Ricci flow of negatively curved incomplete surfaces
aa r X i v : . [ m a t h . A P ] J un ricci flow of negatively curvedincomplete surfaces Gregor Giesen and Peter M. Topping17 June 2009
Abstract
We show uniqueness of Ricci flows starting at a surface of uniformly negativecurvature, with the assumption that the flows become complete instantaneously.Together with the more general existence result proved in [10], this settles the issueof well-posedness in this class.
In 1982, Hamilton [5] introduced the study of Ricci flow, which evolves a Riemannianmetric g on a manifold M under the nonlinear evolution equation ∂∂t g ( t ) = − g ( t )] . (1.1)Hamilton proved that if M is closed (i.e. compact and without boundary) then for anyinitial metric g , there exist T > g ( t ) for t ∈ [0 , T ], with g (0) = g . We also have uniqueness: even if T is reduced, there can be no other suchflow. (See also [4].) Shi [7] and Chen-Zhu [2] generalised this to the case of noncompact M in the case that the initial metric and all flows are assumed to be complete and withbounded curvature.This theory left open the problem of starting a Ricci flow in the more general situationthat the initial metric is incomplete. This possibility springs out when one contemplates,for example, restarting a Ricci flow after a finite-time singularity has occurred in the casethat M has dimension at least 3.In [10], the second author developed a very general existence theorem for Ricci flowswhich produces (as a special case) a Ricci flow starting at any initial Riemannian surfaceof Gauss curvature bounded above – whether complete or not – which distinguishes itselfby being complete at any strictly positive time. Evidence was given in [10] to supportthe idea that this instantaneous completeness should be the right condition to guaranteeuniqueness also.In this paper, we show that this is the case under the additional assumption that theupper bound for the Gauss curvature is negative. We also demonstrate how the existenceissue is simpler in this case. Theorem 1.1.
Suppose M is any surface (i.e. a -dimensional manifold without bound-ary) equipped with a smooth Riemannian metric g whose Gauss curvature satisfies K [ g ] ≤ − η < , but which need not be complete. Then there exists a unique smoothRicci flow g ( t ) for t ∈ [0 , ∞ ) with the following properties: i) g (0) = g ;(ii) g ( t ) is complete for all t > ;(iii) the curvature of g ( t ) is bounded above for any compact time interval within [0 , ∞ ) ;(iv) the curvature of g ( t ) is bounded below for any compact time interval within (0 , ∞ ) .Moreover, this solution satisfies K [ g ( t )] ≤ − η ηt for t ≥ and − t ≤ K [ g ( t )] for t > . Some discussion of what such flows look like can be found in [10]. Generally, as t ↓ Acknowledgements:
Both authors are partially supported by The Leverhulme Trust. A priori estimates on solutions
On a surface, the Ricci curvature of a metric g takes the simple form Ric[ g ] = K [ g ] g ,where K [ g ] represents the Gauss curvature. Therefore the Ricci flow is the conformallyinvariant flow ∂g∂t = − K [ g ] g (which coincides with the Yamabe flow in this dimension).If we choose a local complex coordinate z = x + iy and write the metric locally as g = e u | d z | (where | d z | = d x + d y ) then K [ g ] = − e − u ∆ u (where ∆ := ∂ ∂x + ∂ ∂y is defined in terms of the local coordinates) and we can write the Ricci flow as ∂u∂t = e − u ∆ u = − K [ u ] , (2.1)where we abuse notation here and in the sequel by abbreviating K [e u | d z | ] by K [ u ].The first observation to make about Theorem 1.1 is that without loss of generality, wemay assume that g is a conformal metric e u | d z | on the unit disc D ⊂ C . Indeed,we can lift g to the universal cover of M , and since g has uniformly negative Gausscurvature, the conformal type of this cover must be D (rather than S or C which couldbe ruled out using the Gauss-Bonnet Theorem or by applying Corollary 2.5 below tolarge discs within C , respectively). If we can establish both existence and uniquenesson the disc, then we can be sure to be able to quotient the solution to give ultimately aunique solution on the original surface.Next we observe that by dilating g , and parabolically rescaling g ( t ) (see [9, § η = 1 in Theorem 1.1. These2onsiderations motivate the following: Definition 2.1.
A smooth Ricci flow g ( t ) on D for t ∈ [0 , T ] is called admissible ,provided(i) g ( t ) is complete for t > K [ g ] ≤ C on [0 , T ] × D ;(iii) K [ g ] ≥ − C ε on [ ε, T ] × D for all ε ∈ (0 , T ]. Lemma 2.2.
Suppose e u | d z | is a smooth metric on the disc D with K [ u ] ≤ − and e w ( t ) | d z | is an admissible Ricci flow on [0 , T ] × D with initial condition w (0) = u .Then w satisfies(A) K [ w ] ≥ − t ,(B) w ( t, x ) ≥ ln −| x | + ln(2 t ) ,on (0 , T ] × D , while on [0 , T ] × D we have(C) w ( t, x ) ≤ ln −| x | + ln (2 t + 1) ,(D) w ( t, x ) ≥ u ( x ) − Ct . The proof relies on the following special case of the Schwarz lemma of S.-T. Yau. Forconvenience we give a proof in Appendix B.
Theorem 2.3 (Schwarz-Pick-Ahlfors-Yau [11]) . Let ( M , g ) and ( M , g ) be two Rie-mannian surfaces without boundary. If(a) ( M , g ) is complete,(b) K [ g ] ≥ − a for some number a ≥ , and(c) K [ g ] ≤ − a < ,then any conformal map f : M → M satisfies f ∗ ( g ) ≤ a a g . Setting M = M to be a disc, a = a = C > f = id and either g or g to be H , thecomplete metric of constant curvature − C , one obtains barriers for metrics of uniformlynegative curvature. The most significant consequence is the following lower bound. Corollary 2.4.
Let g be a complete conformal Riemannian metric on a disc, whoseGauss curvature is bounded from below by a constant − C < . If H is the completeconformal metric of constant curvature − C on that disc, then H ≤ g. A further consequence (which also follows by a more elementary comparison argument)is an upper bound for (possibly incomplete) negatively curved surfaces:
Corollary 2.5.
Let g be a conformal Riemannian metric on a disc, whose curvature isbounded from above by a constant − C < . If H is the complete conformal metric ofconstant curvature − C on that disc, then g ≤ H. roof of Lemma 2.2. The Gauss curvature obeys the equation ∂K∂t = ∆ K + 2 K , under Ricci flow (see for example [9, Proposition 2.5.4]) and one may apply the compar-ison principle (for example [3, Theorem 12.14]) if the flow is complete and its curvatureis bounded. For ε > K [ w ( t )] restricted to the time interval t ∈ [ ε, T ]. Comparing to the solutionof the ODE ∂∂t k = 2 k with the lower curvature bound − C ε as initial condition at time t = ε , yields K (cid:2) w | [ ε,T ] (cid:3) ≥ − t − ε ) + C − ε ≥ − t − ε ) . Letting ε →
0, one obtains (A).For any time t >
0, the metric e w ( t ) | d z | is complete (i) and has curvature boundedfrom below (A). Using Corollary 2.4 one obtains directly (B), since the conformal metricof constant curvature − t on the disc is2 t (cid:18) − | x | (cid:19) | d z | . For small δ >
0, consider w | D − δ and write the conformal factor of the Ricci flow on thedisc of radius 1 − δ with Gauss curvature initially − h δ ( t, x ) := ln 2(1 − δ )(1 − δ ) − | x | + 12 ln(2 t + 1) . By Corollary 2.5 we have w | D − δ (0 , · ) ≤ h δ (0 , · ). Furthermore w | D − δ and h δ fulfil therequirements for the direct comparison principle (Theorem A.1), thus w | D − δ ≤ h δ holdsthroughout [0 , T ] × D − δ . Since h δ is continuous in δ , letting δ → w gives ∂∂t w ( t, x ) = − K [ w ( t, x )] ≥ − C, which integrates to give (D). The existence of the solution given by Theorem 1.1 is a special case of the more generalexistence theory developed in [10]. However, the proof can be streamlined in the casethat ( M , g ) is conformally hyperbolic (by which we mean that it can be made hyperbolicby a conformal change of metric) and in this section we sketch this simplified proof inthe particular case that ( M , g ) has K [ g ] ≤ − D . (Inthis paper we can reduce to this case by virtue of our uniqueness result as described inSection 2.) Theorem 3.1 (Existence, special case of [10, Theorem 1.1]) . Let g be a smooth con-formal metric on D (possibly incomplete) with K [ g ] ≤ − . Then there exists a smoothRicci flow G ( t ) on D , for t ∈ [0 , ∞ ) with G (0) = g , such that G ( t ) is complete for every t > . The Gauss curvature of this instantaneously complete solution satisfies − t < K (cid:2) G ( t ) (cid:3) ≤ − t + 1 for t > . oreover, G ( t ) is maximal in the sense that if g ( t ) for t ∈ [0 , ε ] ⊂ [0 , ∞ ) is another Ricciflow with g (0) = g , then g ( t ) ≤ G ( t ) for all t ∈ [0 , ε ] . The properties described in Lemma 2.2 also apply to these solutions.
Proof.
We follow the basic strategy of [10], constructing G ( t ) as a limit of approximatingRicci flows on smaller base manifolds. We make some simplification of the convergence,and exploit what we know about the conformal type to simplify the proof of instantaneouscompleteness in this special case.Let u : D → R be the conformal factor of g , that is, g = e u | d z | .For each k ∈ N , define D k := D − k +1 to be the disc of radius 1 − k +1 , and let h k : D k → R defined by h k ( x ) := ln k +1 (cid:0) − k +1 (cid:1) − | x | be the conformal factor of the complete conformal metric of curvature − k on D k . Notethat h k is pointwise (weakly) decreasing in the sense that for all x ∈ D , and k sufficientlylarge so that x ∈ D k , the sequence h k ( x ) is weakly decreasing.Loosely following [10], given η > R → R withthe properties that Ψ( s ) = 0 for s ≤ − η , Ψ( s ) = s for s ≥ η , and Ψ ′′ ( s ) ≥ s .Then 0 ≤ Ψ ′ ≤ s ) ≥ s for all s . We use Ψ to define the metric¯ g k = e h k − u ) g on D k , which can be viewed as a smoothed-out ‘pointwise maximum’ of the metricsrepresented by u and h k . Writing ¯ u k : D k → R for the conformal factor of ¯ g k , that is,¯ g k = e u k | d z | , we have ¯ u k ≥ u | D k and ¯ u k ≥ h k . Just as in [10, § K [¯ g k ] is bounded below (with lower bound dependent on k ) and abbreviating w k := h k − u | D k , we compute the uniform upper curvature bound K [¯ g k ] = − e − (cid:0) Ψ( w k )+ u (cid:1) ∆ (cid:0) Ψ( w k ) + u (cid:1) = − e − (cid:0) Ψ( w k )+ u (cid:1) (cid:16) Ψ ′′ ( w k ) (cid:12)(cid:12) ∇ w k (cid:12)(cid:12) + Ψ ′ ( w k )∆( h k − u ) + ∆ u (cid:17) ≤ − e − (cid:0) Ψ( w k )+ u (cid:1) (cid:0) Ψ ′ ( w k )∆ h k + (cid:0) − Ψ ′ ( w k ) (cid:1) ∆ u (cid:1) = e − (cid:0) Ψ( w k )+ u (cid:1)(cid:16) Ψ ′ ( w k ) (cid:0) e h k K [ h k ] (cid:1) + (cid:0) − Ψ ′ ( w k ) (cid:1) (cid:0) e u K [ u ] (cid:1)(cid:17) = Ψ ′ ( w k ) e − (cid:0) Ψ( w k ) − w k (cid:1) K [ h k ] + (cid:0) − Ψ ′ ( w k ) (cid:1) e − w k ) K [ u ] ≤ e − η (cid:16) Ψ ′ ( w k ) K [ h k ] + (cid:0) − Ψ ′ ( w k ) (cid:1) K [ u ] (cid:17) ≤ − e − η . In the last-but-one line we used the facts that both K [ h k ] and K [ u ] are negative, andalso that Ψ( s ) − s ≤ η where Ψ ′ ( s ) = 0 (i.e. for s ≥ − η ) and Ψ( s ) ≤ η where Ψ ′ ( s ) = 1(i.e. for s ≤ η ). The last line follows from the fact that both K [ h k ] ≤ − K [ u ] ≤ − u k are (weakly) decreasing (as the h k are decreasing) andlim k →∞ ¯ u k ( x ) = u ( x ) . Let g k ( t ) be the Ricci flow as given by Shi [7] with g k (0) = ¯ g k , on D k over a maximaltime interval [0 , T ). Since these Ricci flows are each complete with bounded curvature,the maximum principle (as in Lemma 2.2) tells us that K [ g k ( t )] ≥ − t for t > K [ g k ( t )] ≤ − t +e η for t ≥
0. In particular, we must have T = ∞ – i.e. long-timeexistence for each g k ( t ) – since the curvature is known to blow up at a singularity of aRicci flow.Let u k : [0 , ∞ ) × D k → R be the conformal factor of g k , that is, g k = e u k | d z | . Sincethe conformal factors u k (0) = ¯ u k are decreasing in k , we can compare u k | D k − ( t ) and u k − ( t ) using Theorem A.1 to find that the sequence u k ( t ) is (weakly) decreasing in thesense that at each point x ∈ D and t ≥
0, for sufficiently large k so that x ∈ D k we have u k ( t, x ) (weakly) decreasing.By virtue of (2.1) we have ∂u k ( t ) ∂t = − K [ u k ( t )] ≥ , and hence u ( x ) ≤ ¯ u k ( x ) ≤ u k ( t, x ) for all x ∈ D k and t >
0, so it makes sense to define u : [0 , ∞ ) × D → R by u ( t, x ) = lim k →∞ u k ( t, x ) , and consider the corresponding metric flow G ( t ) := e u ( t ) | d z | .By parabolic regularity theory we can see that G ( t ) will be a smooth Ricci flow inheritingthe curvature estimates of g k ( t ) and satisfying G (0) = g .To see the instantaneous completeness of G ( t ), we compare u with the conformal factor h ( t, x ) := ln −| x | + ln(2 t ) of the ‘big-bang’ Ricci flow which is the metric on D ofconstant curvature − t at time t >
0. Indeed, the comparison principle of Theorem A.1tells us that h ( t, x ) ≤ u k ( t, x ) for all x ∈ D k and t >
0, and therefore, by taking the limit k → ∞ , h ( t, x ) ≤ u ( t, x ) for all x ∈ D and t > G ( t ) follows from a similar comparison argument. If ˜ u : [0 , ε ] × D → R is the conformal factor of any other Ricci flow with ˜ u (0 , · ) = u , then the comparisonprinciple of Theorem A.1 tells us that ˜ u | D k ( t, x ) ≤ u k ( t, x ) for all x ∈ D k and t ∈ [0 , ε ],and therefore (taking k → ∞ ) ˜ u ( t, x ) ≤ u ( t, x ) for all x ∈ D and t ∈ [0 , ε ].We have almost finished, except that we appear to have constructed a flow G ( t ) for each η >
0, and each of these is guaranteed only to have its Gauss curvature bounded above by − t +e η . However, it is not hard to see that there can exist only one maximal solution,and so all of the flows G ( t ) must be identical. At this point we may take the limit η ↓ K [ G ( t )] ≤ − t +1 for t ≥
0, which completes the proof.
The following theorem states the uniqueness part of the main Theorem 1.1 and concludesits proof. 6 heorem 4.1.
Let e u | d z | be a smooth metric on the unit disk D satisfying the uppercurvature bound K [ u ] ≤ − . For some T > let e v ( t ) | d z | be an admissible Ricci flow(Definition 2.1) with v (0) = u . Then e v ( t ) | d z | is unique among such instantaneouslycomplete solutions. The proof relies on the following geometric comparison result.
