Non-uniqueness of infinitesimally weakly non-decreasable extremal dilatations
aa r X i v : . [ m a t h . C V ] N ov Non-uniqueness of infinitesimally weaklynon-decreasable extremal dilatations
GUOWU YAODepartment of Mathematical Sciences, Tsinghua UniversityBeijing, 100084, People’s Republic of ChinaE-mail: [email protected]
November 18, 2019
Abstract
In this paper, it is shown that a weakly non-decreasable dilatation in an infinites-imal Teichm¨uller equivalence class can be not a non-decreasable one. As an ap-plication, we prove that if an infinitesimal equivalence class contains more thanone extremal dilatation, then it contains infinitely many weakly non-decreasableextremal dilatations.
Let S be a plane domain with at least two boundary points. Denote by Bel ( S ) theBanach space of Beltrami differentials µ = µ ( z ) d ¯ z/dz on S with finite L ∞ -norm.Let Q ( S ) be the Banach space of integrable holomorphic quadratic differentials on S with L − norm k ϕ k = Z Z S | ϕ ( z ) | dxdy < ∞ . Two Beltrami differentials µ and ν in Bel ( S ) are said to be infinitesimally Teich-m¨uller equivalent if Z Z S ( µ − ν ) ϕ dxdy = 0 , for any ϕ ∈ Q ( S ) . The infinitesimal Teichm¨uller space Z ( S ) is defined as the quotient space of Bel ( S )under the equivalence relation. Denote by [ µ ] Z the equivalence class of µ in Z ( S ). Keywords: Teichm¨uller space, quasiconformal map, weakly non-decreasable, non-decreasable.2010
Mathematics Subject Classification.
Primary 30C75; 30C62.The work was supported by the National Natural Science Foundation of China (Grant No.11271216). GUOWU YAO
Especially, we use N ( S ) to denote the set of Beltrami differentials in Bel ( S ) that isequivalent to 0. Z ( S ) is a Banach space and its standard sup-norm is defined by k [ µ ] Z k = k µ k := sup ϕ ∈ Q ( S ) Re Z Z S µϕ dxdy = inf {k ν k ∞ : ν ∈ [ µ ] Z } . We say that µ is extremal (in [ µ ] Z ) if k µ k ∞ = k [ µ ] Z k , uniquely extremal if k ν k ∞ > k µ k ∞ for any other ν ∈ [ µ ] Z . Z ( S ) is a Banach space and its standard norm satisfies k [ µ ] Z k = k µ k := sup ϕ ∈ Q ( S ) Re Z Z S µϕ dxdy = inf {k ν k ∞ : ν ∈ [ µ ] Z } . We say that µ is extremal (in [ µ ] Z ) if k µ k ∞ = k [ µ ] Z k , uniquely extremal if k ν k ∞ > k µ k ∞ for any other ν ∈ [ µ ] Z .A Beltrami differential µ (not necessarily extremal) is called to be non-decreasable in its class [ µ ] Z if for ν ∈ [ µ ] Z ,(1.1) | ν ( z ) | ≤ | µ ( z ) | a.e. in S, implies that µ = ν ; otherwise, µ is called to be decreasable .The notion of non-decreasable dilatation was firstly introduced by Reich in [1] whenhe studied the unique extremality of quasiconformal mappings. A uniquely extremalBeltrami differential is obviously non-decreasable.Let ∆ denote the unit disk { z ∈ C : | z | < } and Z (∆) be the infinitesimalTeichm¨uller space on ∆. In [3], Shen and Chen proved the following theorem. Theorem A.
For every µ ∈ Bel (∆) , there exist infinitely many non-decreasable di-latations in the infinitesimal equivalence class [ µ ] Z unless [ µ ] Z = [0] Z . The author [4] proved that an infinitesimal Teichm¨uller class may contain infinitelymany non-decreasable extremal dilatations. The existence of a non-decreasable ex-tremal in a class is generally unknown.In [7], Zhou et al. defined weakly non-decreasable dilatation in a Teichm¨uller equiv-alence class and proved that there always exists a weakly non-decreasable extremaldilataion in a Teichm¨uller class. Following their defintion, we say that µ ∈ Bel ( S ) isa strongly decreasable dilatation in [ µ ] Z if there exists ν ∈ [ µ ] Z satisfying the followingconditions:(A) | ν ( z ) | ≤ | µ ( z ) | for almost all z ∈ S ,(B) There exists a domain G ⊂ S and a positive number δ > | ν ( z ) | ≤ | µ ( z ) | − δ, for almost all z ∈ G. Otherwise, µ is called weakly non-decreasable . In other words, a Beltrami differential µ is called weakly non-decreasable if either µ is non-decreasable or µ is decreasable butis not strongly decreasable. For the sake of mathematical precision, we call a Beltramidifferential µ to be a pseudo non-decreasable dilatation if it is a weakly non-decreasabledilatation but not a non-decreasable dilatation.In [6], the author proved the following theorem. on-uniqueness of infinitesimally weakly non-decreasable extremal dilatations Theorem B.
