Bartnik Mass Research Articles

Approximations of the quasilocal Bartnik mass in general relativity

Approximations of the quasilocal Bartnik mass in general relativity

Staticity of asymptotically hyperbolic minimal mass extensions

In this paper, we define the Bartnik mass of a domain whose boundary is connected and compact, has scalar curvature bounded below −n(n − 1), and whose extensions are asymptotically hyperbolic manifolds. With this definition, we show that asymptotically hyperbolic admissible extensions of a domain that achieve the Bartnik mass must admit a static potential. Given a non-static admissible extension of a domain, we are able to construct a one-parameter family of metrics that are close to the original metric, have smaller mass, share the same bound on the scalar curvature, and contain the domain isometrically.

Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass

In [52], Mantoulidis and Schoen constructed 3-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be “far away” from being round. The resulting manifolds, called extensions, are geometrically not “close” to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality as discussed in [53,19]. Their construction was later adapted to n+1 dimensions by Cabrera Pacheco and Miao [22]. In recent papers by Alaee, Cabrera Pacheco, and Cederbaum [1] and by Cabrera Pacheco, Cederbaum, and McCormick [21], a similar construction was performed for 3-dimensional asymptotically Euclidean, electrically charged Riemannian manifolds and for asymptotically hyperbolic Riemannian manifolds, respectively.This paper combines and generalizes all the aforementioned results by constructing suitable asymptotically hyperbolic or asymptotically Euclidean extensions with electric charge in n+1 dimensions for n≥2. We study in detail the sub-extremality of these manifolds, consider the so far unstudied case of extremality in extensions with electric charge and allow more general conditions for the metric of our extensions.Besides suggesting instability of a naturally conjecturally generalized Riemannian Penrose Inequality, the constructed extensions give insights into an ad hoc generalized notion of Bartnik mass, similar to the Bartnik mass estimate for minimal surfaces proven by Mantoulidis and Schoen via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.

Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity

We prove rigidity results involving the Hawking mass for surfaces immersed in a 3-dimensional, complete Riemannian manifold ( M , g ) $(M,g)$ with non-negative scalar curvature (respectively, with scalar curvature bounded below by − 6 $-6$ ). Roughly, the main result states that if an open subset Ω ⊂ M $\Omega \subset M$ satisfies that every point has a neighbourhood U ⊂ Ω $U\subset \Omega$ such that the supremum of the Hawking mass of surfaces contained in U $U$ is non-positive, then Ω $\Omega$ is locally isometric to Euclidean R 3 $\mathbb {R}^3$ (respectively, locally isometric to the Hyperbolic 3-space H 3 ${\mathbb {H}}^3$ ). Under mild asymptotic conditions on the manifold ( M , g ) $(M,g)$ (which encompass as special cases the standard ‘asymptotically flat’ or, respectively, ‘asymptotically hyperbolic’ assumptions) the previous quasi-local rigidity statement implies a global rigidity: if every point in M $M$ has a neighbourhood U $U$ such that the supremum of the Hawking mass of surfaces contained in U $U$ is non-positive, then ( M , g ) $(M,g)$ is globally isometric to Euclidean R 3 $\mathbb {R}^3$ (respectively, globally isometric to the Hyperbolic 3-space H 3 ${\mathbb {H}}^3$ ). Also, if the space is not flat (respectively, not of constant sectional curvature − 1 $-1$ ), the methods give a small yet explicit and strictly positive lower bound on the Hawking mass of suitable spherical surfaces. We infer a small yet explicit and strictly positive lower bound on the Bartnik mass of open sets (non-locally isometric to Euclidean R 3 $\mathbb {R}^{3}$ ) in terms of curvature tensors. Inspired by these results, in the appendix we propose a notion of ‘sup-Hawking mass’ which satisfies some natural properties of a quasi-local mass.

Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions

In the context of the Bartnik mass, there are two fundamentally different notions of an extension of some compact Riemannian manifold $(\Omega,\gamma)$ with boundary. In one case, the extension is taken to be a manifold without boundary in which $(\Omega,\gamma)$ embeds isometrically, and in the other case the extension is taken to be a manifold with boundary where the boundary data is determined by $\partial \Omega$. We give a type of convexity condition under which we can say both of these types of extensions indeed yield the same value for the Bartnik mass. Under the same hypotheses we prove that the Bartnik mass varies continuously with respect to the boundary data. This also provides a method to use estimates for the Bartnik mass of constant mean curvature (CMC) Bartnik data, to obtain estimates for the Bartnik mass of non-CMC Bartnik data. The key idea for these results is a method for gluing Bartnik extensions of given Bartnik data to other nearby Bartnik data.

4-Dimensional manifolds with nonnegative scalar curvature and CMC boundary

In this paper, we will consider 4-dimensional manifolds with nonnegative scalar curvature and constant mean curvature (CMC) boundary. For compact manifolds with boundary, the influence of the nonnegativity of the region scalar curvature to the geometry of the boundary is considered. Some inequalities are established for manifolds with inner boundary and outer boundary. Even for compact manifolds without inner boundary, we can obtain some inequalities involving the geometric quantities of the boundary and give some obstruction. We also discuss the 4-dimensional asymptotically flat extension of the 3-dimensional Bartnik data with CMC boundary and provide the upper bound of the Bartnik mass.

On Hawking mass and Bartnik mass of CMC surfaces

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 Bartnik mass of such surfaces, without assumptions on the integral of the squared mean curvature. If the ambient manifold has negative scalar curvature, our method also applies and yields estimates on the hyperbolic Bartnik mass of these surfaces.

Bartnik mass via vacuum extensions

We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284].

Asymptotically flat extensions with charge

The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, a la Mantoulidis and Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.

Smoothing the Bartnik boundary conditions and other results on Bartnik’s quasi-local mass

Quite a number of distinct versions of Bartnik’s definition of quasi-local mass appear in the literature, and it is not a priori clear that any of them produce the same value in general. In this paper we make progress on reconciling these definitions. The source of discrepancies is two-fold: the choice of boundary conditions (of which there are three variants) and the non-degeneracy or “no-horizon” condition (at least six variants). To address the boundary conditions, we show that given a 3-dimensional region Ω of nonnegative scalar curvature (R≥0) extended in a Lipschitz fashion across ∂Ω to an asymptotically flat 3-manifold with R≥0 (also holding distributionally along ∂Ω), there exists a smoothing, arbitrarily small in C0 norm, such that R≥0 and the geometry of Ω are preserved, and the ADM mass changes only by a small amount. With this we are able to show that the three boundary conditions yield equivalent Bartnik masses for two reasonable non-degeneracy conditions. We also discuss subtleties pertaining to the various non-degeneracy conditions and produce a nontrivial inequality between a no-horizon version of the Bartnik mass and Bray’s replacement of this with the outward-minimizing condition.

Asymptotically hyperbolic extensions and an analogue of the Bartnik mass

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant. Following the ideas of Mantoulidis and Schoen (2016), of Miao and Xie (2018), and of joint work of Miao and the authors (Cabrera Pacheco et al., 2017), we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

Asymptotically flat extensions of CMC Bartnik data

Let g be a metric on the 2-sphere 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 curvature, with outer-minimizing boundary isometric to and having mean curvature H, such that near infinity M is isometric to a spatial Schwarzschild manifold whose mass m can be made arbitrarily close to a constant multiple of the Hawking mass of . Moreover, this constant multiplicative factor depends only on (g, H) and tends to 1 as H tends to 0. The result provides a new upper bound of the Bartnik mass associated with such boundary data.

Bartnik’s Mass and Hamilton’s Modified Ricci Flow

We provide estimates on the Bartnik mass of constant mean curvature 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 notion we call the asphericity mass. The asphericity mass is defined by applying Hamilton’s modified Ricci flow and depends only upon the restricted metric of the surface and not on its mean curvature. The theorem is proven by studying a class of asymptotically flat Riemannian manifolds foliated by surfaces satisfying Hamilton’s modified Ricci flow with prescribed scalar curvature. Such manifolds were first constructed by the first author in her dissertation conducted under the supervision of M. T. Wang. We make a further study of this class of manifolds which we denote Ham3, bounding the ADM masses of such manifolds and analyzing the rigid case when the Hawking mass of the inner surface of the manifold agrees with its ADM mass.

A short note on the Bartnik mass

A short note on the Bartnik mass

On the evolution of the spacetime Bartnik mass

It is conjectured that the full (spacetime) Bartnik mass of a surface Sigma is realised as the ADM mass of some stationary asymptotically flat manifold with boundary data prescribed by Sigma. Assum ...