Theorem 4.2 (Geometric comparison principle) . Suppose (cid:0) M , g ( t ) (cid:1) and (cid:0) M , g ( t ) (cid:1) are two conformally equivalent Ricci flows on some time interval [0 , T ] , and define Q :[0 , T ] × M → R to be the function for which g ( t ) = e Q ( t ) g ( t ) . Suppose further that g ( t ) is complete for each t ∈ [0 , T ] and that for some constant C ≥ we have(i) (cid:12)(cid:12) K [ g ] (cid:12)(cid:12) ≤ C, (ii) K [ g ] ≤ C, (iii) Q ≤ C on [0 , T ] × D . If g (0) ≤ g (0) , then g ( t ) ≤ g ( t ) for all t ∈ [0 , T ] .Proof. With respect to a local complex coordinate z , let us write g ( t ) = e u ( t ) | d z | and g ( t ) = e v ( t ) | d z | for some locally defined functions u ( t ) and ˜ v ( t ), and note then that Q = u − ˜ v . Since g ( t ) and g ( t ) are Ricci flows, we get ∂ ( u − ˜ v ) ∂t = e − u ∆ u − e − v ∆˜ v = (cid:0) e − u − e − v (cid:1) ∆ u + e − v ∆( u − ˜ v ) . Writing ∆ g ( t ) for the Laplace-Beltrami operator with respect to the metric g ( t ), weobtain, where Q > u > ˜ v ) (cid:18) ∂∂t − ∆ g ( t ) (cid:19) Q = (cid:18) ∂∂t − e − v ∆ (cid:19) ( u − ˜ v )= (cid:0) e − u − e − v (cid:1) ∆ u = ( u − ˜ v ) e − u − e − v u − ˜ v ∆ u = ( u − ˜ v )( −
2) e − ξ ∆ u = 2( u − ˜ v ) e u − ξ ) (cid:0) − e − u ∆ u (cid:1) = 2 e u − ξ ) K [ g ] ( u − ˜ v ) ≤ C C ( u − ˜ v ) = (2 C e C ) Q, where at each point, ξ was chosen between u and ˜ v according to the mean value theorem.Applying the weak maximum principle to Q (see for example [3, Theorem 12.10] with g (0) as complete background metric with bounded curvature and g ( t ) as one-parameterfamily of complete metrics) keeping in mind that Q (0 , · ) ≤
0, we conclude that Q ≤ , T ] × M as desired. Proof of Theorem 4.1.
Theorem 3.1 provides the existence of such an admissible solutione u ( t ) | d z | with u (0) = u . Let e v ( t ) | d z | be any another admissible solution with thesame initial condition v (0) = u . From Theorem 3.1 we know that u ( t ) is maximal amongsuch instantaneously complete solutions, and in particular, v ( t ) ≤ u ( t ) for all t ∈ [0 , T ].Hence it remains to show the converse inequality u ( t ) ≤ v ( t ).Let C > v : K [ v ( t )] ≤ C for all t ∈ [0 , T ], which exists since v is admissible. For small δ ∈ (0 , T ) define˜ v ( t, x ) := v (cid:0) e − Cδ ( t + δ ) , x (cid:1) + Cδ for ( t, x ) ∈ [0 , T − δ ] × D ,which is a slight adjustment of v , again a solution to the Ricci flow: (cid:18) ∂∂t ˜ v − e − v ∆˜ v (cid:19) ( t, x ) = e − Cδ (cid:18) ∂∂t v − e − v ∆ v (cid:19) (cid:0) e − Cδ ( t + δ ) , x (cid:1) = 0 . u is a lower bound for ˜ v , and hence (by taking δ ↓
0) also a lowerbound for v as desired. To do this, we wish to apply Theorem 4.2 to the Ricci flows g ( t )and g ( t ) generated by the conformal factors u and ˜ v respectively.First, note that g ( t ) is complete for all t ∈ [0 , T − δ ] since e v | d z | is an admissible Ricciflow and is therefore complete for all t ∈ (0 , T ]. Furthermore, g ( t ) has upper and lowercurvature bounds: (cid:12)(cid:12) K [˜ v ] (cid:12)(cid:12) ≤ sup [e − Cδ δ,T ] ×D e − Cδ (cid:12)(cid:12) K [ v ] (cid:12)(cid:12) < ∞ , (4.1)so hypothesis (i) of Theorem 4.2 is satisfied. The upper bound for the curvature of g ( t ) =e u | d z | required by hypothesis (ii) of Theorem 4.2 follows since g ( t ) was constructedto be admissible.Next we verify hypothesis (iii) of Theorem 4.2, namely that u − ˜ v is bounded from above.Applying (C) of Lemma 2.2 to e u | d z | , we find that u ( t, x ) ≤ ln 21 − | x | + 12 ln (2 t + 1) , (4.2)for t ∈ [0 , T ], while (B) of Lemma 2.2 applied to e v | d z | gives v ( t, x ) ≥ ln 21 − | x | + 12 ln (2 t ) , for t ∈ (0 , T ] and hence that˜ v ( t, x ) ≥ ln 21 − | x | + 12 ln (cid:0) − Cδ ( t + δ ) (cid:1) + Cδ = ln 21 − | x | + 12 ln (cid:0) t + δ ) (cid:1) (4.3)for t ∈ [0 , T − δ ]. Subtracting (4.3) from (4.2), we find that u − ˜ v ≤
12 ln (2 T + 1) −
12 ln (2 δ )as desired.The final hypothesis of Theorem 4.2 to verify is that g (0) ≤ g (0), i.e. that u (0 , · ) ≤ ˜ v (0 , · ). But ˜ v (0 , · ) = v (cid:0) e − Cδ δ, · (cid:1) + Cδ ≥ u − C e − Cδ δ + Cδ ≥ u = u (0 , · )by part (D) of Lemma 2.2 as desired.We may therefore apply Theorem 4.2 over the time interval [0 , T − δ ] to deduce that u ( t ) ≤ ˜ v ( t ) for all t ∈ [0 , T − δ ]. Hence, given any ( t, x ) ∈ [0 , T ) × D , we conclude u ( t, x ) ≤ lim δ ↓ ˜ v ( t, x ) = v ( t, x ) . A Comparison principle
In this appendix we clarify the statement and proof of one of the many variants of thestandard weak maximum principle. 8 heorem A.1 (Direct comparison principle) . Let Ω ⊂ R be an open, bounded domainand for some T > let u ∈ C , (cid:0) (0 , T ) × Ω (cid:1) ∩ C (cid:0) [0 , T ] × ¯Ω (cid:1) and v ∈ C , (cid:0) (0 , T ) × Ω (cid:1) ∩ C (cid:0) [0 , T ] × Ω (cid:1) both be solutions of the Ricci flow equation (2.1) for the conformal factorof the metric. Furthermore, suppose that for each t ∈ [0 , T ] we have v ( t, x ) → ∞ as x → ∂ Ω . If v (0 , x ) ≥ u (0 , x ) for all x ∈ Ω , then v ≥ u on [0 , T ] × Ω .Proof. For every ε > v ε ( t, x ) := v (cid:18) ε ln( εt + 1) , x (cid:19) + 12 ln( εt + 1) for all ( t, x ) ∈ [0 , T ] × Ω , which is well-defined since ε ln( εt + 1) ≤ t for all t ≥
0. Observe that v ε is a slightmodification of v , with v ε (0 , · ) = v (0 , · ), and v ε converges pointwise to v as ε →
0, butin contrast to v it is a strict supersolution of the Ricci flow (2.1): (cid:18) ∂∂t v ε − e − v ε ∆ v ε (cid:19) ( t, x ) = 1 εt + 1 (cid:18) ∂∂t v − e − v ∆ v (cid:19) (cid:18) ε ln( εt + 1) , x (cid:19) + ε εt + 1)= ε εt + 1) > t, x ) ∈ [0 , T ] × Ω . (A.1)We are going to prove ( v ε − u ) ≥ , T ] × Ω and conclude the theorem’s statementby letting ε →
0. Since by hypothesis u is continuous on [0 , T ] × ¯Ω and v ε ( t, · ) blows upnear the boundary ∂ Ω for each t ∈ [0 , T ], we have( v ε − u )( t, x ) → ∞ as x → ∂ Ω , for every time t ∈ [0 , T ] and hence ( v ε − u )( t, · ) attains its infimum in Ω. Now assumethat ( v ε − u ) becomes negative in [0 , T ] × Ω, and define the time t at which ( v ε − u )first becomes negative by t := inf n t ∈ [0 , T ] : min x ∈ Ω ( v ε − u )( t, x ) < o ∈ [0 , T ) . Picking any minimum x ∈ Ω of ( v ε − u )( t , · ), we have( v ε − u )( t , x ) = 0 , ∆( v ε − u )( t , x ) ≥ ∂∂t ( v ε − u )( t , x ) ≤ . Subtracting the Ricci flow equation (2.1) from (A.1) at this point ( t , x ), we find0 < (cid:18) ∂∂t v ε − e − v ε ∆ v ε (cid:19) ( t , x ) − (cid:18) ∂∂t u − e − u ∆ u (cid:19) ( t , x )= ∂∂t ( v ε − u )( t , x ) − e − u ( t ,x ) ∆( v ε − u )( t , x ) ≤ , which is a contradiction. Therefore v ε ≥ u on [0 , T ] × Ω, and the corresponding statementfor v follows by letting ε → B Yau’s Schwarz Lemma