Suppose [ µ ] Z ∈ Z (∆) . Then there is a weakly non-decreasable extremaldilatation ν in [ µ ] Z . In the end of [6], the following question is posed.
Question.
Whether a weakly non-decreasable dilatation in [ µ ] Z is a non-decreasableone? In other words, the question is equivalent to ask whether there exists a pseudonon-decreasable dilatation in some [ µ ] Z .In this paper, we answer the question negatively at first. Theorem 1.
For any given λ > , the basepoint [0] Z contains infinitely many pseudonon-decreasable dilatations ν such that k ν k ∞ = λ and the support set of each ν in ∆ has empty interior. However, is the unique non-decreasable dilatation in [0] Z . By further applying Theorem 1, we prove the following results.
Theorem 2.
Suppose [ µ ] Z ∈ Z (∆) and λ > k [ µ ] Z k . Then [ µ ] Z contains infinitelymany weakly non-decreasable dilatations ν with k ν k ∞ ≤ λ . Theorem 3.
Let [ µ ] Z ∈ Z (∆) . If the extremal in the point [ µ ] Z ∈ Z (∆) is not unique,then [ µ ] Z contains infinitely many weakly non-decreasable extremal dilatations. Since Theorem 2 in [4] indicates that there exists [ µ ] Z ∈ Z (∆) such that [ µ ] Z contains infinitely many extremals but only one non-decreasable extremal, we have thefollowing corollary. Corollary 1.
There exists a Beltrami differential µ ∈ Bel (∆) such that µ is the uniquenon-decreasable extremal dilatation in [ µ ] Z while [ µ ] Z contains infinitely many pseudonon-decreasable extremal dilatations. By use of some technique in [2, 4, 5], we can obtain the following interesting theorem.
Theorem 4.
There exists an extremal Beltrami differential µ ∈ Bel (∆) such that [ µ ] Z contains infinitely many non-decreasable extremal dilatations and [ µ ] Z containsinfinitely many pseudo non-decreasable extremal dilatations. It seems that Theorem 2 is covered by Theorem A. But it is not in such a case. Onthe one hand, [0] Z contains infinitely many weakly non-decreasable dilatations while0 is the unique non-decreasable dilatation. On the other hand, after investigating theproof in [3], we find that Shen and Chen actually proved Theorem A in the followingprecise form. Theorem A’ .
Suppose [ µ ] Z = [0] Z . Then for sufficiently large λ > k [ µ ] Z k , [ µ ] Z contains infinitely many non-decreasable dilatations ν with k ν k ∞ ≥ λ . GUOWU YAO
The first lemma comes Lemma 2.2 in [6].
Lemma 2.1.
Suppose that µ ∈ Bel (∆) . Let α ∈ Bel (∆)
Then for any z ∈ ∆ and ǫ > , there exists ν ∈ [ µ ] Z and a small r > such that k ν | ∆ \ ∆( ζ,r ) k ∞ ≤ k µ k ∞ + ǫ and (2.1) ν ( z ) = α ( z ) , when z ∈ ∆( ζ, r ) = { z : | z − ζ | < r } . In particular, ν vanishes on ∆( ζ, r ) when α = 0 . In particular, ν vanishes on ∆( ζ, r ) when α = 0 . The second lemma is actually Theorem 2 in [6].
Lemma 2.2.
Suppose [ µ ] Z ∈ Z (∆) . Let χ ∈ [ µ ] Z and U = { α ∈ [ µ ] Z : | α ( z ) | ≤| χ ( z ) | a.e. on ∆ } . Then then there is a weakly non-decreasable dilatation ν in U . Lemma 2.3.
