Martin compactification of a complete surface with negative curvature

0.98 MB·English

by  Huai-Dong Cao

MARTIN COMPACTIFICATION OF A COMPLETE SURFACE
WITH NEGATIVE CURVATURE
HUAI-DONGCAOANDCHENXUHE
5
1
0
2

b Abstract. InthispaperweconsidertheMartincompactification,associated
e with the operator L = ∆−1, of a complete non-compact surface Σ2 with
F negative curvature. In particular, we investigate positive eigenfunctions with
eigenvalueoneoftheLaplaceoperator∆ofΣ2 andproveauniquenessresult:
6
sucheigenfunctionsareuniqueuptoapositiveconstantmultipleiftheyvanish

] on the part of the geometric boundary S∞(Σ2) of Σ2 where the curvature is
G boundedabovebyanegativeconstant,andsatisfysomegrowthestimateonthe
other part of S∞(Σ2) where the curvature approaches zero. This uniqueness
D
result plays an essential role in our recent paper [CH] in which we prove an
. infinitesimal rigidity theorem for deformations of certain three-dimensional
h
collapsedgradientsteadyRiccisolitonwithanon-trivialKillingvectorfield.
t
a
m
Contents
[

1. Introduction 1
2
v 2. Preliminaries 6
1 2.1. Martin compactification of complete Riemannian manifolds 6
5 2.2. Positive L-harmonic functions and Green’s function 7
7
3. Geometric compactification of the surface Σ2 8
3
4. The minimal Green’s function of the operator L=∆−1 12
0
. 5. Asymptotic expansions of the Green’s function 22
1
5.1. Asymptotic expansion at η =0 23
0
5 5.2. Asymptotic expansion at η =∞ 25
1 5.3. Asymptotic expansions along the rays η =m|ξ| 30
: 6. The Martin kernel and Martin boundary 33
v
i 7. Proof of Theorem 1.8 37
X
Appendix A. Singular Sturm-Liouville problem and a Spectral Theorem 39
r Appendix B. A second order linear ordinary differential equation 43
a
References 45
1. Introduction
Inthispaper,weconsidertheMartincompactificationofacompletenon-compact
surface (Σ2,ds2) with negative curvature. In particular, we investigate positive
eigenfunctionswitheigenvalueoneoftheLaplaceoperator∆of(Σ2,ds2)andprove
2000 Mathematics Subject Classification. 53C21,58G20,31C35,33C05,34B25
TheresearchofthefirstauthorwaspartiallysupportedbyNSFGrantDMS-0909581.
1
2 HUAI-DONGCAOANDCHENXUHE
a certain uniqueness result. The problem arises in our study of deformations of
three-dimensional collapsed gradient steady Ricci solitons [CH].
AclassicalresultduetoG.Herglotz[He]saysthatanypositiveharmonicfunction
u on the unit disk D2 has an integral representation
(cid:90)
u(x)= K(x,Q)dσ(Q),
S1
where K(x,Q) is the Poisson kernel of D2 and σ is a finite positive Borel measure.
A fundamental generalization of the above formula to bounded domains in Rn was
givenbyR.Martinin[Ma]. Heshowedthatananalogueoftheaboverepresentation
formula holds in complete generality, where the integral is taken over an ideal
boundary, called the Martin boundary, defined in terms of the limiting behavior
of a Green’s function. Later, Martin’s results were extended to the second order
ellipticdifferentialoperatorsoncompleteRiemannianmanifolds, see, e.g., [Ta]and
the references therein.
OnaRiemannianmanifoldofnon-positivecurvature,thereisawell-knowncom-
pactificationbyattachingitsgeometricboundarythatisdeterminedbytheasymp-
toticbehaviorofgeodesicsatinfinity,see[EO]. AcelebratedresultbyM.Anderson
andR.Schoen[AS]statesthatonanegativelycurvedmanifold(Mn,g),itsMartin
boundary with respect to the Laplace operator is homeomorphic to its geometric
boundary, provided the sectional curvature is pinched between two negative con-
stants. Thus the Martin compactification of such a manifold associated to the
Laplaceoperatorcoincideswithitsgeometriccompactification. Anderson-Schoen’s
result has been generalized by A. Ancona in [An1] to weakly coercive elliptic oper-
ators L using different techniques. In [Ba3], W. Ballmann constructed examples of
non-positively curved manifolds containing flats of dimension k ≥ 2 such that the
Martin compactification agrees with the geometric compactification.
The Cartan-Hadamard surface (Σ2,ds2) we are concerned with is diffeomorphic
to the upper half-plane R×(0,∞) with the length element given by
(1.1) ds2 = e4y+10e2y+1(cid:0)dx2+dy2(cid:1)
4(e2y−1)2
for (x,y)∈R×(0,∞). We are especially interested in characterizing non-negative
functions W =W(x,y) on Σ2 such that

W +W −P(y)W = 0
 xx yy
(1.2)
 W(x,0)=0 & ∂ W − 1coth(y)W ≥ 0,
y 2
where
e4y+10e2y+1
(1.3) P(y)= .
4(e2y−1)2
NotethattheLaplaceoperatorof(Σ2,ds2)isgivenby∆=P−1(y)(∂2+∂2),hence
x y
such a function W is in fact an eigenfunction of ∆ with eigenvalue one,
∆W =W.
Thisproblemofcharacterizingnon-negativesolutionsof(1.2)arisesinourstudyof
theinfinitesimalrigidityofdeformationsofthecollapsed3DcigarsolitonN2×R,the
productofHamilton’scigarsolitonN2 andthereallineRwiththeproductmetrics,
MARTIN COMPACTIFICATION OF A NEGATIVELY CURVED SURFACE 3
as follows: consider any one-parameter family of complete three-dimensional gra-
dient steady Ricci solitons (M3(t),g(t),f(t)) (0 ≤ t < ε), with (M3(0),g(0),f(0))
being the 3D cigar soliton N2×R, satisfying the following two conditions:
• the metric g(t) admits a non-trivial Killing vector field for all t∈[0,ε),
• the scalar curvature R(t) of g(t) attains its maximum at some point on
M3(t) for each t∈[0,ε).
Then,thefirstvariationofthesectionalcurvaturesof(M3(t),g(t),f(t))producesa
certainnon-negativefunctionW =W(x,y)onΣ2satisfying(1.2). Theinfinitesimal
rigidity of the deformation essentially reduces to proving the uniqueness of such
nonnegative eigenfunctions up to a positive constant multiple; see [CH] for the
details. This led us to consider the Martin compactification of (Σ2,ds2) associated
with the operator L=∆−1.
It turns out that the Gauss curvature of (Σ2,ds2) is negative, bounded from
below, but approaches zero along some paths to infinity. So we cannot apply the
results of Anderson-Schoen [AS] and Ancona [An1] in this case. Nevertheless, we
shall see that the Martin boundary with respect to the operator L is the same as
thegeometricboundaryof(Σ2,ds2). OurfirstresultsarethefollowingMartinand
geometric compactifications of Σ2.
Theorem1.1. TheMartincompactificationofΣ2 associatedwiththeoperatorL=
∆−1ishomeomorphictotheclosedhalfdiskΣˆ =(cid:8)(u,v)∈R2 :u2+v2 ≤1,v ≥0(cid:9).
On the Martin boundary ∂Σˆ we have
(1) the real line (cid:8)(x,y)∈R2 :y =0(cid:9) is identified with {−1<u<1,v =0};
(2) the y-axis with y →∞ is identified with the point (0,1);
(3) the asymptotic ray y = xtanθ is identified with the point ω on the semi-
θ
circle
{ω =(cosθ,sinθ):θ ∈(0,π/2)∪(π/2,π)}⊂∂Σˆ.
θ
Remark 1.2. (a) Eachω incase(3)canbeapproachedbygeodesicthatisasymp-
θ
totic to the ray y =xtanθ as x→∞.
(b) Under the topology in the Martin compactification, we have
limω =(1,0) and lim ω =(−1,0).
θ π
θ→0 θ→π
These two points can be approached by geodesics which are asymptotic to
y =log|x| as |x|→∞.
