Related papers: Stable 3-spheres in $\mathbb{C}^3$

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…

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we…

Let $M$ be a compact constant mean curvature surface either in $\mathbb{S}^3$ or $\mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a…

In this paper, we prove a general halfspace theorem for constant mean curvature surfaces. Under certain hypotheses, we prove that, in an ambient space M^3, any constant mean curvature H_0 surface on one side of a constant mean curvature H_0…

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere…

We report on some recent work on deformation of spaces, notably deformation of spheres, describing two classes of examples. The first class of examples consists of noncommutative manifolds associated with the so called $\theta$-deformations…

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature…

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the…

We study Lagrangian submanifolds of the nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$ with respect to their, so called, angle functions. We show that if all angle functions are constant, then the submanifold is either totally geodesic or…

We prove a CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three dimensional manifold with non-negative CR-Panietz operator which satisfies a…

Consider a surface, enclosing a fixed volume, described by a free-energy depending only on the local geometry; for example, the Canham-Helfrich energy quadratic in the mean curvature describes a fluid membrane. The stress at any point on…

We provide a proof of effective uniformization for nearly round 2-spheres, utilizing an identity related to the third-order differential of the conformal factor. This identity is connected to the geometry of the embedded spacelike surface…

We examine the space of surfaces in $\RR^{3}$ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space $\Mk$ of…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature…

Let $f\colon X\to\mathrm{Spec}\, R$ be a 3-fold flopping contraction, where $X$ has at worst Gorenstein terminal singularities and $R$ is complete local. We describe the space of Bridgeland stability conditions on the null subcategory…

We make observations about constant mean curvature surfaces in Euclidean 3-space and their dual surfaces, and the resulting pairs of surfaces in hyperbolic 3-space under the Lawson correspondence.

We characterize normal $3$-pseudomanifolds with $g_2\leq4$. We know that if a $3$-pseudomanifold with $g_2\leq4$ does not have any singular vertices then it is a $3$-sphere. We first prove that a normal $3$-pseudomanifold with $g_2\leq4$…

We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient…

We prove an estimate for spherical functions $\varphi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian…