Flow Of Metrics Research Articles

Bochner–Kodaira Formulas and the Type IIA Flow

A new derivation of the flow of metrics in the Type IIA flow is given. It is better adapted to the formulation of the flow as a variant of a Laplacian flow, and it uses the projected Levi–Civita connection of the metrics themselves instead of their conformal rescalings.

New metric structures on [formula omitted]-foliations

In the paper we introduce new metric structures on g-foliations that are less rigid than the well-known structures: almost contact and 3-Sasakian structures as well as f-structures with parallelizable kernel and almost para-ϕ-structures with complemented frames. We discuss the properties of the new structures in order to demonstrate similarities with the corresponding classical structures. Then using the flow of metrics on a g-foliation, we build deformation retraction of our structures with positive partial Ricci curvature onto the subspace of the aforementioned classical structures.

Outer entropy equals Bartnik-Bray inner mass and the gravitational ant conjecture

Entropy and energy are found to be closely tied on our quest for quantum gravity. We point out an interesting connection between the recently proposed outer entropy, a coarse-grained entropy defined for a compact spacetime domain motivated by the holographic duality, and the Bartnik-Bray quasilocal mass long known in the mathematics community. In both scenarios, one seeks an optimal spacetime fill-in of a given closed, connected, spacelike, codimension-two boundary. We show that for an outer-minimizing mean-convex surface, the Bartnik-Bray inner mass matches exactly with the irreducible mass corresponding to the outer entropy. The equivalence implies that the area laws derived from the outer entropy are mathematically equivalent as the monotonicity property of the quasilocal mass. It also gives rise to new bounds between entropy and the gravitational energy, which naturally gives the gravitational counterpart to Wall's ant conjecture. We also observe that the equality can be achieved in a conformal flow of metrics, which is structurally similar to the Ceyhan-Faulkner proof of the ant conjecture. We compute the small sphere limit of the outer entropy and it is proportional to the bulk stress tensor as one would expect for a quasilocal mass. Lastly, we discuss some implications of taking quantum matter into consideration in the semiclassical setting.

Holomorphic differentials, thermostats and Anosov flows

We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian two-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.

The Conformal Flow of Metrics and the General Penrose Inequality

The conformal flow of metrics has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the conformal flow of metrics, so that it may be applied to the Penrose inequality for general initial data sets of the Einstein equations. The Penrose conjecture without the assumption of time symmetry is then reduced to solving a system of PDE with desirable properties.

Proof of the Riemannian Penrose inequality with charge for multiple black holes

We present a proof of the Riemannian Penrose inequality with charge in the context of asymptotically flat initial data sets for the Einstein–Maxwell equations, having possibly multiple black holes with no charged matter outside the horizon, and satisfying the relevant dominant energy condition. The proof is based on a generalization of Hubert Bray’s conformal flow of metrics adapted to this setting.

Proof of the Riemannian Penrose Inequality Using the Positive Mass Theorem

We prove the Riemannian Penrose Conjecture, an important case of a con- jecture (41) made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a space-time in the context of general relativity, then outermost minimal spheres with total area A correspond to apparent horizons of black holes contributing a massA/16π, scalar cur- vature corresponds to local energy density at each point, and the rate at which the metric becomes flat at infinity corresponds to total mass (also called the ADM mass). The Riemannian Penrose Conjecture then states that the total mass of an asymptotically flat 3-manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes. The flow of metrics we define continuously evolves the original 3-metric to a Schwarzschild 3-metric, which represents a spherically symmetric black hole in vacuum. We define the flow such that the area of the minimal spheres (which flow outward) and hence the mass contributed by the black holes in each of the metrics in the flow is constant, and then use the Positive Mass Theorem to show that the total mass of the metrics is nonincreasing. Then since the total mass equals the mass of the black hole in a Schwarzschild metric, the Riemannian Penrose Conjecture follows. We also refer the reader to the beautiful work of Huisken and Ilmanen (30), who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole. In Sections 1 and 2, we motivate the problem, discuss important quantities like total mass and horizons of black holes, and state the Positive Mass Theorem and the Penrose Conjecture for Riemannian 3- manifolds. In Section 3, we give the proof of the Riemannian Penrose