(c) We show that ∂Σˆ is the minimal Martin boundary and determine the kernel
function K(·,ω), see Proposition 6.1 and Theorem 6.6.
Theorem 1.3. The geometric compactification Σ˜ of Σ2 with the metric in (1.1) is
homeomorphic to the Martin compactification Σˆ.
The detailed description of Σ˜ is given in Theorem 3.5.
Remark 1.4. In[CL],usinganelementarymethod,L.A.CaffarelliandW.Littman
showed that for any positive solution u to the equation (∆−1)u = 0 on the
EuclideanspaceRn thereexistsauniquenon-negativeBorelmeasureµontheunit
sphere Sn−1 such that
(cid:90)
u(x)= ex·ωdµ(ω).
Sn−1
4 HUAI-DONGCAOANDCHENXUHE
Itfollowsthatthe(minimal)MartinboundaryofRn withrespectto∆−1isSn−1.
Thisissimilartothesemi-circlepartof∂Σˆ whenn=2;notethat,onΣ2,alongthe
ray y = xtanθ the Gauss curvature approaches zero as |x| → ∞. The geometric
boundary of Rn with the flat metric is also Sn−1, i.e., any point on the Martin
boundary can be reached by a geodesic.
Remark 1.5. Since (cid:0)Σ2,ds2(cid:1) is conformal to the hyperbolic plane, the Martin
boundary of Σ2 associated with the Laplace operator ∆ is given by the union
of the real line {y =0} and {∞}, the point at infinity, thus is different from the
geometric boundary ∂Σ˜. Note that the Laplace operator ∆ has zero as the bottom
of the L2-spectrum, i.e., λ (Σ2)=0 (see Remark 2.6), and is not weakly coercive.
1
The Martin compactification of a complete Riemannian manifold M with re-
specttoanoperatorLisalsorelatedtotheDirichletproblematinfinity, i.e.,given
a continuous function f on the geometric boundary S (M) of M, whether there
∞
is a unique L-harmonic function w on M such that w = f on S (M). When L
∞
is the Laplace operator ∆, the Dirichlet problem at infinity is always solvable if
M has negatively pinched curvature −b2 < K < −a2 (which was first proved
M
independently by M. Anderson [And] and D. Sullivan [Su]), or if M is one of the
examples in [Ba3]. Note that in both cases, as we mentioned before, there holds
the stronger conclusion that Martin and geometric compactifications are homeo-
morphic. We remark that the Dirichlet problem at infinity for ∆ on a symmetric
spaceM ofnoncompacttypewasinvestigatedbyH.Fu¨rstenberg[Fu];inparticular,
he showed that the problem can be solved if and only if M has rank one. See also
theearlierworkofL.-K.Hua[Hu1,Hu2,Hu3]onboundedsymmetricdomains. For
more general Cartan-Hadamard manifolds of rank one (in the sense of [Ba1, BS]),
the solvability of the Dirichlet problem at infinity was proved by Ballmann [Ba2].
Moreover,thePoissonintegralrepresentationformulawasestablishedbyBallmann-
Ledrappier [BL]. Meanwhile, H. I. Choi [Ch], Ding-Zhou [DZ], and E. P. Hsu [Hs]
haveshownthattheDirichletproblematinfinityfortheLaplaceoperator∆issolv-
able on certain negatively curved manifolds whose curvature approaches zero with
certain rate. Very recently, R. Neel [Ne] has shown that the asymptotic Dirichlet
problem for the Laplace operator on a Cartan-Hadamard surface is solvable under
the curvature condition K ≤−(1+(cid:15))/(r2logr) (in polar coordinates with respect
to a pole) outside of a compact set, for some (cid:15)>0.
However, in our case, Theorem 1.3 and Remark 1.5 imply the following
Corollary 1.6. The Dirichlet problem at infinity for the Laplace operator ∆ on Σ2
is not always solvable.
Remark 1.7. (a) We also characterize those functions f ∈ C0(cid:0)S (Σ2)(cid:1) for which
∞
the Dirichlet problem at infinity has a unique solution, see Remark 6.9.
(b)Corollary1.6givesanewexampleofaCartan-Hadamardmanifoldforwhich
theDirichletproblematinfinityisnotsolvableingeneral. Notethatthecurvature
of Σ2 is bounded from below by a negative constant, but approaches zero expo-
nentially fast as y → ∞ for each fixed x (see Remark 3.2). Previously, when the
curvatureofaCartan-Hadamardmanifoldisassumedtobeboundedfromaboveby
anegativeconstantbutnotfrombelow, Ancona[An2]constructedacounterexam-
ple to the solvability of Dirichlet problem at infinity. See also the work of Borb´ely
MARTIN COMPACTIFICATION OF A NEGATIVELY CURVED SURFACE 5
[Bo2]. On the other hand, there are several papers on the solvability of the Dirich-
let problem at infinity when the curvature lower bound has a quadratic growth
condition [HM], or certain exponential growth conditions [Bo1, Hs, Ji].
An eigenfunction w ∈ C∞(Σ) of the Laplace operator ∆ with eigenvalue one is
alsoreferredasanL-harmonicfunctionsinceLw =0. Eachboundarypointω ∈∂Σˆ
associates a kernel function K(·,ω) that is positive on Σ and L-harmonic. The
Martin integral representation theorem implies that for any positive L-harmonic
functionw,thereisa(unique)finitenon-negativeBorelmeasureν on∂Σˆ suchthat
(cid:90)
w(x,y)= K(x,y,ω)dν(ω).
∂Σˆ
From this integral representation we derive the following uniqueness result of posi-
tive eigenfunctions.
Theorem 1.8. Suppose that W is a non-negative L-harmonic function on Σ which
vanishes on the boundary {y =0} and satisfies the following inequality:
y−b
(1.4) W(a,y)≥W(a,b)e 2 when y ≥b
for some point (a,b)∈Σ2. Then either W =0, or it is a positive constant multiple
of
(ey−1)2
W (x,y)= √ .
0 e21y e2y−1
Remark 1.9. (a) Theorem 1.8 is used in [CH] to prove an infinitesimal rigidity
resultofthegradientsteadyRiccisolitonM3 =N2×R,whereN2isHamilton’s
cigar soliton [Ha].
(b) In [CH] we used the statement of Theorem 1.8 with (1.4) replaced by the
inequality
1
∂ W − coth(y)W ≥0,
y 2
which is a stronger assumption.
Nowweoutlinethemainstepsintheproofs. SincethemetricofΣ2 isexplicitwe
obtain an integral formula of Green’s function G(x,y) of L by the classical work of
E.Titchmarshin[Ti2]. TheMartinkernelfunctionisderivedbyasymptoticexpan-
sions of G along various paths. In turn, it determines the Martin compactification
of Σ. See Theorem 6.6 for Theorem 1.1 and Theorem 3.5 for Theorem 1.3, while
Corollary1.6isprovedattheendofSection6,andTheorem1.8isprovedinSection
7. We refer to the table of contents for an overview of the paper’s organization.
Acknowledgement. We are grateful to Werner Ballmann, Ovidiu Munteanu,
Christian Remling, Jiaping Wang, Xiaodong Wang and Meijun Zhu for helpful
discussions. Partoftheworkwascarriedoutwhilethefirstauthorwasvisitingthe
University of Macau, where he was partially supported by Science and Technology
DevelopmentFund(MacaoS.A.R.)GrantFDCT/016/2013/A1,aswellasRDG010
of University of Macau.
6 HUAI-DONGCAOANDCHENXUHE
2. Preliminaries
In this section we collect some basics of positive solutions to linear elliptic equa-
tionsoncompleteRiemannianmanifoldsandMartincompactificationofacomplete
Riemannian manifold (Mn,g) with respect to an elliptic operator L.
2.1. MartincompactificationofcompleteRiemannianmanifolds. Let(Mn,g)
beann-dimensionalcompletenon-compactRiemannianmanifoldandconsiderthe
operator L=∆−1. The results in this subsection hold for any second order linear
elliptic operator L with uniformly H¨older continuous coefficients and L(1)≤0, see
for example [Ta].
Denote ∆M ={(x,x):x∈M}⊂M×M the diagonal set. Recall the following
Definition 2.1. Let L be a second order linear elliptic operator on (Mn,g). A
Green’sfunction ofLisafunctionG:M×M\∆M →[0,∞)suchthatthefollowing
two equations hold:
(cid:90)
−L G(x,y)φ(y)dy =φ(x)
x
M
and
(cid:90)
− G(x,y)L φ(y)dy =φ(x)
y
M
for any smooth function φ with compact support on M.
Remark 2.2. Let {U }∞ be an exhaustion of M by relatively compact subsets
n i=1
with C2 boundary, and let G be the unique Green’s function of L on U with the
i i
Dirichlet boundary condition, then we have
G(x,y)= lim G (x,y) for all (x,y)∈M ×M\∆M
i
i→∞
whenevertheGreen’sfunctionGexists,seeforexample[Ta,Proposition5.6]. Such
a Green’s function is called minimal, see [LT] when L=∆.
We assume that L admits a Green’s function G. In the following we describe
the Martin compactification from the limiting behavior of the Green’s function G,
see also [Ba3]. Fix a reference point x ∈ M and consider the normalized Green’s
0
function

1 if x =x=y
 0
(2.1) K(x,y)=
G(x,y)
 G(x ,y) otherwise.
0
For any fixed y ∈ M, the function K(·,y) is harmonic on M\{y} and equals to 1
at x . By the Harnack principle, any sequence {x } ⊂ M with dist(x ,x ) → ∞
0 i 0 i
has a subsequence {x } such that {K(·,x )} converges. The limit is a positive
ik ik
L-harmonic function on M with value 1 at x . Now consider the space of all
0
sequences {x } in M with dist(x ,x ) → ∞ such that {K(·,x )} converges. Two
i 0 i i
such sequences {x } and {x(cid:48)} in M are equivalent if their corresponding limit
i i
functionscoincide. TheMartinboundary ∂ M isdefinedasthespaceofequivalence
L
classes. The Martin topology on MˆL = M ∪∂ M induces the given topology on
L
M and is such that a sequence {x } ⊂ MˆL converges to ω ∈ ∂ M if and only
n L
if {K(·,x )} converges to K(·,ω). The space MˆL is compact with respect to the
i
MARTIN COMPACTIFICATION OF A NEGATIVELY CURVED SURFACE 7
Martin topology and is called Martin compactification of M. The Martin topology
on MˆL is equivalent to the one induced by the following metric:
(cid:90)
d(y,z)= min{1,|K(x,y)−K(x,z)|}f(x)dv ,
x
M
where f : M → (0,1] is any positive continuous function and integrable on M.
Recall
Definition 2.3. A positive L-harmonic function w on Mn is called minimal if for
any L-harmonic function u ≥ 0, u ≤ w on M implies u = Cw for some constant
0 < C ≤ 1. A boundary point ω ∈ ∂ M is called minimal if K(·,ω) is a minimal
L
L-harmonic function. Denote ∂ M ⊂∂ M the set of all minimal boundary points.
e L
Next we collect some basic results on Martin kernels and Martin integral repre-
sentations, see, e.g., [Mu, Theorem 1.10] and [Ta, Section 6].
Theorem 2.4. Let (Mn,g) be a complete Riemannian manifold and L an elliptic
operator on M with the Green’s function G. Then,
(i) Any positive minimal L-harmonic function is a positive constant multiple of
K(·,ω) for some ω ∈∂ M.
e
(ii) ∂ M is a G set, i.e. a countable intersection of open subsets of ∂ M.
e δ L
(iii) K(x,y) is continuous on (M ×MˆL)\∆M.
(iv) For any positive L-harmonic function u, there exists a unique finite Borel
measure ν on ∂ M such that ν(∂ M\∂ M)=0 and
L L e
(cid:90)
(2.2) u(x)= K(x,ω)dν(ω).
∂eM
2.2. Positive L-harmonic functions and Green’s function. Let U ⊂M be a
boundeddomain. Then,thefirsteigenvalueof∆onU withtheDirichletboundary
condition is given by
(cid:26)(cid:90) (cid:90) (cid:27)
λ (U)=inf |∇f|2dv :suppf ⊂U, f2dv =1 .
1
U U
Denote λ (M) the bottom of the L2-spectrum of ∆, then we have
1
λ (M)= lim λ (U )
1 1 i
i→∞
where{U }∞ isanyexhaustionofM byrelativelycompactsubsetswithC2bound-
i i=1
ary. We recall the following well-known result of existence of positive L-harmonic
function for L=∆+λ, see for example, [FS, Theorem 1].
Proposition 2.5. The equation Lu = ∆u + λu = 0 for λ ∈ R has a positive
solution u on M if and only if λ≤λ (M).
1
Remark 2.6. NotethatonthesurfaceΣ2,themetricdefinedby(1.1)isasymptotic
to the flat one as y approaches infinity, so we have λ (cid:0)Σ2(cid:1)=0.
1
For an elliptic operator L, while a Green’s function always exists locally, the
existence of global Green’s function requires extra conditions. The result below
follows from [Ta, Corollary 5.13].
Proposition 2.7. A Riemannian manifold (Mn,g) admits a Green’s function if
either one of the following conditions holds:
(1) The function 1 is not L-harmonic, or
8 HUAI-DONGCAOANDCHENXUHE
(2) there are two non-proportional positive L-harmonic functions on M.
InsomespecialcasewhereMn isdiffeomorphictoRn andLu=0isaseparable
equation,TitchmarshshowedthattheGreen’sfunctionofLhasanexplicitintegral
form, see [Ti1] or [Ti2, Chapter 15].
Theorem 2.8. Let q(x,y) = q (x) + q (y) be a continuous function on R2 =
1 2
{(x,y):x,y ∈R}. Denote the Green’s functions G (x,ξ,λ) and G (y,η,λ) of the
1 2
differential operators L and L respectively with
1 2
d2 d2
L = −q (x)+λ and L = −q (y)+λ.
1 dx2 1 2 dy2 2
Assume that the spectra of L , L are bounded below at λ = α and λ = β respec-
1 2
tively. Then for (cid:60)(λ) < α+β, the Green function G(x,y,ξ,η,λ) associated with
the operator
∂2 ∂2
L= + −q(x,y)+λ
∂x2 ∂y2
has the following integral form
1 (cid:90) c+i∞
(2.3) G(x,y,ξ,η,λ)= G (x,ξ,µ)G (y,η,λ−µ)dµ
2πi 1 2
c−i∞
where the integral is taken along a straight line {c+iy} with (cid:60)(λ)−β <c<α.
Remark 2.9. (a) The Green’s function G(x,y,ξ,η,λ) is constructed using the ex-
haustionofR2 byrectanglesoffinitesize. Itagreeswiththeconstructionofthe
minimal positive Green’s function in Definition 2.1 for complete Riemannian
manifolds.
(b) Similar formulae hold when there are more than two independent variables.
3. Geometric compactification of the surface Σ2
In this section we determine all geodesics on Σ2 and then the geometric com-
pactification of Σ2, see Proposition 3.3 and Theorem 3.5.
Recall that the surface Σ2 = {(x,y)∈R×(0,∞)} has the length element, see
(1.1), of the form
ds2 =P(y)(cid:0)dx2+dy2(cid:1)
with
e4y+10e2y+1
(3.1) P(y)= .
4(e2y−1)2
Clearly, ds2 is a positive definite warped product metric on R × (0,∞) so it is
complete in x-direction for any fixed y. It is also complete as y → ∞ since P(y)
converges to 1. When y →0 we have
4
√
(cid:112) 3 y3
P(y)= + √ +O(y5),
2y 20 3
hence it follows that the metric is also complete as y → 0. Therefore, (Σ2,ds2) is
a complete surface. Moreover, its Gauss curvature is given by
96e2y(cid:0)e8y+2e6y+18e4y+2e2y+1(cid:1)
(3.2) K(y)=− <0,
(e4y+10e2y+1)3
MARTIN COMPACTIFICATION OF A NEGATIVELY CURVED SURFACE 9
thus
4
limK(y)=− and lim K(y)=0.
y→0 3 y→∞
√
Note that the minimal value K =−5 of K(y) is achieved at y =log(2+ 3).
min 3
To summarize, we have the following
Proposition 3.1. (Σ2,g), with the metric g given by (1.1), is a complete surface
with negative Gauss curvature bounded below by −5.
3
Remark 3.2. Let r(y) denote the distance function to a fixed horizontal line l, say
l = {(x,1):x∈R}. Then, by (3.1) and (3.2), we have the following asymptotic
properties:
1
r(y)∼ y and K(y)∼−96e−4r(y)
2
as y →∞. In particular, K(y) approaches zero exponentially fast when y →∞.
Figure 3.1 shows the typical geodesics on Σ2. We sketch the geodesics passing
through the y-axis at the point (0,a) with a > 0. The others can be obtained by
translation in x-direction.
• The vertical dashed blue line is of type (i) and it has constant value of x.
• The red curves are of type (ii) and they have horizontal tangent vector.
• Thegreencurvesareoftype(iii)andtheyareasymptotictoy =log|x|for
large |x|.
• The purple curves are of type (iv) and they are asymptotic to the rays
y =xtanθ for large |x| with 0<θ <π.
y
i
iv iv
iii
iii
(cid:97)
ii
ii
x
0
Figure 3.1. Typical geodesics on Σ2
For the analytic formula of the geodesics of each type, see Proposition 3.3 below.
10 HUAI-DONGCAOANDCHENXUHE
Proposition 3.3. All geodesics of Σ2 can be obtained by translation in x-direction,
thereflectionaboutthey-axisortheircombinationsfromthefollowingonesthrough
some point (0,a) with a>0:
(i) The y-axis.
(ii) The geodesic has the horizontal tangent vector at (0,a) and it is given by the
following formula
√ (cid:32)√ (cid:112) (cid:33)
e2a+10+e−2a 2coshy cosh(2a)−cosh(2y)
(3.3) x(y)=± √ tan−1 ,
4 3 cosh(2y)−sinh2a
with y ∈(0,a].
√
(iii) The geodesic has the slope of the tangent vector m= 3 at (0,a) and it is
sinha
given by the following formula
1
(3.4) x(y)= √ (coshy−cosha)
3
with y ∈(0,∞).
√
(iv) The geodesic has the slope of the tangent vector m> 3 at (0,a) and it is
sinha
given by the following formula
(3.5) x(y)=F(y)−F(a)
with y ∈(0,∞) and
(cid:112) (cid:40)
3+sinh2a
F(y) = log (−3+m2sinh2a)cosh(y)
(cid:112)
−3+m2sinh2a
(cid:41)
(cid:112) (cid:113)
(3.6) + −3+m2sinh2a (−3+m2sinh2a)cosh2y+3cosh2a+2m2sinh2a .
Remark 3.4. The geodesics of type (i), (iii) and (iv) have no horizontal tangent
vector at any point. If a geodesic has a horizontal tangent vector somewhere, then
it can be obtained by translation in x-direction from a geodesic of type (ii).
Proof. First of all, the nonzero Christoffel symbols at any point (x,y) are given by
Γ1 =Γ1 =−Γ2 =Γ2 =k(y)
12 21 11 22
with
12e2y(e2y+1)
k(y)=− .
e6y+9e4y−9e2y−1
So the geodesic equations are given by
(3.7) x(cid:48)(cid:48)(t)+2k(y)x(cid:48)(t)y(cid:48)(t) = 0
(3.8) y(cid:48)(cid:48)(t)+k(y)(cid:0)y(cid:48)(t)2−x(cid:48)(t)2(cid:1) = 0.
It is obvious that all vertical (half) lines are geodesics. Next, for any x ∈ R and
0
a > 0 we consider the geodesic γ(t) passing through γ(0) = (x ,a) ∈ Σ2 with
0
tangent vector γ(cid:48)(0) = (1,m). Since the metric is invariant under the translation
in x-coordinate and the reflection about the y-axis, we may assume that x = 0
0
and m≥0.
From equation (3.7) we get
C (e2y−1)2
(3.9) x(cid:48)(t)= 1
e4y+10e2y+1