Hyperbolic Geometric Flow

Hyperbolic Geometric Flow Kefeng Liu Zhejiang University UCLA Page 1 of 41

Outline Introduction Hyperbolic geometric flow Local existence and nonlinear stability Wave character of metrics and curvatures Exact solutions and Birkhoff theorem Dissipative hyperbolic geometric flow Open problems Page 2 of 41

1. Introduction - Ricci flow - Structure of manifolds - Singularities in manifold and space-time - Einstein equations and Penrose conjecture - Wave character of metrics and curvatures - Applications of hyperbolic PDEs to differential geometry J. Hong, D. Christodoulou, S. Klainerman, M. Dafermos, I. Rodnianski, H. Lindblad, N. Ziper Kong et al (Comm. Math. Phys.; J. Math. Phys. 2006) Page 3 of 41

2. Hyperbolic Geometric Flow Let (M, g ij ) be n-dimensional complete Riemannian manifold. The Levi-Civita connection Γ k ij = 1 { gil 2 gkl x + g il i x g ij j x l } The Riemannian curvature tensors R k ijl = Γk jl x i The Ricci tensor Γk il x j + Γk ip Γp jl Γk jp Γp il, R ijkl = g kp R p ijl R ik = g jl R ijkl Page 4 of 41 The scalar curvature R = g ij R ij

Hyperbolic geometric flow (HGF) 2 g ij 2 = 2R ij (1) for a family of Riemannian metrics g ij (t) on M. General version of HGF 2 g ij 2 + 2R ij + F ij ( g, g ) = 0 (2) Page 5 of 41 -De-Xing Kong and Kefeng Liu: Wave Character of Metrics and Hyperbolic Geometric Flow, 2006

Physical background Relation between Einstein equations and HGF Consider the Lorentzian metric ds 2 = dt 2 + g ij (x, t)dx i dx j Einstein equations in vacuum, ie., G ij = 0 become 2 g ij 2 + 2R ij + 1 2 gpq g ij g pq gpq g ip g kq = 0 (3) This is a special example of general version (2) of HGF. Neglecting the terms of first order gives the HGF (1). (3) is named as Einstein s hyperbolic geometric flow Applications to cosmology: singularity of universe Page 6 of 41

Geometric background - Structure of manifolds - Singularities in manifolds - Wave character of metrics and curvatures - Long-time behavior and stability of manifolds Page 7 of 41

Laplace equation, heat equation and wave equation - Laplace equation (elliptic equations) u = 0 - Heat equation (parabolic equations) u t u = 0 - Wave equation (hyperbolic equations) u tt u = 0 Page 8 of 41

Einstein manifold, Ricci flow, hyperbolic geometric flow - Einstein manifold (elliptic equations) R ij = λg ij - Ricci flow (parabolic equations) g ij = 2R ij - Hyperbolic geometric flow (hyperbolic equations) 2 g ij 2 = 2R ij Page 9 of 41 Laplace equation, heat equation and wave equation on manifolds in the Ricci sense

Geometric flows α ij 2 g ij 2 where α ij, β ij, γ ij which may depend on t. In particular, + β g ij ij + γ ijg ij + 2R ij = 0, are certain smooth functions on M α ij = 1, β ij = γ ij = 0: hyperbolic geometric flow α ij = 0, β ij = 1, γ ij = 0: Ricci flow α ij = 0, β ij = 0, γ ij = const.: Einstein manifold Page 10 of 41

Birkhoff Theorem holds for geometric flows - Fu-Wen Shu and You-Gen Shen: Geometric flows and black holes, arxiv: gr-qc/0610030 All of the known explicit solutions of the Einstein solutions, such as the Schwartzchild solution, Kerr solution, satisfy HGF. At least for short time solutions, there should be a 1 1 correspondence between solutions of HGF and the Einstein equation. Page 11 of 41

Complex geometric flows If the underlying manifold M is a complex manifold and the metric is Kähler, a ij 2 g ij 2 + b g ij ij + c ijg ij + 2R ij = 0, where a ij, b ij, c ij are certain smooth functions on M which may also depend on t. Page 12 of 41

