Related papers: Generic non-uniqueness of complete $H$-surfaces em…

We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in $\R^n$ or non-positively curved n-dimensional…

We show that the mean curvature flow of a generic closed surface in $\mathbb{R}^3$ avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical…

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the…

We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.

The problem of existence of spacelike hypersurfaces with constant mean curvature in asymptotically flat spacetimes is considered for a class of asymptotically Schwarzschild spacetimes satisfying an interior condition. Using a barrier…

We present new examples of complete embedded self-similar surfaces under mean curvature by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of self-shrinkers in $\mathbb R^3$ that are not rotationally…

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of…

In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive…

This paper is a sequel to arXiv:1109.4986, where we proved that a general smooth curve of odd genus, canonically or bicanonically embedded, has semistable finite Hilbert points. Here, we prove that a generic canonically embedded curve of…

We construct, for any $H\in \mathbb{R}$, infinitely many free boundary annuli in geodesic balls of $\mathbb{S}^3$ with constant mean curvature $H$ and a discrete, non-rotational, symmetry group. Some of these free boundary CMC annuli are…

We show that for an immersed two-sided minimal surface in $R^3$, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in $R^3$ of index $2$, as…

We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.

In this paper, we study factorable surfaces in a 3-dimensional isotropic space. We classify such surfaces with constant isotropic Gaussian (K) and mean curvature (H). We provide a non-existence result related with the surfaces satisfying…

This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by…

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In…

In this paper we prove that the unit ball $\mathbb{B}$ of $\mathbb{C}^2$ admits complete properly embedded complex curves of any given topological type. Moreover, we provide examples containing any given closed discrete subset of…

We classify hypersurfaces with rotational symmetry and positive constant $r$-th mean curvature in $\mathbb H^n \times \mathbb R$. Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also…

We consider the smooth inverse mean curvature flow of strictly convex hypersurfaces with boundary embedded in $\mathbb{R}^{n+1},$ which are perpendicular to the unit sphere from the inside. We prove that the flow hypersurfaces converge to…

We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in…

We consider the sub-Riemannian metric $g_{h}$ on $\mathbb{S}^3$ provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the…