Related papers: Complete biconservative surfaces in the hyperbolic…
Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing…
We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp…
We prove that, given $|H|<1$, a generic simple closed curve embedded in the asymptotic boundary of $\mathbb{H}^3$ (with respect to the supremum metric) bounds more than one complete surface embedded in $\mathbb{H}^3$ which has constant mean…
Hypersurfaces are studied and classified under multiple additional assumptions in any Riemannian homogeneous space $(\mathbb{C}P^3, g_a)$, including nearly K\"ahler $\mathbb{C}P^3$. Notably, all extrinsically homogeneous hypersurfaces are…
This paper proves that every finite volume hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover…
We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves,…
We consider surfaces of constant Gaussian curvature immersed in 3-dimensional manifolds, and we strengthen the compactness result of Labourie in the case where the ambient manifold is 3-dimensional hyperbolic space. This allows us to prove…
We study the geometry of complete immersed surfaces in $\mathbb{R}^3$ with constant anisotropic mean curvature (CAMC). Assuming that the anisotropic functional is uniformly elliptic, we prove that: (1) planes and CAMC cylinders are the only…
We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves $C_1\times C_2$ by the diagonal action of either the group $\Z/p\Z$ or the group $\Z/2p\Z$. These K3 surfaces admit a non-symplectic…
Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic…
In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to…
On a finite-volume hyperbolic $3$-manifold, we establish an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and the genus bounds. This area bound implies…
This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity…
In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates…
We classify the hypersurfaces of $\mathbb{Q}^3\times\mathbb{R}$ with three distinct constant principal curvatures, where $\varepsilon \in \{1,-1\}$ and $\mathbb{Q}^3$ denotes the unit sphere $\mathbb{S}^3$ if $\varepsilon = 1$, whereas it…
We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive…
We supply a proof of the fact that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics is topologically tame. This proves the Marden's conjecture. Our approach is to form an exhaustion $M_i$ of $M$…
In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher,…
We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and…
Examples of complete minimal surfaces properly embedded in H^2 x R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces…