Hyperbolic Geometric Flow Kefeng Liu CMS and UCLA August 20, 2007, Dong Conference Page 1 of 51
Outline: Joint works with D. Kong and W. Dai Motivation Hyperbolic geometric flow Local existence and nonlinear stability Wave Nature of Curvatures Exact solutions and Birkhoff theorem Dissipative hyperbolic geometric flow Riemann surfaces Some results related to HGF - Time-periodic solutions of Einstein equations - Hyperbolic mean curvature flow Open problems Page 2 of 51
1. Motivation - Ricci flow: Geometric structure of manifolds - Einstein equations and Penrose conjecture: Singularities in manifold and space-time - Applications of hyperbolic PDEs to differential geometry: Wave character of metrics and curvatures D. Christodoulou, S. Klainerman, M. Dafermos, I. Rodnianski, H. Lindblad, N. Ziper J. Hong (Nonlinear Analysis, 1995) Page 3 of 51 Kong et al (Comm. Math. Phys.; J. Math. Phys. 2006)
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 51 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 + 2R 2 ij + F ij ( g, g ) = 0 (2) Page 5 of 51 - Kong and 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, i.e., G ij = 0 become 2 g ij 2 + 2R ij + 1 2 gpq g ij g pq gpq g ip g jq = 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 Page 6 of 51
Three important types - Hyperbolic geometric flow 2 g ij 2 = 2R ij Wave type equation - Einstein s hyperbolic geometric flow 2 g ij 2 + 2R ij + 1 2 gpq g ij g pq gpq g ip g kq Wave type equation satisfying null condition - Dissipative hyperbolic geometric flow Wave type equation with dissipative terms = 0 Page 7 of 51
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 51
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 51 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 51
Birkhoff Theorem holds for geometric flows - Fu-Wen Shu and You-Gen Shen: Geometric flows and black holes, arxiv: gr-qc/0610030 (Locally) any solutions of the Einstein equation are also the solutions of the Einstein hyperbolic geometric flow, which is relatively easier to deal with. Page 11 of 51
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 51
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. Dai, Kong and Liu: Hyperbolic geometric flow (I): short-time existence and nonlinear stability Page 13 of 51
Method of proof: Two proofs. 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 51
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 51 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), where ε > 0 is a small parameter. on M, we consider g ij (x, 0) = εg1 ij (x), 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); Page 16 of 51
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. 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: equations. The proof uses general theory of nonlinear wave Page 17 of 51
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 51
Einstein s hyperbolic geometric flow 2 g ij + 2R 2 ij + 1 g 2 gpq ij g pq g pq g ip g kq = 0 satisfy the null condition. Existence and uniqueness for small initial data hold. Einstein hyperbolic flow has nonlinear stability for all dimensions. Global existence and nonlinear stability for small initial data (Dai, Kong and Liu) Page 19 of 51
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 51
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 51
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 51
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 51
5. Exact Solutions and Birkhoff theorem 5.1. Exact solutions with the Einstein initial metrics Definition: (Einstein metric on 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 51
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. Note that we can choose suitable velocity to avoid singularity. Page 25 of 51
Let g ij (t, x) = ρ(t)g ij (0, x) By the definition 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 51
General solution formula is g ij (t, x) = ( λt 2 + vt + c)g ij (0, x) Remark: This is different from 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 51
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 51
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 29 of 51
Case III: The initial metric has a negative scalar curvature, i.e., λ < 0. In this case, we divide 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 30 of 51
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 31 of 51
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. It is easy to see 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 has interesting meaning in cosmology. Page 32 of 51
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 33 of 51
Hyperbolic geometric flow 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! Page 34 of 51 Angle speed µ is independent of t!
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 35 of 51 This equation also has strong feature of Ricci flow.
