Initial Hypersurface Research Articles

Asymptotic Convergence for a Class of Fully Nonlinear Curvature Flows

In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space $${\mathbb {R}}^{n+1}$$ with speed $$r^{\alpha } \sigma _k$$, where $$\sigma _k$$ is the k-th elementary symmetric polynomial of the principal curvatures, $$\alpha \in {\mathbb {R}}^1$$, and r is the distance from the hypersurface to the origin. If $$\alpha \ge k+1$$, we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If $$\alpha <k+1$$, a counterexample is given for the above convergence. In the case $$k=1$$ and $$\alpha \ge 2$$, we also prove that the flow converges to a round point if the initial hypersurface is weakly mean-convex and star-shaped.

Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

AbstractWe prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form , where and F is a positive, strictly monotone and 1‐homogeneous curvature function. In particular this class includes the mean curvature . We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews–McCoy–Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.

Contracting Convex Hypersurfaces in Space Form by Non-homogeneous Curvature Function

This paper concerns the evolution of closed convex hypersurfaces in space form $${\mathbb {N}}^{n+1}_\kappa $$ of constant sectional curvature $$\kappa =0, \pm 1$$ by curvature flow, for which the normal speed is given by a general non-homogeneous function $$\varPhi $$ of curvature function F which is monotone, symmetric, homogeneous of degree one. Under the assumption that F is inverse concave and its dual function $$F_*$$ approaches zero on the boundary of positive cone, we prove that the flow converges to a point in finite time. For special flow with $$\varPhi =F^\beta ~(\beta >1)$$, without the assumption of inverse concavity on F, we deduce that if the initial hypersurface is pinched enough, then the flow converges smoothly and exponentially to the unit sphere of $${\mathbb {R}}^{n+1}$$ after some suitable rescaling.

Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case

We make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. We consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3D Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions.

Eigenvalues of a (p, q)-Laplacian System Under the Mean Curvature Flow

In this paper, we derive the evolution equation for the first eigenvalue of a class of (p, q)-Laplacian system. Moreover, by imposing some conditions on the mean curvature of the initial hypersurface, we can show that the first eigenvalue of a class of (p, q)-Laplacian system is monotonic under the volume-preserving mean curvature flow and unnormalized mean curvature flow.

Monotonicity of the first eigenvalue of the Laplace and the $p$-Laplace operators under a forced mean curvature flow

In this paper, we would like to give an answer to \textbf{Problem 1} below issued firstly in [J. Mao, Eigenvalue estimation and some results on finite topological type, Ph.D. thesis, IST-UTL, 2013]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the $p$-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicated constraint, we can give lower bounds for the first nonzero closed eigenvalue of the Laplacian provided additionally the second fundamental form of the initial hypersurface satisfies a pinching condition.

The Maximal Development of Near-FLRW Data for the Einstein-Scalar Field System with Spatial Topology $${\mathbb{S}^3}$$

The Friedmann--Lema\^{\i}tre--Robertson--Walker (FLRW) solution to the Einstein-scalar field system with spatial topology $\mathbb{S}^3$ models a universe that emanates from a singular spacelike hypersurface (the Big Bang), along which various spacetime curvature invariants blow up, only to re-collapse in a symmetric fashion in the future (the Big Crunch). In this article, we give a complete description of the maximal developments of perturbations of the FLRW data at the chronological midpoint of its evolution. We show that the perturbed solutions also exhibit curvature blowup along a pair of spacelike hypersurfaces, signifying the stability of the Big Bang and the Big Crunch. Moreover, we provide a sharp description of the asymptotic behavior of the solution up to the singularities, showing in particular that various time-rescaled solution variables converge to regular tensorfields on the singular hypersurfaces that are close to the corresponding FLRW tensorfields. Our proof crucially relies on $L^2$-type approximate monotonicity identities in the spirit of the ones we used in our joint works with Rodnianski, in which we proved similar results for nearly spatially flat solutions with spatial topology $\mathbb{T}^3$. In the present article, we rely on new ingredients to handle nearly round spatial metrics on $\mathbb{S}^3$, whose curvatures are order-unity near the initial data hypersurface. In particular, our proof relies on i) the construction of a globally defined spatial vectorfield frame adapted to the symmetries of a round metric on $\mathbb{S}^3$; ii) estimates for the Lie derivatives of various geometric quantities with respect to the elements of the frame; and iii) sharp estimates for the asymptotic behavior of the FLRW solution's scale factor near the singular hypersurfaces.

