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

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 give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each $H\in\mathbb{R}$, there exists a constant mean curvature $H$-sphere in the…

We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a…

On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $\Omega$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the…

We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl(2, R). In particular, all constant mean curvature spheres in those spaces are described…

In this paper we consider the equiform motion of a sphere in Euclidean space $\mathbf{E}^7$. We study and analyze the corresponding kinematic three dimensional surface under the hypothesis that its scalar curvature $\mathbf{K}$ is constant.…

We establish an explicit expression for the smallest non-zero eigenvalue of the Laplace--Beltrami operator on every homogeneous metric on the 3-sphere, or equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of…

We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} \alpha^{-1} (\lambda_1 +\cdots +\lambda_{\alpha}) > - \theta \bar{\lambda}, \end{equation*}…

We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for…

In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature outside a fixed compact…

In this note we prove that a constant mean curvature surface is proper-biharmonic in the unit Euclidean sphere $\mathbb{S}^4$ if and only if it is minimal in a hypersphere $\mathbb{S}^3(\frac{1}{\sqrt{2}})$.

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some…

We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension $3$. We prove a stability inequality and provide sufficient conditions ensuring instability of…

Let $M$ be an $n(\geq3)$-dimensional oriented compact submanifold with parallel mean curvature in the simply connected space form $F^{n+p}(c)$ with $c+H^2>0$, where $H$ is the mean curvature of $M$. We prove that if the Ricci curvature 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…

For constant mean curvature surfaces of class $C^2$ immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of…

In the present paper we give a geometric proof for the existence of cylinders with constant mean curvature $H>H(X)$ in certain simply connected homogeneous three-manifolds $X$ diffeomorphic to $\mathbb{R}^3$, which always admit a Lie group…

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is…

Llarull's Theorem states that any Riemannian metric on the $n$-sphere which has scalar curv{\-}ature greater than or equal to $n(n-1)$, and whose distance function is bounded below by the unit sphere's, is isometric to the unit sphere.…

Let $(M, g)$ be a complete Riemannian $3$-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We \textsl{unconditionally} characterize the large, embedded stable constant mean curvature…