Non-variational extrema of exponential Teichmüller spaces
aa r X i v : . [ m a t h . C V ] O c t Non-variational extrema of exponentialTeichm¨uller spaces
Gaven Martin Cong Yao ∗ to Pekka Koskela on the ocassion of his 60 th birthday Abstract
The exponential Teichm¨uller spaces E p , 0 ≤ p ≤ ∞ , interpolatebetween the classical Teichm¨uller space ( p = ∞ ) and the space ofharmonic diffeomorphisms ( p = 0). In this article we prove the exis-tence of non-variational critical points for the associated functional:mappings f of the disk whose distortion is p -exponentially integrable,0 < p < ∞ , yet for any diffeomorphism g ( z ) of D with g | ∂ D = identity and g = identity we have f ◦ g is not of p -exponentially integrable dis-tortion. In 1939 Teichm¨uller stated his famous theorem [8]: In the homotopy classof a diffeomorphism between closed Riemann surfaces, there is a unique ex-tremal quasiconformal mapping of smallest maximal distortion K ( z, f ). Fur-thermore, this function f is either conformal or has a Beltrami coefficient µ f = f z /f z of the form µ f ( z ) = k Ψ( z ) | Ψ( z ) | , (1)where Ψ is a holomorphic function and 0 ≤ k < ∗ Both author’s research supported in part by the New Zealand Marsden FundThis work forms part of the second authors PhD Thesis. L p -norm of the distortion and letting p → ∞ . What is remarkable hereis that in Ahlfors’ approach the p -minimisers have essentially no regularityor good topological properties - they may not even be continuous (and 70years later we are just beginning to establish some of these facts) but theassociated holomorphic Hopf differentials converge as p → ∞ to yield theBeltrami coefficient in the limit.In our work [7], following initial explorations in [5], we developed theexponential Teichm¨uller spaces where instead of minimising the maximaldistortion K ( z, f ) or its L p -norm we minimize the p -exponential norm E p [ f ] = Z Σ e p K ( z,f ) dz (2)where f is in a given homotopy class or f has fixed boundary data.Due to the many recent and significant advancements in the theory ofmappings of finite distortion (see [3] as a starting point) we were able to showthat for each p a homeomorphic minimiser f p exists, that as p → ∞ we obtainthe extremal quasiconformal mapping (giving a new proof for Teichm¨uller’stheorem) as a limit f p → f ∞ and that as p → f
7→ E p [ f ] forthe functional defined at (2) on the disk, so Σ = D and f | ∂ D = f a givenhomeomorphism. Lifting to the universal cover will give the result on arbi-trary Riemann surfaces. In particular our main result here is the following. Theorem 1
Let < p < ∞ . Then there is a homeomorphism f : D → D such that E p [ f ] = Z D e p K ( z,f ) dz < ∞ , ∈ W , loc ( D ) , the Sobolev space of mappings with locally square integrablefirst derivatives, and which has the following property: If g : D → D is adiffeomorphism which is not conformal, then E p [ f ◦ g ] = + ∞ For the variational problem we would want the diffeomorphism to be theidentity on the boundary, but that is not necessary here. In fact our solutionis entirely local. If U ⊂ D is open and g | U is not conformal, then we willprove that Z U e p K ( z,f ◦ g ) dz = + ∞ The reader will easily see that this is covered in the reduction to the followinglemma and its proof, however we do make some comments. We set thelemma up in a variational fashion to be more useful in other applications.The restriction 0 < | t | < k∇ ϕ k < Lemma 1
Let < p < ∞ . Then there is a homeomorphism f : D → D suchthat E p [ f ] = Z D e p K ( z,f ) dz < ∞ ,f ∈ W , loc ( D ) , the Sobolev space of mappings with locally square integrablefirst derivatives, and which has the following property. Let ϕ ∈ C ∞ ( D ) bea non-zero smooth test function with k∇ ϕ k L ∞ ( D ) < . For all < t < set f t ( z ) = f ( z + tϕ ( z )) . Then f t : D → D is a homeomorphism with f t | ∂ D = f | ∂ D , f t ∈ W , loc ( D ) , E q [ f t ] < ∞ for all q ≤ q t < q with q t → p as t → , However E p [ f t ] = + ∞ , < t < . We start with the function F ( r ) = 1 r log ( r ) , < r < . (3)3his function satisfies Z F ( r ) rdr < ∞ , and for any q > , Z F q ( r ) rdr = + ∞ . The indefinite integral can be evaluated in terms of special functions, butwith a bit of work one can find Z F ( r ) r dr = 1log 2 . (4)We would therefore like e K ( z,f ) = er log ( r ) , r = | z | . (5)and so K ( z, f ) = 1 − r log r ) . Given an increasing surjective function ρ : [0 , → [0 , radial stretching f ( z ) = z | z | ρ ( | z | ) . From [3, Section 2.6]) we compute the Beltrami coefficient µ f ( z ) = zz | z | ˙ ρ ( | z | ) − ρ ( | z | ) | z | ˙ ρ ( | z | ) + ρ ( | z | ) . (6)Then, K ( z, f ) = 1 + | µ ( z ) | − | µ ( z ) | = ( r ˙ ρ + ρ ) + ( r ˙ ρ − ρ ) ( r ˙ ρ + ρ ) − ( r ˙ ρ − ρ ) = 12 ( r ˙ ρρ + ρr ˙ ρ ) . Equivalently r ˙ ρρ = K ( z, f ) ± p K ( z, f ) − . Choosing the larger answer, which is no smaller than 1, we have r ˙ ρρ = 1 − r log 2 r ) + r [1 − r log 2 r )] − , log ρ ( r ) = Z r − s log s ) + q [1 − s log s )] − s ds,ρ ( r ) = exp[ Z r − s log s ) + q [1 − s log s )] − s ds ] . ρ (1) = e = 1, and log ρ (0) ≤ R
01 1 s ds = −∞ , so ρ (0) = 0as required. This gives us a self-homeomorphism of D f ( z ) = z | z | ρ ( | z | ) , (7)whose distortion K ( z, f ) is exponentially integrable, E [ f ] < ∞ . As f is ahomeomorphism with exponentially integrable distortion, it extends homeo-morphically to the boundary circle.Next we recall the composition formula. Given g t : D → D , K ( g t ( z ) , f ◦ ( g t ) − ) = K ( z, f ) K ( z, g t )[1 − ℜ e ( µ f µ g t )(1 + | µ f | )(1 + | µ g t | ) ] . We are interested in the sign of the term ℜ e ( µ f µ g t ). If it is non-positive, wethen have K ( g t ( z ) , f ◦ ( g t ) − ) ≥ K ( z, f ) K ( z, g t ) . Observe that for each pair of complex numbers z = a + bi and w = c + di ,we have ℜ e ( z ¯ w ) = ac + bd. Thus, at least one of the following is non-positive: ℜ e ( z ¯ w ) , ℜ e ( zw ) , ℜ e ( − z ¯ w ) , ℜ e ( − zw ) . Next, for a radial stretching f ( z ) = z | z | ρ ( | z | ), µ f ( z ) = zz | z | ˙ ρ ( | z | ) − ρ ( | z | ) | z | ˙ ρ ( | z | ) + ρ ( | z | ) , µ f ( iz ) = iziz | z | ˙ ρ ( | z | ) − ρ ( | z | ) | z | ˙ ρ ( | z | ) + ρ ( | z | ) = − µ f ( z ) ,µ f (¯ z ) = zz | z | ˙ ρ ( | z | ) − ρ ( | z | ) | z | ˙ ρ ( | z | ) + ρ ( | z | ) = µ f ( z ) , µ f ( i ¯ z ) = i ¯ zi ¯ z | z | ˙ ρ ( | z | ) − ρ ( | z | ) | z | ˙ ρ ( | z | ) + ρ ( | z | ) = − µ f ( z ) . Given ϕ ∈ C ∞ ( D ) with k∇ ϕ k L ∞ ( D ) < g t ( z ) = z + tϕ ( z ) isa diffeomorphism of D which is the identity in a neighbourhood of ∂ D . For | t | < µ g t = tϕ z tϕ z is smooth and compactly supported on D .5ssume that ℜ e ( µ g t (0)) = 0 and ℑ m ( µ g t (0)) = 0. By choosing r ′ smallenough, there is a neighbourhood A = D (0 , r ′ ) in which ℜ e ( µ g t ) and ℑ m ( µ g t )do not change their signs, and further inf z ∈ A | µ g t ( z ) | ≥ ε > . Then, q := inf z ∈ A K ( z, g t ) > . Also, J ( z, g t ) = | tφ z | − t | φ z | shows us that J ( z, g t ) ≥ c t > Z D exp[ K ( w, f ◦ ( g t ) − )] dw = Z D exp[ K ( z, f ) K ( z, g t ) − ℜ e ( µ f µ g t )(1 − | µ f | )(1 − | µ g t | ) ] J ( z, g t ) dz ≥ c t Z A exp[ K ( z, f ) · inf z ∈ A K ( z, g t )] dz = c t Z A [ e | z | log ( | z | ) ] q dz = c t Z π dθ Z r ′ [ er log ( r ) ] q · rdr ≥ πc t Z r ′ [ er log ( r ) ] q dr = + ∞ . We now consider the condition ℜ e ( µ g t (0)) ℑ m ( µ g t (0)) = 0. Since µ g t ( z ) = tϕ z ( z )1+ tϕ z ( z ) we can compute ℜ e ( µ g t ) = 12 (cid:16) tϕ z tϕ z + tϕ z tϕ z (cid:17) = t ℜ e ( ϕ z ) + t ℜ e ( ϕ z ϕ z ) | tϕ z | , ℑ m ( µ g t ) = 12 i (cid:16) tϕ z tϕ z − tϕ z tϕ z (cid:17) = t ℑ m ( ϕ z ) + t ℑ m ( ϕ z ϕ z ) | tϕ z | . So the condition ℜ e ( µ g t (0)) ℑ m ( µ g t (0)) = 0 is satisfied only if ℜ e ( ϕ z (0)) ℑ m ( ϕ z (0)) = 0 . (8)We record this as follows. Theorem 2
The Sobolev homeomorphism f defined at (7) has exponentiallyintegrable distortion K ( z, f ) . However, for any ϕ ∈ C ∞ ( D ) with • k∇ ϕ k < , ℜ e ( ϕ z (0)) ℑ m ( ϕ z (0)) = 0 , • g t defined as above,then for any t ∈ ( − , , t = 0 , Z D exp[ K ( w, f ◦ ( g t ) − )] dw = + ∞ . To complete the proof of the lemma, and thus of the main theorem, we haveto remove the condition ℜ e ( ϕ z (0)) ℑ m ( ϕ z (0)) = 0. We will have to modify f to do so. To this end we need the following lemma: Lemma 2
In the unit disk D , there is a countable dense subset { z k } , disjointBorel sets S , S , S , S , and a positive number δ > , such that at everypoint z k , there is an R k > that for any < r < R k and i = 1 , , , , | S i ∩ D ( z k , r ) || D ( z k , r ) | > δ. We postpone the proof of this lemma to the next section. Now let F ( r ) beas defined in (3). This time we define the distortion function as K ( z, f ) = 1 + 1 p log h X k k F ( | z − z k | ) χ D ( z k ,dist ( z k ,∂ D )) i , where { z k } ⊂ D is a dense subset as in Lemma 2. Then exp[ p K ( z, f )] ∈ L ( D ). Indeed, using (4) we see Z D exp[ p K ( z, f )] dz ≤ X k πe p k Z F ( r ) rdr = 2 πe p log 2 < ∞ . The absolute value of the Beltrami coefficient that corresponds to K ( z, f ) is | µ f ( z ) | = s K ( z, f ) − K ( z, f ) + 1 . (9)By the existence theorem for mappings with exponentially integrable distor-tion, [3, Theorem 20.4.9], we may set µ f ( z ) = | µ f ( z ) | e iθ ( z ) for any measurable7unction θ : D → [0 , π ) and then find a homeomorphism f : D → D thathas Beltrami coefficient µ f on D . As above, using the composition formula K ( g t ( z ) , f ◦ ( g t ) − ) = K ( z, f ) K ( z, g t )[1 − ℜ e ( µ f µ g t )(1 + | µ f | )(1 + | µ g t | ) ] . (10) ℜ e ( µ f µ g t ) = 1 | tϕ z | (cid:16) t ℜ e ( ϕ z µ f ) + t ℜ e ( ϕ z µ f ϕ z ) (cid:17) . (11)We now determine the argument of µ f . Let S , S , S , S as in Lemma 2.3.2,and set µ f ( z ) = | µ f ( z ) | , z ∈ S ; −| µ f ( z ) | , z ∈ S ; i | µ f ( z ) | , z ∈ S ; − i | µ f ( z ) | , z ∈ D − S i =1 S i ⊃ S . So each has density δ near every z i . We put ±| µ f | and ± i | µ f | respectivelyinto the above and get ℜ e ( µ f µ g t ) = | µ f || tϕ z | (cid:16) t ℜ e ( ϕ z ) + t ℜ e ( ϕ z ϕ z ) (cid:17) , z ∈ S ; (12) ℜ e ( µ f µ g t ) = − | µ f || tϕ z | (cid:16) t ℜ e ( ϕ z ) + t ℜ e ( ϕ z ϕ z ) (cid:17) , z ∈ S ; (13) ℜ e ( µ f µ g t ) = | µ f || tϕ z | (cid:16) t ℑ m ( ϕ z ) + t ℑ m ( ϕ z ϕ z ) (cid:17) , z ∈ S ; (14) ℜ e ( µ f µ g t ) = − | µ f || tϕ z | (cid:16) t ℑ m ( ϕ z ) + t ℑ m ( ϕ z ϕ z ) (cid:17) , z ∈ S , (15)and we recall here that 0 < | t | <
1. There are now three cases to considerdepending on ϕ : (1) ϕ z ≡ D . Since ϕ ∈ C ∞ ( D ), this happens only when ϕ ≡ D , andthen g t ( z ) = z , f ◦ ( g t ) − = f . (2) Suppose ℜ e ( ϕ z ) is not the constant 0, say ℜ e ( ϕ z )( z ) > z ∈ D . Then by the smoothness of ϕ there is an open neighbourhood U where ℜ e ( ϕ z ) ≥ ε >
0. Following the basic example, for any t ∈ ( − ,
0) in8 ∩ S we have | µ g t | ≥ ε t > ℜ e ( µ f µ g t ) < , and J ( z, g t ) = | tφ z | − t | φ z | > c t . Then, as before, K ( g t ( z ) , f ◦ ( g t ) − ) ≥ q K ( z, f ) . The density of { z k } implies there is z k ∈ U . We choose a small disk D ( z k , r ) ⊂ A , where r < R k as in Lemma 2. Following our earlier ar-guments we now can compute Z D exp[ p K ( w, f ◦ ( g t ) − )] dw = Z D exp[ p K ( g t ( z ) , f ◦ ( g t ) − ) J ( z, g t )] dz ≥ Z D ( z k ,r ) ∩ S exp[ pq K ( z, f )] dz ≥ δ Z D ( z k ,r ) exp[ pq K ( z, f )] dz ≥ πδ Z r ( e p k F ( r )) q rdr ≥ C Z r F q ( r ) rdr = ∞ . If ℜ e ( ϕ z )( z ) <
0, then of course the same result follows. (3)
Suppose ℑ m ( ϕ z ) is not the constant 0. Then the result follows in anentirely similar fashion.This last observation completes the proof. (cid:3) In fact we can generalise Lemma 1 to the complex coefficient case. Thatis, set g η = z + ηϕ, η ∈ C , ϕ ∈ C ∞ ( D ) . In this case ℜ e ( µ f µ g η ) = 1 | ηϕ z | (cid:16) ℜ e ( ηϕ z µ f ) + ℜ e ( η ϕ z µ f ϕ z ) (cid:17) . (16)Again we choose µ f and S , S , S , S same as before. Write η = | η | e iα , then ℜ e ( µ f µ g η ) = | µ f || ηϕ z | (cid:16) | η |ℜ e ( e iα ϕ z ) + | η | ℜ e ( e iα ϕ z ϕ z ) (cid:17) , z ∈ S , and analogously for S , S , S . 9hen, for any non-constant ϕ , we may find a neighbourhood in D whereeither ℜ e ( e iα ϕ z ) or ℑ m ( e iα ϕ z ) is nonzero. So by the same argument as above Z D exp[ p K ( g η ( z ) , f ◦ ( g η ) − )] dz = ∞ . For every fixed ϕ , the number ε depends only on α . If we let α move on[0 , π ], then e iα ϕ z and e iα ϕ z ϕ z move continuously and thus ε = ε ( α ) can bechosen as a continuous function of α . Since [0 , π ] is compact, ε ( α ) admits apositive minimal value. Then we have established the following theorem. Theorem 3
For every p > , there is a homeomorphism f : D → D suchthat exp[ p K ( z, f )] ∈ L ( D ) , and for any non-constant ϕ ∈ C ∞ ( D ) and forany η ∈ C \ { } , exp[ p K ( g η ( z ) , f ◦ ( g η ) − )] / ∈ L ( D ) , where g η = z + ηϕ . Proof.