3. Local Existence and Nonlinear Stability Local existence theorem (Dai, Kong and Liu, 2006) Let (M, gij 0 (x)) be a compact Riemannian manifold. Then there exists a constant h > 0 such that the initial value problem 2 g ij (x, t) = 2R ij(x, t), 2 g ij (x, 0) = g 0 ij (x), g ij (x, 0) = k0 ij (x), has a unique smooth solution g ij (x, t) on M [0, h], where kij 0 (x) is a symmetric tensor on M. W. Dai, D. Kong and K. Liu: Hyperbolic geometric flow (I): shorttime existence and nonlinear stability Page 13 of 41

Method of proof Strict hyperbolicity Suppose ĝ ij (x, t) is a solution of the hyperbolic geometric flow (1), and ψ t : M M is a family of diffeomorphisms of M. Let g ij (x, t) = ψ t ĝij(x, t) be the pull-back metrics. The evolution equations for the metrics g ij (x, t) are strictly hyperbolic. Page 14 of 41

Symmetrization of hyperbolic geometric flow Introducing the new unknowns g ij, h ij = g ij, g ij,k = g ij x k, we have Rewrite it as g ij = h ij, g kl g ij,k h ij = g kl h ij x k, = g kl g ij,k x l + H ij. A 0 (u) u = Aj (u) u x j + B(u), Page 15 of 41 where the matrices A 0, A j are symmetric.

Nonlinear stability Let M be a n-dimensional complete Riemannian manifold. Given symmetric tensors gij 0 and g1 ij 2 g ij (t, x) = 2R ij(t, x) 2 g ij (x, 0) = g ij (x) + εgij 0 (x), on M, we consider g ij (x, 0) = εg1 ij (x), where ε > 0 is a small parameter. Definition The Ricci flat Riemannian metric g ij (x) possesses the (locally) nonlinear stability with respect to (gij 0, g1 ij ), if there exists a positive constant ε 0 = ε 0 (gij 0, g1 ij ) such that, for any ε (0, ε 0 ], the above initial value problem has a unique (local) smooth solution g ij (t, x); g ij (x) is said to be (locally) nonlinear stable, if it possesses the (locally) nonlinear stability with respect to arbitrary symmetric tensors gij 0 (x) and g1 ij (x) with compact support. Page 16 of 41

Nonlinear stability theorem (Dai, Kong and Liu, 2006) The flat metric g ij = δ ij of the Euclidean space R n with n 5 is nonlinearly stable. Remark The above theorem gives the nonlinear stability of the hyperbolic geometric flow on the Euclidean space with dimension larger than 4. The situation for the 3-, 4-dimensional Euclidean spaces is very different. This is a little similar to the proof of the Poincaré conjecture: the proof for the three dimensional case and n 5 dimensional case are very different, while the four dimensional smooth case is still open. Page 17 of 41

Method of proof Define a 2-tensor h g ij (x, t) = δ ij + h ij (x, t). Choose the elliptic coordinates {x i } around the origin in R n. It suffices to prove that the following Cauchy problem has a unique global smooth solution 2 h ij (x, t) = 2 k=1 h ij (x, 0) = εgij 0 (x), n 2 h ij x k x + H k ij ( h kl, h kl x p, h ij (x, 0) = εg1 ij (x). 2 ) h kl, x p x q Page 18 of 41

Einstein s hyperbolic geometric flow 2 g ij + 2R 2 ij + 1 g 2 gpq ij satisfy the null condition g pq g pq g ip g kq = 0 Global existence and nonlinear stability for small initial data (Dai, Kong and Liu) Page 19 of 41

4. Wave Nature of Curvatures Under the hyperbolic geometric flow (1), the curvature tensors satisfy the following nonlinear wave equations 2 R ijkl 2 = R ijkl + (lower order terms), 2 R ij = R 2 ij + (lower order terms), 2 R = R + (lower order terms), 2 where is the Laplacian with respect to the evolving metric, the lower order terms only contain lower order derivatives of the curvatures. Page 20 of 41

Evolution equation for Riemannian curvature tensor Under the hyperbolic geometric flow (1), the Riemannian curvature tensor R ijkl satisfies the evolution equation 2 2R ijkl = R ijkl + 2 (B ijkl B ijlk B iljk + B ikjl ) g pq (R pjkl R qi + R ipkl R qj + R ijpl R qk + R ijkp R ql ) ( +2g pq Γp il Γq jk ) Γp jl Γq ik, where B ijkl = g pr g qs R piqj R rksl and is the Laplacian with respect to the evolving metric. Page 21 of 41

Evolution equation for Ricci curvature tensor Under the hyperbolic geometric flow (1), the Ricci curvature tensor satisfies 2 2R ik = R ik + 2g pr g qs R piqk R rs 2g pq R pi R qk ( +2g jl g pq Γp il Γq jk ) Γp jl Γq ik 2g jp g lq g pq R ijkl + 2g jp g rq g sl g pq g rs R ijkl Page 22 of 41

Evolution equation for scalar curvature Under the hyperbolic geometric flow (1), the scalar curvature satisfies 2 2R = R + 2 Ric 2 ( +2g ik g jl g pq Γp il Γq jk Γp jl Γq ik 2g ik g jp g lq g pq R ijkl 2g ip g kq g pq R ik + 4R ik g ip g rq g sk g pq ) g rs Page 23 of 41

5. Exact Solutions and Birkhoff theorem 5.1 Exact solutions with the Einstein initial metrics Definition (Einstein metric and manifold) A Riemannian metric g ij is called Einstein if R ij = λg ij for some constant λ. A smooth manifold M with an Einstein metric is called an Einstein manifold. If the initial metric g ij (0, x) is Ricci flat, i.e., R ij (0, x) = 0, then g ij (t, x) = g ij (0, x) is obviously a solution to the evolution equation (1). Therefore, any Ricci flat metric is a steady solution of the hyperbolic geometric flow (1). Page 24 of 41

If the initial metric is Einstein, that is, for some constant λ it holds R ij (0, x) = λg ij (0, x), x M, then the evolving metric under the hyperbolic geometric flow (1) will be steady state, or will expand homothetically for all time, or will shrink in a finite time. Page 25 of 41

Let g ij (t, x) = ρ(t)g ij (0, x) By the defintion of the Ricci tensor, one obtains Equation (1) becomes R ij (t, x) = R ij (0, x) = λg ij (0, x) 2 (ρ(t)g ij (0, x)) 2 = 2λg ij (0, x) This gives an ODE of second order d 2 ρ(t) dt 2 = 2λ One of the initial conditions is ρ(0) = 1, another one is assumed as ρ (0) = v. The solution is given by ρ(t) = λt 2 + vt + 1 Page 26 of 41

General solution formula is g ij (t, x) = ( λt 2 + vt + c)g ij (0, x) Remark This is different with the Ricci flow! Case I The initial metric is Ricci flat, i.e., λ = 0. In this case, ρ(t) = vt + 1. (4) If v = 0, then g ij (t, x) = g ij (0, x). This shows that g ij (t, x) = g ij (0, x) is stationary. Page 27 of 41

If v > 0, then g ij (t, x) = (1 + vt)g ij (0, x). This means that the evolving metric g ij (t, x) = ρ(t)g ij (0, x) exists and expands homothetically for all time, and the curvature will fall back to zero like 1 t. Notice that the evolving metric g ij (t, x) only goes back in time to v 1, when the metric explodes out of a single point in a big bang. If v < 0, then g ij (t, x) = (1 + vt)g ij (0, x). Thus, the evolving metric g ij (t, x) shrinks homothetically to a point as t T 0 = 1 v. Note that, when t T 0, the scalar curvature is asymptotic to 1 T 0 t. This phenomenon corresponds to the black hole in physics. Page 28 of 41

Conclusion: For the Ricci flat initial metric, if the initial velocity is zero, then the evolving metric g ij is stationary; if the initial velocity is positive, then the evolving metric g ij exists and expands homothetically for all time; if the intial velocity is negative, then the evolving metric g ij shrinks homothetically to a point in a finite time. Page 29 of 41

Case II The initial metric has positive scalar curvature, i.e., λ > 0. In this case, the evolving metric will shrink (if v < 0) or first expands then shrink (if v > 0) under the hyperbolic flow by a time-dependent factor. Page 30 of 41

Case III The initial metric has a negative scalar curvature, i.e., λ < 0. In this case, we devide into three cases to discuss: Case 1 v 2 + 4λ > 0. (a) v < 0: the evolving metric will shrink in a finite time under the hyperbolic flow by a time-dependent factor; (b) v > 0: the evolving metric g ij (t, x) = ρ(t)g ij (0, x) exists and expands homothetically for all time, and the curvature will fall back to zero like 1 t 2. Page 31 of 41

