Abstract: 

Abstract: We provide a harmonic level set proof (along the lines of the argument in [6]) of the positive mass theorem for asymptotically flat 3-manifolds with a non-compact boundary first established by Almaraz-Barbosa-de Lima in [2].

Abstract: The purpose of this article is to study rigidity of free boundary minimal two-disks that locally maximize the modified Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature and mean convex boundary. Assuming the strict stability of Σ, we prove that a neighborhood of it in M is isometric to one of the half de Sitter–Schwarzschild space.  

Abstract: The purpose of this article is to prove that, under suitable constraints on time-symmetric initial data set for the Einstein–Maxwell equation M, if Σ⊂M is an embedded strictly stable minimal two-sphere which locally maximizes the charged Hawking mass, then there exists a neighborhood of it in M isometric to the Reissner–Nordström–de Sitter space. At the same time, motivated (Bray et al. in Commun Anal Geom 18(4):821–830, 2010), we will deduce an estimate for the area of a two-sphere which is locally area minimizing on time-symmetric initial data set for the Einstein–Maxwell equation. Moreover, if the equality holds, then there exists a neighborhood of it in M isometric to the charged Nariai space. 

Abstract: We explore the fourth-order Steklov problem of a compact embedded hypersurface Σn with free boundary in a (n+1)-dimensional compact manifold  Mn+1 which has nonnegative Ricci curvature and strictly convex boundary. If  Σ is minimal we establish a lower bound for the first eigenvalue of this problem. When  M=Bn+1 is the unit ball in ℝn+1, if Σ has constant mean curvature HΣ we prove that the first eigenvalue satisfies σ1≤n+∣HΣ∣. In the minimal case (HΣ=0), we prove that σΣ=n. 

Abstract: In this paper, we obtain one sharp estimate for the length L(∂Σ) of the boundary ∂Σ of a capillary minimal surface Σ in M, where M is a compact three-manifolds with strictly convex boundary, assuming Σ has index one. The estimate is in term of the genus of Σ, the number of connected components of ∂Σ and the constant contact angle θ. Making an extra assumption on the geometry of M along ∂M, we characterize the global geometry of M, which is saturated only by the Euclidean three-balls. For capillary stable CMC surfaces, we also obtain similar results. 

Abstract: The main aim of this article is to give the complete classification of critical met-rics of the volume functional on a compact manifold 𝑀 with boundary 𝜕𝑀 andunder the harmonic Weyl tensor condition. In particular, we prove that a criticalmetric with a harmonic Weyl tensor on a simply connected compact manifoldwith the boundary isometric to a standard sphere 𝕊n-1 must be isometric to ageodesic ball in a simply connected space form ℝn, ℍn, and 𝕊n. To this end,we first conclude the classification of such critical metrics under the Bach-flatassumption and then we prove that both geometric conditions are equivalent inthis situation.

Abstract: In this article, we investigate the geometry of critical metrics of the volume functional on an n-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifold. In addition, localized version estimates to the mean curvature and area of the boundary of critical metrics are also obtained. 

Abstract: The purpose of this article is to investigate the structure of complete non-compact quasi-Einstein manifolds. We show that complete noncompact quasi-Einstein manifolds with λ = 0 are connected at infinity. In addition, we provide some conditions under which quasi-Einstein manifolds with λ < 0 are f-non-parabolic. In particular, we obtain estimates on volume growth of geodesic balls for such manifolds. 

Abstract: The aim of this paper is to study compact Riemannian manifolds  (M, g) that admit a non-constant solution to the system of equations 

− f g + Hess f − f Ric = μRic + λg,

 where Ric is the Ricci tensor of g whereas μ and λ are two real parameters. More precisely, under assumption that (M, g) has zero radial Weyl curvature, this means that the interior product of ∇ f with the Weyl tensor W is zero, we shall provide the complete classification for the following structures: positive static triples, critical metrics of volume functional and critical metrics of the total scalar curvature functional.

Abstract: The aim of this note is to characterize Clifford tori as the only Killing invariant surfaces, under an additional hypothesis. Moreover, we build a family of Killing invariant minimal surfaces in 𝕊3, that does not contain Clifford tori as well as we present examples of Killing invariant surfaces whose mean curvature is not identically zero. 

Abstract:  The aim of this paper is to generalize some recent local rigidity results for three-dimensional Riemannian manifolds (M3, g) with a bound on the scalar curvature. More precisely, we study rigidity of strictly stable minimal surfaces Σ ⊂ M which locally maximize the Hawking mass on a Riemannian three-manifoldM whose scalar curvature is bounded from below by a negative constant. Moreover, we conclude that the metric of M near Σ must split as ga = dr2 +ua(r)2gΣ which is one the Kottler-Schwarzschild metric, where gΣ is a metric of constant gaussian curvature.

Abstract: We study the space of smooth Riemannian structures on compact threemanifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao–Tam critical metrics. We provide an estimate to the area of the boundary of Miao–Tam critical metrics on compact threemanifolds. In addition, we obtain a Bochner type formula which enables us to show that a Miao–Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in 𝕊3.

Abstract: We consider a properly embedded minimal hypersurfacewith free boundary in a compact n-dimensional Riemannian manifold M be with nonnegative Ricci curvature and strictly convex boundary. Here, we obtain a new estimate from below for the first nonzero Steklov eigenvalue.

Abstract: The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ, gΣ) embedded into a complete five dimensional manifold (M5, g) with positive scalar curvature R and nonnegative Ricci curvature. Under a suitable choice, we have . Moreover, occurring equality we deduce that (Σ, gΣ) is isometric to a standard sphere (𝕊4, gcan) and in a neighbourhood of Σ, (M5, g) splits as ((-ϵ, ϵ) × 4, dt2 + gcan) and the Riemannian covering of (M5, g) is isometric to ℝ× 𝕊4. 

Abstract: The purpose of this note is to provide some volume estimates for Einstein warped products similar to a classical result due to Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. To do so, we make use of the approach of quasi-Einstein manifolds which is directly related to Einstein warped products. In particular, we present an obstruction for the existence of such a class of manifolds.

Abstract: The aim of this note is to prove that any compact non-trivial almost Ricci soliton (Mn, g, X, λ) with constant scalar curvature is isometric to a Euclidean sphere 𝕊n. As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field X decomposes as the sum of a Killing vector field Y and the gradient of a suitable function.