There is really only one thing to note, and that is that the gradientestimate we used above, namely k∇ ϕ k L ∞ ( D ) <
1, and the condition t ∈ [ − , g η is a homeomorphism. The do not affect thedivergence of the integrand exp[ p K ( g η ( z ) , f ◦ ( g η ) − )] in L ( D ). (cid:3) As an easy consequence, the result claimed in Theorem 1 now followsdirectly. This is because Theorem 3 is entirely local and any diffeomorphism g : D → D which is the identify on the boundary of the disk, and not equalto the identity, must have a point z ∈ D where the distortion is greater than1. Choose a C ∞ ( D ) function ϕ , ϕ ( z ) = 1 near z and note that g ( z ) = z + ϕ ( z )( g ( z ) − z )near z . The result then follows from Theorem 3. (cid:3) As a first step, we start with the point p = (0 ,
0) and choose the disk sectors S = { z ∈ D ( p , ) : 0 < arg( z − p ) < π } ,S = { z ∈ D ( p , ) : π < arg( z − p ) < π } ,S = { z ∈ D ( p , ) : π < arg( z − p ) < π } , = { z ∈ D ( p , ) : 3 π < arg( z − p ) < π } . We construct inductively. At Step n ≥
2, we choose the points p nj,l =( j n − , l n − ), for any integers j, l ∈ [1 − n − , n − −
1] such that D ( p nj,l , n ) ⊂ D , and p nj,l has not been chosen in the previous steps. Define the sector unions S n = [ j,l { z ∈ D ( p nj,l , n ) : 0 < arg( z − p nj,l ) < π } ,S n = [ j,l { z ∈ D ( p nj,l , n ) : π < arg( z − p nj,l ) < π } ,S n = [ j,l { z ∈ D ( p nj,l , n ) : π < arg( z − p nj,l ) < π } ,S n = [ j,l { z ∈ D ( p nj,l , n ) : 3 π < arg( z − p nj,l ) < π } . Write S n = S n ∪ S n ∪ S n ∪ S n . We now define S i as the set that z ∈ S ni for some n but not in S m for any m ≥ n + 1. Precisely, S i = ∞ [ n =1 (cid:16) S ni ∩ ∞ \ m = n +1 ( S m ) c (cid:17) , i = 1 , , , . We claim that the points p , p nj,l and the sets S i satisfy the requirements.We estimate the total area of S n ≥ S n . At each Step n , we have no morethan 2 n points, and each disk has area π n . Thus (cid:12)(cid:12)(cid:12) ∞ [ n =1 S n (cid:12)(cid:12)(cid:12) ≤ ∞ X n =1 n n π < π . (17)Fix any point p = p nj,l . Then { z ∈ D ( p, n ) : 0 < arg( z − p ) < π } ⊂ S n . r < n be an arbitrary number. Let N be the largest integer that N > r . Consider the sector F := { z ∈ D ( p, r ) : 0 < arg( z − p ) < π } ⊂ S n . Note that by the choice of N , F ∩ S m = ∅ for any n + 1 ≤ m ≤ N −
1. Sowe consider the disk D ( p, N ). Analogously we have | D ( p, N ) ∩ [ n ≥ N S n | ≤ N · π . On the other hand, by the choice of N we have N +1 ≤ r < N . So | F | ≥ · π N +2 . Thus | D ( p, r ) ∩ S || D ( p, r ) | ≥ | F ∩ S || D ( p, N ) | ≥ π N +4 − π N +5 π N = 132 . It is symmetric for S , S , S . This completes the proof of the lemma. (cid:3) References [1] L.V. Ahlfors,
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