and $y''=q$, if and only if…

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed in S^n x R or H^n x R in terms of its first and second fundamental forms and of the projection of the vertical vector field…

In the recent paper \cite{DGNP} we have proved that the only stable $C^2$ minimal surfaces in the first Heisenberg group $\Hn$ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result…

We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two…

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in $H^2\times R$. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary $C$ is a Jordan curve homologous to…

Near the end of his life, Bernhard Riemann made the marvelous discovery of a 1-parameter family $R_{\lambda}$, $\lambda\in (0,\infty)$, of periodic properly embedded minimal surfaces in $\mathbb{R}^3$ with the property that every horizontal…

We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with…

Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more…

Let $(M,g)$ be a closed oriented Riemannian $3$-manifold and suppose that there is a strongly irreducible Heegaard splitting $H$. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the stable…

The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan, which treats a spin structure on a Riemann surface M as a complex line bundle S whose square is the…

We consider Lie minimal surfaces, the critical points of the simplest Lie sphere invariant energy, in Riemannian space forms. These surfaces can be characterized via their Euler-Lagrange equations, which take the form of differential…

We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The…