Let J i ⊂ ∆ ( i = 1 , , . . . , m ) be m Jordan domains such that J i ⊂ ∆ , J i ( i = 1 , , . . . , m ) are mutually disjoint and ∆ \ S m J i is connected. Suppose µ, ν ∈ Bel (∆) satisfying µ ( z ) = ν ( z ) a.e. on ∆ \ S mi =1 J i . Then the following two conditionsis equivalent:(a) [ µ ] Z = [ ν ] Z ,(b) [ µ | J i ] Z = [ ν | J i ] Z , where we regard [ µ | J i ] Z , [ ν | J i ] Z as the points in the infinitesimalTeichm¨uller space Z ( J i ) , ( i = 1 , , · · · , m ).Proof. See the proof of Lemma 3 of [5].For µ ∈ Bel (∆), ϕ ∈ Q (∆), let λ µ [ ϕ ] = Re Z Z ∆ µ ( z ) ϕ ( z ) dxdy. The following Construction Theorem is essentially due to Reich [2] and is very usefulfor the study of (unique) extremality of quasiconformal mappings (see [2, ? , 4, 5]). Construction Theorem.
Let A be a compact subset of ∆ consisting of m ( m ∈ N )connected components and such that ∆ \ A is connected and each connected componentof A contains at least two points. There exists a function A ∈ L ∞ (∆) and a sequence ϕ n ∈ Q (∆) ( n = 1 , , . . . ) satisfying the following conditions (2 . − (2 . : (2.2) |A ( z ) | = ( , z ∈ A, , f or a.a. z ∈ ∆ \ A, (2.3) lim n →∞ {k ϕ n k − λ A [ ϕ n ] } = 0 , (2.4) lim n →∞ | ϕ n ( z ) | = ∞ a.e. in ∆ \ A. and as n → ∞ , (2.5) ϕ n ( z ) → uniformly on A. on-uniqueness of infinitesimally weakly non-decreasable extremal dilatations Proof.
See the proof of of Construction Theorem in [5].From the Construction Theorem, we can get
Lemma 2.4.
Let A be as in Construction Theorem and A ( z ) be constructed by Con-struction Theorem. Let ν ( z ) = ( k A ( z ) , z ∈ ∆ \ A, B ( z ) , z ∈ A, where k < is a positive constant and B ( z ) ∈ L ∞ ( A ) with kBk ∞ ≤ k . Then ν ( z ) isextremal in [ ν ] and for any χ ( z ) extremal in [ ν ] Z , χ ( z ) = ν ( z ) for almost all z in ∆ \ A .Proof. See the proof of of Lemma 5 in [5].
Lemma 2.5.
Let J i ⊂ ∆ ( i = 1 , , . . . , m ) be m Jordan domains such that J i ⊂ ∆ , J i ( i = 1 , , . . . , m ) are mutually disjoint and ∆ \ S m J i is connected. Put A = S m J i . Let A ( z ) be constructed by the Construction Theorem. Let ν ( z ) = ( k A ( z ) , z ∈ ∆ \ A, B ( z ) , z ∈ A, where k < is a positive constant and B ( z ) ∈ L ∞ ( A ) with kBk ∞ ≤ k . We regard [ ν | J i ] Z as a point in the Teichm¨uller space T ( J i ) , i = 1 , , . . . , m . Then,(A) ν is a weakly non-decreasable dilatation in [ ν ] Z if and only if every ν | J i is a weaklynon-decreasable dilatation in [ ν | J i ] Z , i = 1 , , . . . , m ;(B) ν is a non-decreasable dilatation in [ ν ] Z if and only if every ν | J i is a non-decreasabledilatation in [ ν | J i ] Z , i = 1 , , . . . , m .Proof. It is evident that ν is extremal in [ ν ] Z by Lemma 2.4.(A) The “only if” part is obvious. Now, assume that every ν | J i is a weakly non-decreasable dilatation in [ ν | J i ] Z , i = 1 , , . . . , m . We show that ν is a weakly non-decreasable dilatation in [ ν ] Z . Suppose to the contrary. Then [ ν ] Z is a strongly de-creasable dilatation in [ ν ] Z . That is, there exists a Beltrami differential η ∈ [ ν ] Z suchthat(1) | η ( z ) | ≤ | ν ( z ) | for almost all z ∈ ∆,(2) there exists a domain G ⊂ ∆ and a positive number δ > | η ( z ) | ≤ | ν ( z ) | − δ, for almost all z ∈ G. Observe that η is extremal in [ ν ] Z and hence η ( z ) = ν ( z ) a.e on ∆ \ A by Lemma2.4. It forces that G is contained in some J i . Furthermore, by Lemma 2.3, we have η | J i ∈ [ ν | J i ] Z . Thus ν | J i is a strongly decreasable dilatation in [ ν | J i ] Z , a contradiction.(B) The “only if” part is also obvious. Assume that every ν | J i is a non-decreasabledilatation in [ ν | J i ] Z , i = 1 , , . . . , m . We show that ν is a non-decreasable dilatationin [ ν ] Z . Suppose to the contrary. Then [ ν ] Z is a decreasable dilatation in [ µ ] Z . Thereexists a Beltrami differential η ∈ [ ν ] Z such that | η ( z ) | ≤ | ν ( z ) | for almost all z ∈ ∆ but η ( z ) = ν ( z ) on a subset E ⊂ ∆ with positive measure. It is no harm to assume that E ∩ J has positive measure. Since η ( z ) = ν ( z ) a.e on ∆ \ A , it follows from Lemma 2.3that η | J ∈ [ ν | J ] Z . Thus, ν | J is decreasable in [ ν | J ] Z , a contradiction. GUOWU YAO
The following lemma comes from Lemma 7 in [4].
