Wave Character of Metrics and Hyperbolic Geometric Flow

Transcription

1 Wave Character of Metrics and Hyperbolic Geometric Flow De-Xin Kon and Kefen Liu 2 Center of Mathematical Sciences, Zhejian University, Hanzhou 327, China 2 Department of Mathematics, University of California at Los Aneles, CA 995, USA (Dated: Auust 27, 26. Revised November 29, 26) In this letter, we illustrate the wave character of the metrics and curvatures of manifolds, and introduce a new understandin tool - the hyperbolic eometric flow. This kind of flow is new and very natural to understand certain wave phenomena in the nature as well as the eometry of manifolds. It possesses many interestin properties from both mathematics and physics. Several applications of this method have been found. PACS numbers: 2.4.Vh; 2.3.Jr; 4.2.Jb; 2.4.Ky; 4.3.Nk. Introduction. Let us observe the water in the beautiful Westlake in Hanzhou. If there is no wind, the water surface can be rearded as a plane with a flat Riemmannian metric δ ij (i, j =, 2). When wind blows over, the water wave propaates from one side to another side. In this case, the metric of the water surface is not flat lobally, and chanes alon the time. There exists a front, called wave front, such that the metric is not flat after the front and is still flat before the front. We would like to call this phenomenon the wave character of the metric. Motivated by such wave character of metric as well as the work of Ricci flow, we introduce and study the followin evolution equation which we would like to call the hyperbolic eometric flow: let M be n-dimensional complete Riemannian manifold with Riemannian metric ij, the Levi-Civita connection is iven by the Christoffel symbols Γ k ij = { il 2 kl x i + il x j } ij x l, where ij is the inverse of ij. The Riemannian curvature tensors read R k ijl = Γk jl x i Γk il x j + Γk ipγ p jl Γk jpγ p il, R ijkl = kp R p ijl. The Ricci tensor is the contraction and the scalar curvature is R ik = jl R ijkl R = ij R ij. The hyperbolic eometric flow considered here is the evolution equation 2 = 2R ij. () for a family of Riemannian metrics ij (t) on M. () is a nonlinear system of second order partial differential equations on the metric ij. The hyperbolic eometric flow () is only weakly hyperbolic, since the symbol of the derivative of E = E( ij ) = 2R ij has zero eienvectors. However, usin DeTurck s technique (see 4), instead of considerin the system () we only need to consider a modified evolution system which is strictly hyperbolic (see 2 for the details). The hyperbolic eometric flow is a very natural tool to understand the wave character of the metrics and wave phenomenon of the curvatures. We will prove that it has many surprisinly ood properties, which have essential and fundamental differences from the Einstein field equations (see ) and the Ricci flow (see 7). More applications of hyperbolic eometric flow to both mathematics and physics can be expected. The elliptic and parabolic partial differential equations have been successfully applied to differential eometry and physics (see 8). Typical examples are Hamilton s Ricci flow and Schoen-Yau s solution of the positive mass conjecture (see 7, 9). A natural and important question is if we can apply the well-developed theory of hyperbolic differential equations to solve problems in differential eometry and theoretical physics. This letter is an attempt to apply the hyperbolic equation techniques to study some eometrical problems and physical problems. We have already found interestin results in these directions (see 3). The method may be more important than the results presented in this letter. Our results show that the hyperbolic eometric flow is a natural and powerful tool to study some problems arisin form differential eometry such as sinularities, existence and reularity. 