Quermassintegral preserving curvature flow in Hyperbolic space

We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

Slow-roll corrections in multi-field inflation: a separate universes approach

In view of cosmological parameters being measured to ever higher precision, theoretical predictions must also be computed to an equally high level of precision. In this work we investigate the impact on such predictions of relaxing some of the simplifying assumptions often used in these computations. In particular, we investigate the importance of slow-roll corrections in the computation of multi-field inflation observables, such as the amplitude of the scalar spectrum Pζ, its spectral tilt ns, the tensor-to-scalar ratio r and the non-Gaussianity parameter fNL. To this end we use the separate universes approach and δ N formalism, which allows us to consider slow-roll corrections to the non-Gaussianity of the primordial curvature perturbation as well as corrections to its two-point statistics. In the context of the δ N expansion, we divide slow-roll corrections into two categories: those associated with calculating the correlation functions of the field perturbations on the initial flat hypersurface and those associated with determining the derivatives of the e-folding number with respect to the field values on the initial flat hypersurface. Using the results of Nakamura & Stewart '96, corrections of the first kind can be written in a compact form. Corrections of the second kind arise from using different levels of slow-roll approximation in solving for the super-horizon evolution, which in turn corresponds to using different levels of slow-roll approximation in the background equations of motion. We consider four different levels of approximation and apply the results to a few example models. The various approximations are also compared to exact numerical solutions.

On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds

The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown that, if geometrically well-motivated restrictions are placed on the supports of the (smooth) initial datum and of the (smooth) inhomogeneous term, then there exists a continuous global solution which is smooth "on each side" of the initial value hypersurface. A uniqueness result in Sobolev regularity $H^{1/2+\varepsilon}_\mathrm{loc}$ is proved among solutions supported in the union of the causal past and future of the initial value hypersurface, and whose product with the indicator function of the causal future (resp. past) of the hypersurface is past compact (resp. future compact). An explicit representation formula for solutions is obtained, which prominently features an invariantly defined, densitised version of the null expansion of the hypersurface. Finally, applications to quantum field theory on curved spacetimes are briefly discussed.

On the Occurrence of Mass Inflation for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant and an Exponential Price Law

In this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along the event horizon. More precisely, we construct open sets of characteristic data which, on the outgoing initial null hypersurface (taken to be the event horizon), converges exponentially to a reference Reissner-N\"{o}rdstrom black hole at infinity. We prove the stability of the radius function at the Cauchy horizon, and show that, depending on the decay rate of the initial data, mass inflation may or may not occur. In the latter case, we find that the solution can be extended across the Cauchy horizon with continuous metric and Christoffel symbols in $L^2_{{\rm loc}}$, thus violating the Christodoulou-Chru\'sciel version of strong cosmic censorship.

