Related papers: Second order Contact of Minimal Surfaces

We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat…

Let $\mathbb{Q}_3$ be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in $\mathbb{Q}_3$. By an isotropic curve we mean a nonconstant holomorphic map…

Consider a domain D in R^3 which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f : M --> D. Moreover, if D is smooth and bounded, then…

We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L^2 complex relative to a suitable metric on the bundle and a complete metric on the…

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third order differential equation determined by the conformal…

We provide two examples of smooth projective surfaces of tame CM type, by showing that any parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in projective…

For compact regions Omega in R^3 with generic smooth boundary B, we consider geometric properties of Omega which lie midway between their topology and geometry and can be summarized by the term "geometric complexity". The "geometric…

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first…

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M…

The explicit form of the interface equation of motion derived assuming a minimal surface is extended to general bicontinuous interfaces that appear in the diffusion limited stage of the phase separation process of binary mixtures. The…

The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this…

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.

If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal…

In this paper we classify compact minimal surfaces in $S^5$ with non-negative Gaussian curvature using the notion of a contact angle.

We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some…

We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the…

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as…

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…

Consider a strictly convex bounded regular domain $C$ of $\R^3$. For any arbitrary finite topological type we find a compact Riemann surface $\mathcal{M}$, an open domain $M\subset \mathcal{M}$ with the fixed topological type, and a…

In this paper we develop the theory of properly immersed minimal surfaces in the quotient space $\mathbb H^2\times\mathbb R/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb…