In this section, we consider the geometric properties of solutions of the hyperbolic geo-metric flow on Riemann surface of general initial value. We first give some special explicit solutions of the reduced equation of the hyperbolic geometric flow to highlight some special forms of the solutions. Then we derive the reduced equation of the hyperbolic geometric flow in more general conformal class and give some results about certain important quan-tities like the total mass or volume function which illustrate some geometric obstructions to the long time existence of the reduced problems, as well as the existence of periodic solutions.
Let us consider a simple case, let (M, g) be a complete Riemann surface with a metric of conformal type
g=ρ(x, y)2 dx2+dy2 .
Here we can assume thatM is globally conformal toR2 orT2. If we look for the solutions of the hyperbolic geometric flow (1.1) or (1.12) in this conformal class, as in Section 1, we can reduce this system explicitly to
∂2u
∂t2 − 4lnu= 0, (6.1)
whereu=ρ2 and4is the standard Laplacian. This can be done by a simple observation that the Ricci curvature can be written in terms of the Gaussian curvature asRic(gt) = Kgt, and the Gaussian curvature can be written as K=−ρ124lnρ.
Settingw= lnu, same as (2.9) we have
∂2w
∂t2 +
∂w
∂t
2
−e−w4w= 0. (6.2)
The equation (6.2) is reminiscent of the traditional wave map and semilinear hyperbolic equation in two dimension. In general, it is not easy to handle globally. We first search for some special solutions.
Example 6.1 (Solutions of separation variables)We look for solutions of (6.2) with the following form
w=f(t) +g(x, y). We have
d2f dt2 +
df dt
2
−e−f(t)e−g(x,y)4g= 0,
this means
ef(t) d2f dt2 +
df dt
2!
=e−g(x,y)4g=c,
wherec is a constant independent oft, x, y. So we can solve these two equations by ef(t)= c
2t2+at+b, and
4g=ceg(x,y), (6.3)
where a, b are arbitrary real numbers, and g(x, y) is a solution of (6.3) which is the famous Liouville equation which has arisen in complex analysis and differential geometry on Riemann surfaces, in particular, in the problem of prescribing curvature. We just notice that when the Riemann surface is compact, there is no solution to (6.3) unless
c= 0.
Example 6.2 (Solutions of traveling wave type)We look for solutions of (6.2) of the type
w=f(x−at),
wherea is a real number, then we have the following ordinary differential equation ef a2f00+a2f02
−f00= 0, which can be solved explicitly in the following implicit form
a2ef −f =c1x+c2.
Let (M, g0) be a Riemann surface with metricg0. We shall call metricsg0 andg point-wise conformally equivalent if g = ug0 for some positive function u ∈ C∞(M), whereas we say that metrics g0 and g are conformally equivalent if there is a diffeomorphism Φ of M and a positive function u ∈ C∞(M) such that Φ∗(g) =ug. Although we assume that the Riemann surface is globally conformal to R2 in the above simple setting, we can investigate the hyperbolic geometric flow in the same pointwise conformal class in more general setting. Let (M, g0) be a Riemann surface with metric g0, we would like to search for solutions of the hyperbolic geometric flow in the pointwise conformal class (M, u(t,·)g0), where u(t,·) is a function with parameter t.
To make the problem more tangible, we first prove the following lemma.
Lemma 6.1 Let g0 and g be point-wise conformally equivalent metrics on the Riemann surface M such that g=ug0 for a positive smooth function u, and k(·) is the Gaussian curvature of (M, g0), then the Gaussian curvature of (M, ug0) is
K(x, y) = 1 u
k(x, y)−1
24g0lnu
, (6.4)
where (x, y) ∈ M is any local parameter and 4g0denotes the Laplacian with respect to metric g0.
Proof. Let
ω10, ω20 be a local oriented orthonormal coframe field on (M, g0). If we set ωi =√
uωi0, then {ω1, ω2} is a local oriented orthonormal coframe field for g. It is worth to notice now that the Gaussian curvature k of (M, g0) is determined by the following equation
kω10∧ω20 =dϕ012, whereϕ012 denotes the Riemannian connection formg0.
Now we compute the connection ϕ12 form of (M, u(·)g0). Let du = u1ω10 +u2ω02. Then
dω1 = 1
2√
udu∧ω10−√
uϕ012∧ω02
=−u2
2uω01− u1
2uω02+ϕ012
∧ω2. Thus
ϕ12= u2
2uω10− u1
2uω20+ϕ012=ϕ012− 1
2∗dlnu, where∗ denotes the Hodge star operator. So we conclude
Kω1∧ω2 =dϕ12=dϕ012−1
2d∗dlnu=kω10∧ω20−1
24(lnu)ω10∧ω20, and the result follows immediately.
From (6.4) we can reduce the hyperbolic geometric flow (1.1) in a point-wise conformal class to
∂2u
∂t2 =4g0lnu−2k(x, y), (6.5) and moreover we can consider the Cauchy problem for (6.5)
∂2u
∂t2 =4g0lnu−2k(x, y),
t= 0 : u=u0(x, y), ut=u1(x, y),
(6.6)
whereu0 is a positive function and u1 is an arbitrary function on M.
Let us now consider (6.5) or (6.6). As investigated by Kong and Liu [5], we know that the hyperbolic geometric flow on three dimension has a deep relation with the Einstein equations. It is a challenging problem to find periodic solutions and blowing up phenomena of the hyperbolic geometric flow as in the Einstein equation case. It is interesting that in two dimensional case we can prove the following theorem which gives some obstructions to the existence of smooth long time solutions and periodic solution of (6.6).
In order to state our result, we introduce
Definition 6.1 A solution of (6.6) is regular, if it is positive and smooth.
Theorem 6.1 Let (M, g0) be a compact Riemannian surface with metric g0, χ(M) de-notes its Euler characteristic number. If u(t, x, y) is a regular solution of (6.6). We have
(a) Ifχ(M)>0, then any solution of the reduced problem (6.6) must blow up in finite time for any initial value (u0, u1);
(b) If χ(M)6= 0, there is no periodic regular solution of equation (6.5). Moreover in the case χ(M) = 0, if there is a periodic solution u(t, x, y) of (6.5), then we must have
R
M
u(t, x, y)dVg0 =c for some positive constant c;
(c) Ifχ(M) = 0, andu(t, x, y)is a regular solution of (6.6). Then if R
M
u1(x, y)dVg0 <
0, u(t, x, y) must blow up in finite time.
Proof. Let u(t, x, y) be a regular solution of (6.5) or (6.6). We denote the volume ofM with respect to the metric u(t, x, y)g0 by
V (t) = Z
M
u(t, x, y)dVg0.
Then taking integration on both sides of (6.5) and using Gauss-Bonnet formula, we have d2V (t)
dt2 =−4πχ(M), this means
V (t) =−2πχ(M)t2+c1t+c2 (6.7) for some constants c1 and c2. If we consider the Cauchy problem (6.6), we can easily get the numbers c1 and c2, c2 =
Z
M
u0(x, y)dVg0 and c1 = Z
M
u1(x, y)dVg0. From (6.7) we
conclude (a) by the assumptionχ(M)>0. Otherwise, for some sufficiently large number T, we have V (T)≤0, which is impossible.
To prove (b), we just notice that ifu(t, x, y) is a periodic solution of (6.5), thenV (t) must be a periodic function of parameter t, which is impossible unless χ(M) = 0 and c1 = 0 in (6.7).
The conclusion (c) can also be concluded from the expression (6.7). In this case we have
V (t) =c1t+c2, but
c1 = Z
M
u1(x, y)dVg0 <0,
sou(t, x, y) must blow up in finite time. This proves Theorem 6.1.
From the above theorem, we see that problem (6.6) or equation (6.5) have essential geometric obstruction to the existence of long time solution or periodic solution. However, as in the study of Ricci flow on Riemann surfaces, we consider the following normalized equation for the equation (6.5)
∂2u
∂t2 =4g0lnu−2k(x, y) +4πχ(M) V (g0) , t= 0 : u=u0(x, y), ut=u1(x, y),
(6.8)
whereV (g0) denotes the volume ofM with respect to metricg0. We have
Theorem 6.2 Let (M, g0) be a compact Riemann surface with metricg0,χ(M) denotes its Euler characteristic number. If u(t, x, y) is a regular solution of (6.8), then we have
(i) If Z
M
u1(x, y)dVg0 < 0, then the volume function V (t) is linearly contracting and u(t, x, y) must blow up in a finite time;
(ii) If Z
M
u1(x, y)dVg0 >0, then the volume functionV (t)is linearly expanding.
This theorem can be proved in a manner similar to the proof of (6.7), here we omit the details.
Theorem 6.2 reveals that the hyperbolic geometric flow has very different features from the traditional Ricci flow, because even for the normalized equation (6.8) we can not get the long-time existence of solutions which depends on the velocity u1. This also gives an evidence that, by choosing the initial velocity of the hyperbolic geometric flow, we may
obtain the long time existence of the corresponding solution. We believe that the above results may reveal some interesting features of the Einstein equations about the expansion of the Universe, if one could connect the foliation of solution of the Einstein equations in three dimension with the solution of (6.6) or (6.8). We will pursue this problem in a forthcoming paper.
Acknowledgements.
The work of Kong was supported in part by the NNSF of China (Grant No. 10671124) and the NCET of China (Grant No. NCET-05-0390); the work of Liu was supported in part by the NSF and NSF of China.References
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