Covariant Hamiltonian tetrad approach to numerical relativity

A Hamiltonian approach to the equations of general relativity is proposed using the powerful mathematical language of multivector-valued differential forms. In the approach, the gravitational coordinates are the 12 spatial components of the line interval (the vierbein) including their antisymmetric parts, and their 12 conjugate momenta. A feature of the proposed formalism is that it allows Lorentz gauge freedoms to be imposed on the Lorentz connections rather than on the vierbein, which may facilitate numerical integration in some challenging problems. The 40 Hamilton's equations comprise 12 + 12 = 24 equations of motion, 10 constraint equations (first class constraints, which must be arranged on the initial hypersurface of constant time, but which are guaranteed thereafter by conservation laws), and 6 identities (second class constraints). The 6 identities define a trace-free spatial tensor that is the gravitational analog of the magnetic field of electromagnetism. If the gravitational magnetic field is promoted to an independent field satisfying its own equation of motion, then the system becomes the WEBB system, which is known to be strongly hyperbolic. Some other approaches, including ADM, BSSN, WEBB, and Loop Quantum Gravity, are translated into the language of multivector-valued forms, bringing out their underlying mathematical structure.

Flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space

This paper concerns closed, h-convex hypersurfaces of dimension n≥2 in the hyperbolic space Hκn+1 of constant sectional curvature κ evolving in direction of its normal vector, where the speed equals a power β>1 of a curvature function F, which is monotone, symmetric, homogeneous of degree 1. It is shown that if the initial h-convex hypersurface is pinched, then this is maintained under the flow, and the hypersurfaces shrink to a round point in Hκn+1 in finite time. As a consequence, when rescaling appropriately, the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of Rn+1.

Sachs\u2019 free data in real connection variables

We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs’ constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the first-order formulation is that Sachs’ propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null infinity in terms of a shear of a (non-geodesic) asymptotic null vector field in the physical spacetime. Finally, we compute the modification to the spin coefficients and the null congruence in the presence of torsion.

Robustness of inflation to inhomogeneous initial conditions

We consider the effects of inhomogeneous initial conditions in both the scalar field profile and the extrinsic curvature on different inflationary models. In particular, we compare the robustness of small field inflation to that of large field inflation, using numerical simulations with Einstein gravity in 3+1 dimensions. We find that small field inflation can fail in the presence of subdominant gradient energies, suggesting that it is much less robust to inhomogeneities than large field inflation, which withstands dominant gradient energies. However, we also show that small field inflation can be successful even if some regions of spacetime start out in the region of the potential that does not support inflation. In the large field case, we confirm previous results that inflation is robust if the inflaton occupies the inflationary part of the potential. Furthermore, we show that increasing initial scalar gradients will not form sufficiently massive inflation-ending black holes if the initial hypersurface is approximately flat. Finally, we consider the large field case with a varying extrinsic curvature K, such that some regions are initially collapsing. We find that this may again lead to local black holes, but overall the spacetime remains inflationary if the spacetime is open, which confirms previous theoretical studies.

A Geometric Invariant Characterising Initial Data for the Kerr\u2013Newman Spacetime

We describe the construction of a geometric invariant characterising initial data for the Kerr–Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr–Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr–Newman spacetime in terms of Killing spinors. The space-spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, we study the properties of solutions to the approximate Killing spinor equation and use them to construct the geometric invariant.

Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure.

We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.

Fluids and vortex from constrained fluctuations around C-metric black holes

By foliating the four-dimensional C-metric black hole spacetime, we consider a kind of initial-value-like formulation of the vacuum Einstein's equation, the holographic initial data is a double consisting of the induced metric and the Brown–York energy momentum tensor on an arbitrary initial hypersurface. Then by perturbing the initial data that generates the background spacetime, it is shown that, in an appropriate limit, the fluctuation modes are governed by the continuity equation and the compressible Navier–Stokes equation which describe the momentum transport in non-relativistic viscous fluid on a flat Newtonian space. It turns out that the flat space fluid behaves as a pure vortex and the viscosity to entropy ratio is subjected to the black hole acceleration.

On inverse mean curvature flow in Schwarzschild space and Kottler space

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $\Sigma$ is strictly mean convex and star-shaped, then the flow hypersurface $\Sigma_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken-Ilmanen's result [18].

Non-parametric Inverse Curvature Flows in the AdS-Schwarzschild Manifold

We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.