Construction of vacuum initial data by the conformally covariant split system Naqing Xie Fudan University Shanghai, China Belgrade, 9-14 SEP 2019 Naqing Xie Belgrade, 9-14 SEP 2019
Outline ◮ The geometric (physical) initial data is referred to as a triple ( M , g , K ) where ( M , g ) is a Riemannian 3-fold and K is a symmetric 2-tensor. They cannot be chosen freely; they must satisfy the constraints. Naqing Xie Belgrade, 9-14 SEP 2019
Outline ◮ The geometric (physical) initial data is referred to as a triple ( M , g , K ) where ( M , g ) is a Riemannian 3-fold and K is a symmetric 2-tensor. They cannot be chosen freely; they must satisfy the constraints. ◮ In this talk, we give a brief introduction to the standard conformal method, initiated by Lichnerowicz, and extended by Choquet-Bruhat and York. Naqing Xie Belgrade, 9-14 SEP 2019
Outline ◮ The geometric (physical) initial data is referred to as a triple ( M , g , K ) where ( M , g ) is a Riemannian 3-fold and K is a symmetric 2-tensor. They cannot be chosen freely; they must satisfy the constraints. ◮ In this talk, we give a brief introduction to the standard conformal method, initiated by Lichnerowicz, and extended by Choquet-Bruhat and York. ◮ There is another way to construct vacuum initial data, referred to as ’the conformally covariant split’ or, historically, ’Method B.’ Naqing Xie Belgrade, 9-14 SEP 2019
Outline ◮ The geometric (physical) initial data is referred to as a triple ( M , g , K ) where ( M , g ) is a Riemannian 3-fold and K is a symmetric 2-tensor. They cannot be chosen freely; they must satisfy the constraints. ◮ In this talk, we give a brief introduction to the standard conformal method, initiated by Lichnerowicz, and extended by Choquet-Bruhat and York. ◮ There is another way to construct vacuum initial data, referred to as ’the conformally covariant split’ or, historically, ’Method B.’ ◮ Joint with P. Mach and Y. Wang, we prove existence of solutions of the conformally covariant split system giving rise to non-constant mean curvature vacuum initial data for the Einstein field equations. Naqing Xie Belgrade, 9-14 SEP 2019
Spacetime and the Einstein Field Equations ◮ Let ( N 1 , 3 , ˆ g ) be a Lorentz manifold satisfying the vacuum Einstein field equations g ) − R (ˆ g ) Ric (ˆ g = 0 . ˆ (1) 2 Naqing Xie Belgrade, 9-14 SEP 2019
Spacetime and the Einstein Field Equations ◮ Let ( N 1 , 3 , ˆ g ) be a Lorentz manifold satisfying the vacuum Einstein field equations g ) − R (ˆ g ) Ric (ˆ g = 0 . ˆ (1) 2 ◮ Let ( M 3 , � g ij , K ij ) be a spacelike hypersurface in ( N 1 , 3 , ˆ g ). Here � g ij is the induced 3-metric of M and K ij is the second fundamental form of M in N . Naqing Xie Belgrade, 9-14 SEP 2019
Vacuum Constraint Equations ◮ The triple ( M , � g , K ) satisfies the vacuum Einstein’s constraints . g K ) 2 = 0 � R − | K | 2 g + ( tr � (Hamiltonian cosntraint) , (2a) � g K − d tr � g K = 0 (momentum constraint) , (2b) div � where � R is the scalar curvature of M with respect to the metric � g . Naqing Xie Belgrade, 9-14 SEP 2019
Vacuum Constraint Equations ◮ The triple ( M , � g , K ) satisfies the vacuum Einstein’s constraints . g K ) 2 = 0 � R − | K | 2 g + ( tr � (Hamiltonian cosntraint) , (2a) � g K − d tr � g K = 0 (momentum constraint) , (2b) div � where � R is the scalar curvature of M with respect to the metric � g . ◮ These equations are coming from (the contracted version of) the Gauss-Codazzi-Mainardi equations in submanifold geometry. (Necessary conditions) Naqing Xie Belgrade, 9-14 SEP 2019
◮ Question: How to construct vacuum initial data satisfying the vacuum Einstein’s constraints ? Naqing Xie Belgrade, 9-14 SEP 2019
◮ Question: How to construct vacuum initial data satisfying the vacuum Einstein’s constraints ? ◮ This problem is notoriously difficult ! Naqing Xie Belgrade, 9-14 SEP 2019