Case 2 v 2 + 4λ < 0. In this case, the evolving metric g ij (t, x) = ρ(t)g ij (0, x) exists and expands homothetically (if v > 0) or first shrinks then expands homothetically (if v < 0) for all time. The scalar curvature will fall back to zero like 1. t 2 Case 3 v 2 + 4λ = 0. If v > 0, then evolving metric g ij (t, x) = ρ(t)g ij (0, x) exists and expands homothetically for all time. In this case the scalar curvature will fall back to zero like 1 t 2. If v < 0, then the evolving metric g ij (t, x) shrinks homothetically to a point as t T = v 2λ > 0 and the scalar curvature is asymptotic to 1 T t. Page 32 of 41

Remark A typical example of the Einstein metric is ds 2 = 1 1 κr 2dr2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2, where κ is a constant taking its value 1, 0 or 1. We can prove that { } ds 2 = R 2 1 (t) + r 2 dθ 2 + r 2 sin 2 θdϕ 2 1 κr 2dr2 is a solution of the hyperbolic geometric flow (1), where R 2 (t) = 2κt 2 + c 1 t + c 2 in which c 1 and c 2 are two constants. This metric plays an important role in cosmology. Page 33 of 41

5.2 Exact solutions with axial symmetry Consider ds 2 = f(t, z)dz 2 t g(t, z) [ (dx µ(t, z)dy) 2 + g 2 (t, z))dy 2], where f, g are smooth functions with respect to variables. Since the coordinates x and y do not appear in the preceding metric formula, the coordinate vector fields x and y are Killing vector fields. The flow x (resp. y ) consists of the coordinate translations that send x to x + x (resp. y to y + y), leaving the other coordinates fixed. Roughly speaking, these isometries express the x-invariance (resp. y-invariance) of the model. The x-invariance and y-invariance show that the model possesses the z-axial symmetry. Page 34 of 41

HGF gives g t = µ t = 0 where g z and µ z satisfy f = 1 [ g 2 2g 2 z + ] 1 µ2 z + g 4µ2 z (c 1t + c 2 ), gg 2 z gg zµ zz µ 1 z + g 2 z + µ2 z = 0 Birkhoff Theorem holds for axial-symmetric solutions! Angle speed µ is independent of t! Page 35 of 41

6. Dissipative hyperbolic geometric flow Let M be an n-dimensional complete Riemannian manifold with Riemannian metric g ij. Consider the hyperbolic geometric flow 2 g ij 2 = 2R ij + 2g pq g ip ( c + 1 n 1 g jq ( g pq g pq + ) 2 + 1 ( d 2g pq g pq n 1 g pq g pq ) gij + ) for a family of Riemannian metrics g ij (t) on M, where c and d are arbitrary constants. g ij Page 36 of 41

By calculations, we obtain the following evolution equation of the scalar curvature R with respect to the metric g ij (x, t) 2 R = R + 2 Ric 2 + 2 ( c + 1 n 1 2g ik g jl g pq Γ p ij 8g ik Γq ip Γ p kq ( d 2g pq g pq ( g pq g pq Γ q kl ) 2 + 1 n 1 ) R g pq 2g ik g jl g pq Γ p ik 8g ik Γp ip Γ q kq g pq ) Γ q jl + 8g ik Γq pq R+ Γ p ik Page 37 of 41

Introduce and By (1), we have y = g pq g pq z = g pq g rs g pr = T r g qs { gpq } = g pq y = 2R n 2 n 1 y2 + dy 1 n 1 z + cn Global stability of Euclidean metric - Dai, Kong and Liu, 2006 2. Page 38 of 41

7. Open Problems Yau has several conjectures about asymptotically flat manifolds with nonnegative scalar curvature. HGF supplies a promising way to approach these conjectures. Penrose cosmic censorship conjecture. Given initial metric gij 0 and symmetric tensor k ij, study the singularity of the HGF with these initial data. These problems are from general relativity and Einstein equation in which HGF has root. Page 39 of 41

Global existence and singularity; HGF has global solution for small initial data. HGF flow and (minimal) hypersurface. Study HGF with initial data given by initial metric and second fundamental form h ij. There was an approach of geometrization by using Einstein equation which is too complicated to use. HGF may simplify and even complete the approach. Page 40 of 41

Thank you Page 41 of 41