Parabolicity of complete spacelike hypersurfaces in certain GRW spacetimes. Applications to uniqueness of complete maximal hypersurfaces.
Juan J. Salamanca
Departamento de Matemáticas Universidad de Córdoba
14071 - Córdoba Email: jjsalamanca@uco.es
7th International Meeting on Lorentzian Geometry Sao Paulo 2013
J.J. Salamanca (U. Córdoba) 1 / 26
Based on:
A. Romero, R.M. Rubio and J.J. Salamanca, Uniqueness of complete maximal hypersurfaces in spatially parabolic Generalized
Robertson-Walker spacetimes,Class. Quantum Grav.,V. 30, N. 11 (2013).
Maximal Hypersurfaces
Physics
They serve as initial values set to build the spacetime solution of the Einstein’s equation of General Relativity.
They have an important role in the analysis of the Cauchy problem.
The existence of such that hypersurfaces implies, in several cases, that the universe change from a expansive phase to a contractive one.
Differential Geometry
They arose as critical points of a variational problem given by the area functional.
Historical importance.
J.J. Salamanca (U. Córdoba) 3 / 26
GRW Spacetimes
Consider(I,−dt2),I ⊆R, joint with a Riemannian manifold(F, gF).
Letf :I →(0,∞)be a smooth function.
We callGeneralized Robertson-Walker (GRW) spacetime1to the product manifold,I×F, endowed with the Lorentzian metric
g=−π∗
I(dt2) +f2(t)πF∗(gF), whereπI andπF denote the projections ontoI andF.
We denote this manifold byM =I ×f F. The manifold(F, gF)is called fiber; the manifold(I,−dt2),base; andf is thewarping function.
Examples of GRW spacetimes
The Lorentz-Minkowski spacetime: f ≡1and fiber the Euclidean space.
Friedmann models (3-dimensional fiber with constant sectional curvature).
Einstein-de-Sitter spacetime.
Whenf is non-locally constant, the GRW spacetime is calledproper.
In contrast, a GRW spacetime isstaticprovided that its warping function is constant.
J.J. Salamanca (U. Córdoba) 5 / 26
Some properties of GRW spacetimes
Any GRW spacetime with complete fiber is globally hyperbolic2. The physical space for a suitable family of comoving observers (from∂t) is represented by homothetic copies of the fiber(F, gF).
When this physical space is a homogeneus and isotropic
Riemannian manifold, then the spacetime obeys theCosmological Principle3.
Spatially closed GRW spacetimes
If the fiber of a GRW spacetime is compact, then it is calledspatially closed.
This family has been useful to get closed cosmological models.
Some observational and theoretical arguments on the total mass balance of the universe suggest the convenience of regarding open cosmological models4.
A spatially closed GRW spacetime violates the holographic principle5.
4H.Y. Chiu, A cosmological model of universe,Annals of Physics43(1967), 1–41.
5R. Bousso, The holographic principle,Rev. Mod. Phys.,74(2002), 825–874.
J.J. Salamanca (U. Córdoba) 7 / 26
Spatially parabolic GRW spacetimes
We introduce the following notion: a GRW spacetime isspatially parabolicif its fiber is a parabolic Riemannian manifold; that is, it is a noncompact Riemannian manifold such that the only superharmonic functions on it which are bounded from below are the constants.
For example,M =I×f R2, withR2endowed with the Euclidean metric. Also,M =I×f (R×Sn).
Parabolicity of Riemannian manifolds
In the two-dimensional case, there exists some results relating parabolicity with Gaussian curvature. For example, if the Gaussian curvature of a complete Riemannian surface is non-negative, then it must be parabolic.
In higher dimensions, parabolicity is quite different, and it seems not to have a clear relationship with curvature. In fact, the Euclidean space, Rnis parabolic if and only ifn≤2.
Of course, there exists parabolic Riemannian manifolds whose Ricci curvature is not bounded from below.
J.J. Salamanca (U. Córdoba) 9 / 26
Parabolicity of Riemannian manifolds
In the two-dimensional case, there exists some results relating parabolicity with Gaussian curvature. For example, if the Gaussian curvature of a complete Riemannian surface is non-negative, then it must be parabolic.
In higher dimensions, parabolicity is quite different, and it seems not to have a clear relationship with curvature. In fact, the Euclidean space, Rnis parabolic if and only ifn≤2.
Of course, there exists parabolic Riemannian manifolds whose Ricci curvature is not bounded from below.
Parabolicity of Riemannian manifolds
In the two-dimensional case, there exists some results relating parabolicity with Gaussian curvature. For example, if the Gaussian curvature of a complete Riemannian surface is non-negative, then it must be parabolic.
In higher dimensions, parabolicity is quite different, and it seems not to have a clear relationship with curvature. In fact, the Euclidean space, Rnis parabolic if and only ifn≤2.
Of course, there exists parabolic Riemannian manifolds whose Ricci curvature is not bounded from below.
J.J. Salamanca (U. Córdoba) 9 / 26
Parabolicity of Riemannian manifolds
In the two-dimensional case, there exists some results relating parabolicity with Gaussian curvature. For example, if the Gaussian curvature of a complete Riemannian surface is non-negative, then it must be parabolic.
In higher dimensions, parabolicity is quite different, and it seems not to have a clear relationship with curvature. In fact, the Euclidean space, Rnis parabolic if and only ifn≤2.
Of course, there exists parabolic Riemannian manifolds whose Ricci curvature is not bounded from below.
Quasi-isometries
Given Riemannian manifolds(P, g)and(P0, g0), a diffeomorphismφ fromP ontoP0 is called a quasi-isometry if there exists a constant c≥1such that
c−1|v|g ≤ |dφ(v)|
g0 ≤c|v|g for allv∈TpP,p∈P.
A key fact in the consecution of the results is thatparabolicity is invariant under a quasi-isometry6.
6M. Kanai, Rough isometries and the parabolicity of Riemannian manifolds,J.
Math. Soc. Japan,38(1986), 227–238.
J.J. Salamanca (U. Córdoba) 10 / 26
Quasi-isometries
Given Riemannian manifolds(P, g)and(P0, g0), a diffeomorphismφ fromP ontoP0 is called a quasi-isometry if there exists a constant c≥1such that
c−1|v|g ≤ |dφ(v)|
g0 ≤c|v|g for allv∈TpP,p∈P.
A key fact in the consecution of the results is thatparabolicity is invariant under a quasi-isometry6.
Spacelike Hypersurfaces in GRW spacetimes I
LetM be a GRW spacetime. Given ann-dimensional manifoldS, an immersionx:S →M is said to bespacelikeif the Lorentzian metric ofM induces, viax, a Riemannian metricgS onS. In this caseS is called a spacelike hypersurface.
A spacelike hypersurfacelies between two slicesif there exist t1, t2 ∈I,t1 < t2, such that
x(S)⊂[t1, t2]×F.
J.J. Salamanca (U. Córdoba) 11 / 26
Spacelike Hypersurfaces in GRW spacetimes I
LetM be a GRW spacetime. Given ann-dimensional manifoldS, an immersionx:S →M is said to bespacelikeif the Lorentzian metric ofM induces, viax, a Riemannian metricgS onS. In this caseS is called a spacelike hypersurface.
A spacelike hypersurfacelies between two slicesif there exist t1, t2 ∈I,t1 < t2, such that
x(S)⊂[t1, t2]×F.
Spacelike Hypersurfaces in GRW spacetimes II
AsM is time-orientable, it can be takenN ∈X⊥(S)as the only globally defined unitary timelike vector field normal toSin the same time-orientation of the vector field−∂t.
Thehyperbolic anglebetweenN andSis given by coshθ=g(N, ∂t).
J.J. Salamanca (U. Córdoba) 12 / 26
Spacelike Hypersurfaces in GRW spacetimes II
AsM is time-orientable, it can be takenN ∈X⊥(S)as the only globally defined unitary timelike vector field normal toSin the same time-orientation of the vector field−∂t.
Thehyperbolic anglebetweenN andSis given by coshθ=g(N, ∂t).
Spacelike Hypersurfaces in GRW spacetimes III
The shape operator associated withN is AX =−∇XN,
where∇is the Levi-Civita connection ofM.
Themean curvature functionrelative toN is H :=−(1/n)trace(A).
A spacelike hypersurface withH = 0is called amaximal hypersurface.
J.J. Salamanca (U. Córdoba) 13 / 26
Spacelike slices
In any GRW spacetimeM =I×f F, the level hypersurfaces of the functiontconstitute a distinguished family of spacelike hypersurfaces, thespacelike slices.
A spacelike slicet=t0 is totally umbilical and has constant mean curvatureH =−f0(t0)/f(t0). Hence a spacelike slice is maximal when f0(t0) = 0.
Observe that the family of spacelike slices foliate the GRW spacetime.
A technique result
The results are based on the following proposition, Proposition
LetSbe a complete spacelike hypersurface in a GRW spacetimeM, whose fiber has a parabolic universal Riemannian covering.
If the hyperbolic angle ofSis bounded and the warping function onS satisfies:
supf(τ)<∞and inff(τ)>0, then,S is parabolic.
J.J. Salamanca (U. Córdoba) 15 / 26
Sketch of the proof
1 First, we prove that the universal Riemannian covering of the hypersurface is quasi-isometric to the universal Riemannian covering of the fiber.
2 Topologically, the universal Riemannian covering of the hypersurface is diffeomorphic to the universal Riemannian covering of the fiber.
3 Then, under the hypothesis, the following inequalites hold c−1gF(dπ(X), dπ(X))≤gS(X, X)≤c gF(dπ(X), dπ(X)), for certain constantc≥1.
4 Finally, we conclude using the parabolicity of the universal Riemannian covering ofS that the hypersurface is parabolic.
Sketch of the proof
1 First, we prove that the universal Riemannian covering of the hypersurface is quasi-isometric to the universal Riemannian covering of the fiber.
2 Topologically, the universal Riemannian covering of the hypersurface is diffeomorphic to the universal Riemannian covering of the fiber.
3 Then, under the hypothesis, the following inequalites hold c−1gF(dπ(X), dπ(X))≤gS(X, X)≤c gF(dπ(X), dπ(X)), for certain constantc≥1.
4 Finally, we conclude using the parabolicity of the universal Riemannian covering ofS that the hypersurface is parabolic.
J.J. Salamanca (U. Córdoba) 16 / 26
Maximal Hypersurfaces in GRW spacetimes
For a spacelike hypersurface,S, we will denoteτ :=πI ◦xand∆the Laplacian operator associated toS. If the hypersurface is maximal, the following equation holds,
∆τ =−f0(τ) f(τ)
n+|∇τ|2 (1) Denotingf(τ) =f◦τ andf0(τ) =f0◦τ,
∆f(τ) =−nf0(τ)2
f(τ) +f(τ) (logf)00(τ)|∇τ|2. (2)
Uniqueness results
Theorem 1
LetSbe a complete maximal hypersurface of a proper GRW spacetimeM =I×f F whose fiber has parabolic Riemannian
universal covering and whose warping function satisfies(logf)00(t)≤0.
If the hyperbolic angle ofSis bounded and f(τ)is bounded, and
inf(f(τ))>0,
then,S must be a spacelike slicet=t0, withf0(t0) = 0.
J.J. Salamanca (U. Córdoba) 18 / 26
Sketch of the proof
By the Proposition, the hypersurfaceSmust be a parabolic Riemannian manifold.
From (2) and the hypothesis of the theorem, the functionf(τ)is a positive superharmonic function onS.
By parabolicity,f(τ)must be constant. As the GRW spacetime is proper, thenS is a spacelike slice.
Uniqueness results
Theorem 2
LetSbe a complete maximal hypersurface of a GRW spacetime M =I×fF whose fiber has parabolic Riemannian universal covering and whose warping function satisfies(logf)00(t)≤0.
If the hyperbolic angle ofSis bounded andSlies between two spacelike slices, thenSmust be a spacelike slicet=t0, with f0(t0) = 0.
J.J. Salamanca (U. Córdoba) 20 / 26
Solving new Calabi-Bernstein’s type problems I
Let(F, gF)a (non-compact) Riemannian manifold and letf :I →Rbe a positive smooth function. For eachu∈C∞(Ω),Ωopen subdomain in F, such thatu(Ω)⊂I we can consider its graph
Σu ={(u(p), p) :p∈Ω}in the GRW spacetimeM =I ×f F. The induced metric,gu is Riemannian if and only ifusatisfies
|Du|< f(u), everywhere onF, whereDudenotes the gradient ofuin (F, gF).
Solving new Calabi-Bernstein’s type problems I
Let(F, gF)a (non-compact) Riemannian manifold and letf :I →Rbe a positive smooth function. For eachu∈C∞(Ω),Ωopen subdomain in F, such thatu(Ω)⊂I we can consider its graph
Σu ={(u(p), p) :p∈Ω}in the GRW spacetimeM =I ×f F. The induced metric,gu is Riemannian if and only ifusatisfies
|Du|< f(u), everywhere onF, whereDudenotes the gradient ofuin (F, gF).
J.J. Salamanca (U. Córdoba) 21 / 26
Solving new Calabi-Bernstein’s type problems II
WhenΣu is spacelike, we can consider the corresponding mean curvature function
H(u) = div
Du nf(u)√
f(u)2−|Du|2
− f0(u)
n√
f(u)2−|Du|2
n + |Du|f(u)22
.
The differential equationH(u) = 0with the constraint|Du|< f(u)is called themaximal hypersurface equationinM, and its solutions are the maximal graphs inM.
Solving new Calabi-Bernstein’s type problems III
Concretely, we will determine, in some cases, all the entire (i.e., defined on allF) solutions of
div
Du f(u)p
f(u)2− |Du|2
=− f0(u) pf(u)2− |Du|2
n+|Du|2 f(u)2
(E1)
|Du|< λf(u), 0< λ <1. (E2)
J.J. Salamanca (U. Córdoba) 23 / 26
Solving new Calabi-Bernstein’s type problems IV
Theorem
Letf :I −→Rbe a non-locally constant positive smooth function.
Assume(logf)00 ≤0,f is bounded andinff >0.
Then, the only entire solutions to the equation(E)on a parabolic Riemannian manifoldF are the constant functionsu=c, with f0(c) = 0.
M. Caballero, A. Romero and R.M. Rubio
Uniqueness of maximal surfaces in Generalized Robertson-Walker spacetimes and Calabi-Bernstein type problems
J. Geom. Phys.,60(2010), 394–402.
M. Kanai
Rough isometries and the parabolicity of Riemannian manifolds J. Math. Soc. Japan,38(1986), 227–238.
S. Nishikawa
On maximal spacelike hypersurfaces in a Lorentzian manifold Nagoya Math. J.,95(1984), 117–124.
J.J. Salamanca (U. Córdoba) 25 / 26