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 36 of 51
Introduce and y = g pq g pq z = g pq g rs g pr { } gpq = T r g g qs Then we have proved the following = g pq y = 2R n 2 n 1 y2 + dy 1 n 1 z + cn Dissipative Hyperbolic Geometric Flow - Dai, Kong and Liu, 2006 2. Page 37 of 51
7. Riemann surfaces 7.1. Global existence Consider the evolution of a Riemannian metric g ij on a complete non-compact surface M under the hyperbolic geometric flow equation 2 g ij = 2R 2 ij. Let us consider the case R 2 with the following initial metric t = 0 : ds 2 = u 0 (x)(dx 2 + dy 2 ), where u 0 (x) is a smooth function with bounded C 2 norm and satisfies 0 < m u 0 (x) M <, in which m, M are two positive constants. Page 38 of 51
Theorem 1: (Kong and Liu, 2007) Given the above initial metric, for any smooth function u 1 (x) satisfying (1) u 1 (x) has bounded C 1 norm; (2) it holds that u 1 (x) u 0 (x), x R, u0 (x) the Cauchy problem 2 g ij 2 = 2R ij (i, j = 1, 2), t = 0 : g ij = u 0 (x)δ ij, g ij = u 1 (x)δ ij (i, j = 1, 2) has a unique smooth solution for all time, and the solution metric g ij possesses the form g ij = u(t, x)δ ij. Page 39 of 51
Theorem 2: (Kong and Liu, 2007) Under the assumptions mentioned in Theorem 1, suppose that there exists a small positive constant ε such that u 1 (x) u 0 (x) u0 (x) + ε, x R, then the Cauchy problem has a unique smooth solution with the above form for all time, moreover the solution metric g ij converges to flat metric at an algebraic rate, where k 2 1 (1 + t) k is a positive constant depending on ε, M, the C 2 norm of u 0 and C 1 norm of u 1. Page 40 of 51
Method of proof. (1) Cauchy problem for a kind of nonlinear wave equation { utt ln u = 0, t = 0 : u = u 0 (x), u t = u 1 (x) In the present situation, { utt (ln u) xx = 0, t = 0 : u = u 0 (x), u t = u 1 (x) Page 41 of 51 Let φ = ln u, then the equation is reduced to φ tt e φ φ xx = φ 2 t.
(2) Quasilinear hyperbolic system Let v = φ t, w = φ x. Then Introduce Lemma: φ t = v, w t v x = 0, v t e φ w x = v 2 p = v + e φ 2 w, q = v e φ 2 w. p, q satisfy p t λp x = 1 4 {p2 + 3pq}, Page 42 of 51 q t + λq x = 1 4 {q2 + 3pq}, where λ = e φ 2.
Lemma: It holds that { pt (λp) x = pq, q t + (λq) x = pq. Let r = p x, s = q x. Lemma: r and s satisfy { rt λr x = 1 [(2q + 3p)r + 3ps], 4 s t + λs x = 1 [(2p + 3q)s + 3qr]. 4 Page 43 of 51
(3) Uniform estimates on p, q and r, s Maximum principle for hyperbolic systems (4) Global existence of HGF. (5) Estimate on R R = (ln u) xx u = 1 2 { (r s)e φ 2 + ( p q 2 ) 2 } 1 R(t, x) C (1 + t) min{2,c}, (t, x) R+ R.. Page 44 of 51
7.2. Formation of singularities Consider the metric t = 0 : ds 2 = u 0 (x)(dx 2 + dy 2 ). Without loss of generality, we assume that there exists a point x 0 R such that u 0 (x 0) < 0. Page 45 of 51 We choose u 1 (x) u 0 (x) u0 (x), x R.
Theorem 3: For the initial data u 0 (x) and u 1 (x) mentioned above, the Cauchy problem has a unique smooth solution only in [0, T max ) R, where 4 T max = inf x R {p 0 (x)} where p 0 (x) = 2u 0 (x)u 3 2 0 (x). Moreover, there exists some point ( T max, x ) such that the scalar curvature R(t, x) satisfies R(t, x) as (t, x) ( T max, x ). By choosing suitable velocity, we may avoid doing surgery! We have precise control on Blowup set. Page 46 of 51 Solution character close to blowup point.
8. Some results related to HGF 8.1. Time-periodic solutions of Einstein equations Einstein equation: G µν R µν 1 2 g µνr = 0 A new solution: Breather (1) time periodic space-time, (2) asymptotically flat in space. Page 47 of 51 Physical characters: Time periodic solution, asymptotical flatness of t-sections, naked singularity (r = 0).
8.2. Hyperbolic mean curvature flow for hypersurfaces 2 X = H n 2 where X = X(t, x 1,, x n ), H mean curvature, n out normal vector. Our recent results: (He-Kong-Liu) Extrinsic flow: short time existence, nonlinear stability, relations between the HGF and the HMCF flow equations for extremal hypersurfaces in space-time. Page 48 of 51 Intrinsic HGF flow and extrinsic HMCF flow, both have interesting physical meaning.
9. Open Problems Yau has several conjectures about complete noncompact manifolds with nonnegative curvature. HGF may provide a promising way to approach these conjectures. Hyperbolic PDE has advantage in dealing with noncompact manifolds. Given initial metric gij 0 and symmetric tensor k ij, study the singularity of the Einstein HGF with these initial data. Singularities of Einstein equation belong to these singularities. This should be related to the Penrose cosmic censorship conjecture. Page 49 of 51
HGF on general manifolds, global existence and singularity. HGF has global solution for small initial data. Choosing velocity to avoid surgery to get geometric structure. There was an approach of geometrization by using Einstein equation which is too complicated to use. May HGF shed some light on this approach, by adjusting speed to avoid surgery? Hyperbolic Yang-Mills flow? Page 50 of 51
Thank You All! Page 51 of 51