Related papers: Rigidity of Hawking mass for surfaces in three man…

In this paper, we present two rigidity results for stable constant mean curvature (CMC) surfaces immersed in $3$-manifolds with positive scalar curvature, assuming that the Hawking mass is zero. In the first result, we assume the surface to…

In this paper, we establish symmetry results for solutions of the mean field equation \[ \frac{\alpha}{2} \Delta u + e^u - 1 = 0 \] on \( \mathbb{S}^2 \) for $\frac{1}{3}\leq \alpha < \frac{1}{2}$.The proofs utilize the Sphere Covering…

Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the…

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main…

In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical…

The study of stable minimal surfaces in Riemannian $3$-manifolds $(M, g)$ with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when $(M, g)$ is asymptotically flat and has horizon…

In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -\Delta _{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in…

In this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3-manifolds. We require the metric have the form g_{ij}=\delta_{ij}+h_{ij} with h_{ij}=O_{4}(r^{-1}) and R=O(r^{-3-\tau}),\tau>0.…

Let $g$ be a smooth Riemannian metric on $\mathbb{S}^2$ and $H>0$ a constant. We establish an upper bound for the corresponding Bartnik mass $\mathfrak m_B(\mathbb{S}^2, g, H)$ assuming that the Gauss curvature $K_g$ is non-negative. Our…

Let $g$ be a metric on the $2$-sphere $\mathbb{S}^2$ with positive Gaussian curvature and $H$ be a positive constant. Under suitable conditions on $(g, H)$, we construct smooth, asymptotically flat $3$-manifolds $M$ with non-negative scalar…

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 paper, we prove that the even solution of the mean field equation $\Delta u=\lambda(1-e^u) $ on $S^2$ must be axially symmetric when $4<\lambda \leq 8$. In particular, zero is the only even solution for $\lambda=6$. This implies the…

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…

A surface of constant mean curvature (CMC) equal to $H$ in a sub-Riemannian $3$-manifold is strongly stable if it minimizes the functional $\text{area}+2H\,\text{volume}$ up to second order. In this paper we obtain some criteria ensuring…

We study rigidity of minimal two-spheres $\Sigma$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of $\Sigma$, we prove that a…

Let (M,g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K of M, all volume preserving stable constant mean curvature surfaces of sufficiently large…

In this paper, we prove a rigidity result for three-dimensional Riemannian manifolds with boundary, under the assumption that a free boundary minimal two-disk, which locally maximizes a modified Hawking mass, is embedded in a…

We provide estimates on the Bartnik mass of constant mean curvature (CMC) surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new…

In this paper, we introduce a non linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with…

We prove sharp lower bounds for the charged Hawking mass of stable surfaces in electrostatic space-times in various contexts. An upper bound for the genus of stable surfaces in the electrostatic system is provided. We also study the…