For convenience, we prove now the Schwarz lemma of Yau (Theorem 2.3). The proofuses the following generalised maximum principle by Omori, whose proof was simplifiedby Yau in [12, Theorem 1, p. 206]: 9 heorem B.1. [6, Theorem A ′ , p. 211] On a complete Riemannian surface ( M , g ) withGaussian curvature bounded from below, let f be a C -function which is bounded above.Then, for an arbitrarily point p ∈ M and for any ε > , there exists a point q dependingon p such that(i) ∆ g f ( q ) < ε ; (ii) |∇ f ( q ) | g < ε ; (iii) f ( q ) ≥ f ( p ) . The essential idea [12] to find q is to imagine the point (cid:0) q, f ( q ) (cid:1) on the graph of f in M × R which is closest to the point ( p, k ) for some enormous k ≫ Proof of Theorem 2.3.
By dilating g and g , we may assume a = 1 = a , that is K [ g ] ≥ − ≥ K [ g ]. Since we only need the theorem in the case that f is strictlyconformal, we will assume this in the proof and leave the minor adjustments required forthe full theorem to the reader. Define w ∈ C ∞ ( M ) by f ∗ ( g ) = e w g . (B.1)It remains to show that w ≤
0. Assume instead that there exists p ∈ M with w ( p ) > ε ∈ (0 ,
1) such that ε < e w ( p ) − e − w ( p ) w ( p ) . (B.2)Now define ˜ w ( x ) := − e − w ( x ) for all x ∈ M . Since ( M , g ) is complete with curvaturebounded from below and ˜ w is bounded above, we may apply Theorem B.1 to find a point q ∈ M with ∆ g ˜ w ( q ) < ε, |∇ ˜ w | g ( q ) < ε and ˜ w ( q ) ≥ ˜ w ( p ) . (B.3)Since x
7→ − e − x is strictly increasing, we also have w ( q ) ≥ w ( p ) >
0. Now compute∆ g ˜ w = e − w ∆ g w − e − w |∇ w | g = e − w ∆ g w − e w |∇ ˜ w | g . (B.4)By computing with respect to a local complex coordinate, we find that∆ g w = − e w K [ g ] ◦ f + K [ g ] , which together with the curvature estimates − K [ g ] ≥ K [ g ] ≥ − − w ∆ g w ≥ e w − e − w , and so (B.4) improves to∆ g ˜ w ≥ e w (1 − |∇ ˜ w | g ) − e − w . Evaluating at q , using (B.3) and the fact that w ( q ) ≥ w ( p ), we obtain ε > ∆ g ˜ w ( q ) ≥ e w ( q ) (1 − ε ) − e − w ( q ) ≥ e w ( p ) (1 − ε ) − e − w ( p ) and hence ε > e w ( p ) − e − w ( p ) w ( p ) which contradicts (B.2). In the weakly conformal case f might either be constant (nothing to prove) or have isolated singularpoints P := { p , p , . . . } . The function w we define in (B.1) will then have logarithmic singularities on P , but will be strictly negative close to such singularities and the ˜ w of the proof could be adjusted to asmooth function on the whole of M (including P ) without altering anything where w is positive. eferences [1] M. Bertsch, R. Dal Passo and
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Comm. Pureand Appl. Math. (1975), 201–228. mathematics institute, university of warwick, coventry, CV4 7AL, uk Giesen: [email protected]: