Related papers: Optimal length estimates for stable CMC surfaces i…

We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature…

For appropriately values of $H$, we obtain an area estimate for a complete non-compact $H$-surface of finite topology and finite area, embedded in a three-manifold of negative curvature. Moreover, in the case of equality and under…

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…

We prove a linear upper bound on the Morse index of closed constant mean curvature (CMC) surfaces in orientable three-manifolds in terms of genus, number of branch points and a Willmore-type energy.

In this paper we obtain density estimates for compact surfaces immersed in R^n with total boundary curvature less than 4pi and with sufficiently small L^p norm of the mean curvature, p>2. Our results generalize the main results in [2]. We…

We characterize all compact embedded stable minimal capillary surfaces with capillary angle close to either $0$ or $\pi$ that are supported on a complete embedded minimal surface with finite total curvature that is not an affine plane.…

We construct a new class of complete constant mean curvature surfaces in R^3. These are geometrically different than the surfaces constructed by Kapouleas' gluing technique. These are obtained by piecing together half-Delaunay surfaces to…

In this paper, we investigate the geometric properties associated with the $\mathfrak{g}$-stability of surfaces with boundary whose null expansion satisfies $\Theta^{+} = h \geq 0$. First, we show that a $\mathfrak{g}$-stable hypersurface…

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…

In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $\nu = \meta{N}{\partial_t}$ does not…

In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean…

We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces…

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface…

The speed of a ball rolling without skidding or spinning on a surface $S$ is the length of the velocity of its center. We show that if the speed depends only on $p\in S$, then $S$ has constant mean curvature; and, conversely, that if the…

With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is…

In homogenous space Sol we study compact surfaces with constant mean curvature and with non-empty boundary. We ask how the geometry of the boundary curve imposes restrictions over all possible configurations that the surface can adopt. We…

A $k$-harmonic map is a critical point of the $k$-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^{n} (n\ge 3)$ is a CMC proper triharmonic hypersurface with at most three distinct…

In this paper we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study…

In this paper we use stable capillary surfaces (analogous to the $\mu$-bubble construction) to study manifolds with strictly mean convex boundary and nonnegative scalar curvature. We give an obstruction to filling 2-manifolds by such…

We study the principal curvatures of properly embedded constant mean curvature hypersurfaces in the Anti-de Sitter space $\mathbb{H}^{n,1}$. We generalize the notion of convex hull and give an upper bound on the principal curvatures which…