◮ Question: How to construct vacuum initial data satisfying the vacuum Einstein’s constraints ? ◮ This problem is notoriously difficult ! ◮ There is a so-called conformal method. (Lichnerowicz, Choquet-Bruhat, York, Isenberg, ...) Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Free data ( M 3 , g , σ, τ ): g - a Riemannian metric on M ; σ - a symmetric trace- and divergence-free (TT) tensor of type (0 , 2); τ a smooth function on M . Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Free data ( M 3 , g , σ, τ ): g - a Riemannian metric on M ; σ - a symmetric trace- and divergence-free (TT) tensor of type (0 , 2); τ a smooth function on M . ◮ Consider the following system of equations for a positive function φ and a one-form W : − 8∆ φ + R φ = − 2 3 τ 2 φ 5 + | σ + LW | 2 φ − 7 , (3a) ∆ L W = 2 3 φ 6 d τ. (3b) Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Free data ( M 3 , g , σ, τ ): g - a Riemannian metric on M ; σ - a symmetric trace- and divergence-free (TT) tensor of type (0 , 2); τ a smooth function on M . ◮ Consider the following system of equations for a positive function φ and a one-form W : − 8∆ φ + R φ = − 2 3 τ 2 φ 5 + | σ + LW | 2 φ − 7 , (3a) ∆ L W = 2 3 φ 6 d τ. (3b) ◮ Here ∆ = ∇ i ∇ i and R are the Laplacian and the scalar curvature computed with respect to metric g , and ∆ L W is defined as ∆ L W = div g ( LW ), where L is the conformal Killing operator, ( LW ) ij = ∇ i W j + ∇ j W i − 2 3(div g W ) g ij . (4) Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Equation (3a) is called the Lichnerowicz equation , and equation (3b) is called the vector equation . Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Equation (3a) is called the Lichnerowicz equation , and equation (3b) is called the vector equation . ◮ System (3) is referred to as the vacuum conformal constraints . Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Equation (3a) is called the Lichnerowicz equation , and equation (3b) is called the vector equation . ◮ System (3) is referred to as the vacuum conformal constraints . ◮ A dual to a form W satisfying the equation LW = 0 is called a conformal Killing vector field. Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Equation (3a) is called the Lichnerowicz equation , and equation (3b) is called the vector equation . ◮ System (3) is referred to as the vacuum conformal constraints . ◮ A dual to a form W satisfying the equation LW = 0 is called a conformal Killing vector field. ◮ Proposition A. Suppose that a pair ( φ, W ) solves the g = φ 4 g , and vacuum conformal constraints (3). Define � K = τ 3 φ 4 g + φ − 2 ( σ + LW ). Then the triple ( M , � g , K ) becomes an initial data set satisfying the vacuum Einstein’s constraints and tr � g K = τ . Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Assume that M is closed and has no conformal Killing vector fields. Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Assume that M is closed and has no conformal Killing vector fields. ◮ For τ being a constant, then the vector equation implies that W ≡ 0 and one has to focus on the Lichnerowicz equation. Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Assume that M is closed and has no conformal Killing vector fields. ◮ For τ being a constant, then the vector equation implies that W ≡ 0 and one has to focus on the Lichnerowicz equation. ◮ The following table [Isenberg] summarizes whether or not the Lichnerowicz equation admits a positive solution. σ 2 ≡ 0 , τ = 0 σ 2 ≡ 0 , τ � = 0 σ 2 �≡ 0 , τ = 0 σ 2 �≡ 0 , τ � = 0 Y + No No Yes Yes Y 0 Yes No No Yes Y − No Yes No Yes Here Y denotes the Yamabe constant. For data in the class ( Y 0 , σ 2 ≡ 0 , τ = 0), any constant is a solution. For data in all other classes for solutions exist, the solution is unique. Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ In general, the case of non-constant τ remains still open. Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ In general, the case of non-constant τ remains still open. ◮ Some results are obtained when d τ/τ or σ are small. Isenberg, ´ O Murchadha, Maxwell, ... Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ In general, the case of non-constant τ remains still open. ◮ Some results are obtained when d τ/τ or σ are small. Isenberg, ´ O Murchadha, Maxwell, ... ◮ Recently, Dahl, Gicquaud, and Humbert proved the following criterion for the existence of solutions to Eqs. (3). Assume that ( M , g ) has no conformal Killing vector fields and that σ �≡ 0, if the Yamabe constant Y ( g ) ≥ 0. Then, if the limit equation � 3 | LW | d τ 2 ∆ L W = α (5) τ has no nonzero solutions for all α ∈ (0 , 1], the vacuum conformal constraints (3) admit a solution ( φ, W ) with φ > 0. Naqing Xie Belgrade, 9-14 SEP 2019
Conformal Method - A ◮ Moreover, they provided an example on the sphere S 3 such that the limit equation (5) does have a nontrivial solution for some α 0 ∈ (0 , 1]. Naqing Xie Belgrade, 9-14 SEP 2019
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