2. Hyperbolic eometric flow. Hyperbolic eometric flow considered here is the evolution equation (), it describes the wave character of the Riemannian metrics ij (t) on an n-dimensional complete Riemannian manifold M. The version () of the hyperbolic eometric flow is the unnormalized evolution equation. We next consider the normalized version of hyperbolic eometric flow (), which preserves the volume of the flow. The hyperbolic eometric flow and the normalized hyperbolic eometric flow differ only by a chane of scale in space and a chane of parametrization in time. We now derive the normalized version of (). Assume that ij (t) is a solution of the (unnormalized) hyperbolic eometric flow () and choose the normalization factor ϕ = ϕ(t)

2 2 such that ij = ϕ 2 ij and M dṽ =. (2) Next we choose a new time scale t = ϕ(t)dt. (3) Then, for the normalized metric ij, we have R ij = R ij, R = ϕ 2 R, r = r, (4) ϕ2 where r = M RdV/ dv is the averae scalar curvature. Notin the second equation in (2) ives N dv = ϕ n. (5) Then ij 2 ij 2 M = ϕ ij + 2 dϕ dt ij, ( = 2 ij d dt loϕ ( ) ( d d 2 dt loϕ ) ij dϕ lo dt dt + ) ij = 2 R ij + 3ϕ ( d dt loϕ ) ij ( ) { d d 2ϕ 2 dt loϕ dϕ lo dt dt 3 d dt loϕ = 2 R ij + a ij + b ij. By (5) and calculations, we observe that a and b are certain functions of t. The followin evolution equation 2 ij 2 = 2 R ij + a ij + } ij + b ij (6) is called the normalized version of the hyperbolic eometric flow (). Thus, studyin the behavior of the hyperbolic eometric flow near the maximal existence time is equivalent to studyin the lon-time behavior of normalized hyperbolic eometric flow. Motivated by (6), we may consider the followin more eneral evolution equations 2 + 2R ij + F ij (, ) =, (7) where F ij are some iven smooth functions of the Riemannian metric and its first order derivative with respect to t. We name the evolution equations (7) as eneral version of hyperbolic eometric flow. Obviously, when F ij, the system (7) oes back to the standard hyperbolic eometric flow (). In particular, consider the space-time R M with the Lorentzian metric ds 2 = dt 2 + ij (x, t)dx i dx j. (8) The Einstein equations in the vacuum, which correspond to the metric (8), read 2 + 2R ij + 2 pq ij pq pq ip jq =. (9) Clearly, the system (9) is a special case of (7) in which F ij = 2 pq ij pq pq ip jq. (9) is called Einstein s hyperbolic eometric flow. We will study the eometric and physical meanins and applications of the Einstein s hyperbolic eometric flow later. Remark. Nelectin the lower order terms in (9) leads to the hyperbolic eometric flow (). In this sense, () can be rearded as the resultin equations takin leadin terms of the Einstein equations in the vacuum with respect to the metric (8). We will see that the intrinsically defined hyperbolic eometric flow equations have many interestin new features. Remark 2. Notice that the the equations for Einstein manifolds, i.e., R ij = κ ij (κ is a constant) () are elliptic, the Ricci flow equations ij = 2R ij () are parabolic, and the hyperbolic eometric flow (see ()) are hyperbolic. In some sense, the above three kinds of equations can be rearded as the eneralization on manifolds of the famous Laplace equation, heat equation and wave equation, respectively. We may also consider the followin field equations α ij 2 + β ij ij + γ ij ij + 2R ij =, (2) where α ij, β ij, γ ij are certain smooth functions on M which may depend on t. In particular, if α ij =, β ij = γ ij =, then (2) oes back to the hyperbolic eometric flow; if α ij =, β ij =, γ ij =, then (2) is nothin but the famous Ricci flow; if α ij =, β ij =, γ ij = 2 n r, then (2) is the normalized Ricci flow (see 7). In this sense, we name the evolution equations (2) as hyperbolic-parabolic eometric flow. At the end of this section, we remark that if the underlyin manifold M is a complex manifold and the metric is Kähler, similar to (2) the followin complex evolution equations are very natural to consider a ij 2 + b ij ij + c ij ij + 2R ij =, (3)

3 3 where a ij, b ij, c ij are certain smooth functions on M which may also depend on t. The evolution equations (3) are called the complex hyperbolic-parabolic eometric flow. 3. Exact solutions. This section is devoted to studyin the exact solutions for the hyperbolic eometric flow (). These exact solutions are useful to understand the basic features of the hyperbolic eometric flow. They will also be useful to understand the eneral Einstein equations. We believe that there is a correspondence between the solutions of the hyperbolic eometric flow and the eneral Einstein equations. First recall that a Riemannian metric ij is called Einstein if R ij = λ ij for some constant λ. A smooth manifold M with an Einstein metric is called an Einstein manifold. If the initial metric ij (, x) is Ricci flat, i.e., R ij (, x) =, then ij (t, x) = ij (, x) is obviously a solution to the evolution equation (). Therefore, any Ricci flat metric is a stationary solution of the hyperbolic eometric flow (). If the initial metric is Einstein, that is, for some constant λ it holds R ij (, x) = λ ij (, x), x M, (4) then the evolvin metric under the hyperbolic eometric flow () will be steady state, or will expand homothetically for all time, or will shrink in a finite time. Indeed, since the initial metric is Einstein, (4) holds for some constant λ. Let ij (t, x) = ρ(t) ij (, x). (5) By the defintion of the Ricci tensor, one obtains R ij (t, x) = R ij (, x) = λ ij (, x). (6) In the present situation, the equation () becomes 2 (ρ(t) ij (, x)) 2 = 2λ ij (, x). (7) This ives an ODE of second order d 2 ρ(t) dt 2 = 2λ. (8) Obviously, one of the initial conditions for (8) is Another one is assumed as ρ() =. (9) ρ () = v, (2) where v is a real number standin for the initial velocity. The solution of the initial value problem (8)-(2) is iven by ρ(t) = λt 2 + vt +. (2) A typical example of the Einstein metric is ds 2 = κr 2 dr2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2, (22) where κ is a constant takin its value, or. We can prove that { } ds 2 = R 2 (t) κr 2 dr2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 (23) is a solution of the hyperbolic eometric flow (), where R 2 (t) = 2κt 2 + c t + c 2 in which c and c 2 are two constants. The metric (23) plays an important role in cosmoloy. More interestin examples are the exact solutions with axial symmetry. We are interested in the exact solutions with the followin form for the hyperbolic eometric flow () ds 2 = f(t, z)dz 2 + h(t) (dx µ(t, z)dy) (t, z))dy 2, (t, z) (24) where f, h, are smooth functions with respect to variables. Clearly, ( ij ) = and the inverse of ( ij ) is ( ij ) = h µh µh (µ )h f µ h µ h µ h h f (25). In order to uarantee that the metric ij is Riemannian, we assume f(t, z) >, h(t) >. (26) (t, z) Since the coordinates x and y do not appear in the precedin metric formula, the coordinate vector fields x and y are Killin vector fields. The flow x (resp. y ) consists of the coordinate translations that send x to x + x (resp. y to y + y), leavin the other coordinates fixed. Rouhly speakin, 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. By a direct calculation, we obtain the Ricci curvature

4 4 correspondin to the metric (24) R = h 4f 2 2fµ 2 3 z + 2f zz z f z 2fz 2, R 2 = h 4f 2 3 µfz z + 2fµz 2 + 2f 2 µ zz 2 f z µ z 4f z µ z 2fµµ 2 z 2fµ zz, R 3 =, R 22 = h 4f µf 3 z µ z + 8fµ z µ z + 2f(µ 2 2 )µ 2 z +2f(µ 2 2 ) zz (µ 2 2 )f z z 2f(µ 2 2 )z 2 4f 2 µµ zz, R 23 =, R 33 = 2 ( 2 2 z + µ 2 z). (27) Notin (25) and (27), we obtain from () that 2 h 3 tt + 2ht 2 h tt 2h t t (28) 2f 2 2fµ 2 3 z + 2f zz z f z 2fz 2, hµ 2 3 tt + 2h t µ t + 2µ t h t 2 hµ tt 2 µh tt 2 2 h t µ t 2hµt 2 2f 2 3 µfz z + 2fµ 2 z + 2f 2 µ zz 2 f z µ z 4f z µ z 2fµµ 2 z 2fµ zz, (29) 3 2 (µ )h tt hµµ tt µh t µ t + 2hµ 2 2 t hµ 2 t h(µ 2 2 ) tt 2(µ 2 2 )h t t 4hµ t µ t and 2f µf z µ z + 8fµ z µ z + 2f(µ 2 2 )µ 2 z +2f(µ 2 2 ) zz (µ 2 2 )f z z 2f(µ 2 2 ) 2 z 4f 2 µµ zz, (3) f tt = 2 (2 z + µ 2 z). (3) Multiplyin (28) by µ then summin (29) ives 2h t µ t 2 hµ tt 2 2 h t µ t ( ) (32) 2f 2f 2 µ 2 zz 4f z µ z 2 f z µ z. Multiplyin (28) by ( 2 µ 2 ), (32) by 2µ then summin these two resultin equations and (3) leads to 2 h tt + h ( 2 t + µ 2 t ) =. (33) Consider the linear expandin of the rotation, i.e., h(t) = t. In this case, it follows from (33) that t = µ t =. (34) This implies that and µ are independent of t, that is, = (z) and µ = µ(z). Therefore, (28)-(3) reduce to 2f zz + 2fµ 2 z 2fz 2 f z z =, 2fµ zz 4f z µ z f z µ z =, (35) ( ) f tt = 2 = 2 z + µ 2 z F (z). By the third equation in (35), we have f(t, z) = 2 F (z)t2 + c (z)t + c 2 (z), (36) where c (z) and c 2 (z) are two arbitrary smooth functions of z. Substitutin (36) into the first equation in (35) yields a quadratic equation on t A(z)t 2 + B(z)t + C(z) =, (37) where A(z) = µ (µ + µ µ µ ), B(z) = 2c ( + µ 2 2 ) c, C(z) = 2c 2 ( + µ 2 2 ) c 2, (38) where stands for the derivative of with respect to z. Notin the arbitrariness of t ives the followin system of ODEs µ (µ + µ µ µ ) =, 2c ( + µ 2 2 ) c =, (39) 2c 2 ( + µ 2 2 ) c 2 =. Similarly, substitutin (36) into the second equation in (35) leads to ( µ µ 3 µ 2 µ ) =, 2c (µ 2 µ ) µ c =, (4) 2c 2 (µ 2 µ ) µ c 2 =. Case I = In this case, it follows from the first equation in (39) that µ =. This implies that = a, µ = b, where a and b are constants. Then the solution of (35) is f = c (z)t + c 2 (z), = a, µ = b. (4) Therefore, the desired solution reads ds 2 = (c (z)t + c 2 (z))dz 2 + t a (dx bdy)2 + a 2 dy 2, where a is a positive constant, and c (z), c 2 (z) are two positive smooth functions of z.

5 5 Case II µ =, i.e., µ = µ = const. In this case, (4) is always true, and (39) becomes { 2c ( 2 ) = c, (42) 2c 2 ( 2 ) = c 2. It follows from (42) that c (z) = a (ln ) 2, c (z) = a 2 (ln ) 2, where a, a 2 are two constants, and is an arbitrary positive smooth function of z. Hence, the desired solution reads ds 2 = (ln ) 2 (t 2 /2+a t+a 2 )dz 2 + t (dx µ dy) dy 2. In order to uarantee the above metric is Riemannian, we assume that the constants a and a 2 satisfy a 2 < 2a 2 and is an arbitrary positive smooth function satisfyin (ln ). Case III, µ Obviously, the first equation in (39) is equivalent to the first one in (4), and they are equivalent to µ + µ µ µ =. (43) On the one hand, it follows from the second equation in (39) that c = 2 + µ 2 2 c. (44) On the other hand, by the second equation in (4) we have c = 2 µ 2 µ c µ. (45) Notin (43), we observe that (44) is equivalent to (45). Thus, we have c ( ) µ = 2 c µ 2. (46) It follows from (46) that c (z) = b 4 (µ ) 2, (47) where b is a constant. Similarly, we et c 2 (z) = b 2 4 (µ ) 2, (48) where b 2 is a constant. Therefore, the desired metric reads ds 2 = f(t, z)dz 2 + t (dx µ(z)dy) (z))dy 2, (z) (49) where f(t, z) = 2 + µ µ 2 4 (b t + b 2 ),, µ are smooth functions satisfyin (43). Moreover, in order to uarantee the metric (49) is Riemannian, we require >, b and b 2 >. Remark 3. Recently, Shu and Shen prove that Birkhoff theorem is still true for the hyperbolic eometric flow (see ). 4. Short-time existence and uniqueness. In this section we state the short-time existence and uniqueness result for the hyperbolic eometric flow () on a compact n- dimensional manifold M. We can show that the hyperbolic eometric flow is a system of second order nonlinear weakly hyperbolic partial differential equations. The deeneracy of the system is caused by the diffeomorphism roup of M which acts as the aue roup of the hyperbolic eometric flow. Because the hyperbolic eometric flow () is only weakly hyperbolic, the short-time existence and uniqueness result on a compact manifold does not come from the standard PDEs theory. However we can still prove the followin short-time existence and uniqueness theorem. Theorem. Let (M, ij (x)) be a compact Riemannian manifold. Then there exists a constant ε > such that the initial value problem 2 ij 2 (t, x) = 2R ij(t, x), ij (, x) = ij(x), ij (, x) = k ij (x), has a unique smooth solution ij (t, x) on M, ε), where kij (x) is a symmetric tensor on M. The above short-time existence and uniqueness theorem can be proved by the followin two ways: (a) usin the aue fixin idea as in Ricci flow, we can derive a system of second order nonlinear strictly hyperbolic partial differential equations, thus Theorem comes from the standard PDEs theory; (b) we reduce the hyperbolic eometric flow () to a first-order quasilinear symmetric hyperbolic system, then usin the Friedrich s theory 6 of symmetric hyperbolic system (more exactly, the quasilinear version 5) we can also prove Theorem. See Dai, Kon and Liu 2 for the details. 5. Wave property of curvatures. The hyperbolic eometric flow is an evolution equation on the metric ij (t, x). The evolution for the metric implies a nonlinear wave equation for the Riemannian curvature tensor R ijkl, the Ricci curvature tensor R ij and the scalar curvature R. Theorem 2. Under the hyperbolic eometric flow (), the curvature tensors satisfy the evolution equations 2 R ijkl 2 = R ijkl + (lower order terms), (5) 2 R ij 2 = R ij + (lower order terms), (5)

6 6 2 R = R + (lower order terms), (52) 2 where is the Laplacian with respect to the evolvin metric, the lower order terms only contain lower order derivatives of the curvatures. By (), the equations (5)-(52) come from direct calculations. See Dai, Kon and Liu 3 for the details and eometric applications. The equations (5)-(52) show that the curvatures possess interestin wave property. 6. Summary and discussion. Hyperbolic partial differential equations can be used to describe the wave phenomena in the nature. In this letter, the hyperbolic eometric flow is introduced to illustrate the wave character of the metrics, which also implies the wave property of the curvature. Note that the hyperbolic eometric flow possesses very interestin eometric properties and dynamical behavior. A direct application of Theorem 2 ives the stability of solutions to the hyperbolic eometric flow equation on the Euclidean spaces under metric perturbations (see 3). More applications of this flow to differential eometry and physics can be expected. So far there have been many successes of elliptic and parabolic equations applied to mathematics and physics, but by now very few results on the applications of hyperbolic PDEs are known (see 5). We believe that the hyperbolic eometric flow is a new and powerful tool to study eometric problems. Moreover, its physical application has been observed (see 3). In the future we will study several fundamental problems, for examples, lontime existence, formation of sinularities, as well as the physical and eometrical applications. The authors thank Chunlei He and Wenron Dai for their valuable discussions. The work of Kon was supported in part by the NNSF of China and the NCET of China; the work of Liu was supported by the NSF and NSF of China. S. Chandrasekhar, The mathematical theory of black holes, Oxford University Press, New York, W.-R. Dai, D.-X. Kon and K. Liu, Hyperbolic eometric flow (I): short-time existence and nonlinear stability, to appear. 3 W.-R. Dai, D.-X. Kon and K. Liu, The applications of the hyperbolic emotric flow, in preparation. 4 D.M. DeTurck, J. Differential Geom (983). 5 A.E. Fischer and J.E. Marsden, Comm. Math. Phys (972). 6 K.O. Friedrich, Comm. Pure Appl. Math (954). 7 R. Hamilton, J. Differential Geom (982). 8 R. Schoen and S.T. Yau, Lectures on Differential Geometry, International Press, Boston, R. Schoen and S.T. Yau, Comm. Math. Phys (979); (98); Phys. Rev. Lett (979). F.-W. Shu and Y.-G. Shen, Geometric flows and black holes, arxiv.or/abs/r-qc/63.