Lemma 2.6.
Set ∆ s = { z : | z | < s } for s ∈ (0 , . Let χ ( z ) be defined as follows, (2.6) χ ( z ) = ( , z ∈ ∆ − ∆ s , e k z ∈ ∆ s , where e k < is a positive constant. Then [ χ ] Z contains infinitely many non-decreasableBeltrami differentials η with k η k ∞ < e k . Proof of Theorem 1.
Let C be a compact subset with empty interior and positivemeasure meas ( C ) ∈ (0 , C = { re iθ : r ∈ C , θ ∈ [0 , π ) } . Then C is 2-dimensional compact subset in ∆ with empty interior and meas ( C ) ∈ (0 , π ).Note that N (∆) = [0] Z . It is obvious that 0 is the unique non-decreasable dilatationin N (∆). We now show that, for any given λ > N (∆) contains infinitely many pseudonon-decreasable dilatations ν with k ν k ∞ = λ and the support set of each ν in ∆ hasempty interior. Fix a positive integer number m and let γ ( z ) = ( λ z m | z | m , z ∈ C , , z ∈ ∆ \ C . Claim. γ ∈ N (∆) and is a pseudo non-decreasable dilatation in [0] Z .By the definition of N (∆), we need to show that Z Z D γ ( z ) ϕ ( z ) dxdy = 0 , for any ϕ ∈ Q (∆) . Note that { , z, z , · · · , z n , · · · } is a base of the Banach space Q (∆). It suffices to prove Z Z ∆ γ ( z ) z n dxdy = 0 , for any n ∈ N . (3.1)By the definition of C , we see that the open set A = [0 , \C is the union of countablymany disjoint open intervals. Set D = { re iθ : r ∈ A , θ ∈ [0 , π ) } . It is clear that∆ = D ∪ C and Z Z ∆ γ ( z ) z n dxdy = Z Z D γ ( z ) z n dxdy, for any n ∈ N . (3.2)Observe that D is the union of countably many disjoint ring domains each of whichcan be written in the form R = { re iθ : r ∈ ( x, x ′ ) , θ ∈ [0 , π ) } , x, x ′ ∈ (0 , λ ). A simplecomputation shows that Z Z R γ ( z ) z n dxdy = Z Z R λ z m | z | m z n dxdy = λ Z r n +1 dr Z π e i ( m + n ) θ dθ = 0 , for any n ∈ N . (3.3) on-uniqueness of infinitesimally weakly non-decreasable extremal dilatations Z Z D γ ( z ) z n dxdy = 0 , for any n ∈ N . (3.4)Thus, we have prove that γ ∈ N (∆). Since the support set of γ in ∆ has empty interior,by the definition γ is a weakly non-decreasable dilatation in [0] Z . On the other hand,it is obvious that 0 is the unique non-decreasable dilatation in [0] Z and hence γ is apseudo non-decreasable dilatation. When m varies over N or the set C varies suitably,we obtain infinitely many pseudo non-decreasable dilatations in [0] Z . The completesthe proof of Theorem 1. Proof of Theorem 4.
Choose J = { z ∈ ∆ : | z | < } and J = { z ∈ ∆ : | z − | < } . Let A = J ∪ J . Let A ( z ) be constructed by the Construction Theorem and let µ ( z ) = k A ( z ) where k < s = { z ∈ ∆ : | z | < s } where s ∈ (0 , ) and let e k ∈ (0 , k ] be a constant. Set µ ( z ) = k A ( z ) , z ∈ ∆ \ A, e k z ∈ ∆ s , , z ∈ J − ∆ s , , z ∈ J . By Theorem 1, [0 | J ] Z contains infinitely many pseudo non-decreasable dilatationsand 0 | J is the unique non-decreasable dilatation in [0 | J ] Z . Applying Lemma 2.6 to J ,we see that [0 | J ] Z contains infinitely many non-decreasable dilatations with L ∞ − normof at most k . By the foregoing reason, it derives readily from Lemma 2.5 that [ µ ] Z isthe desired Teichm¨uller class. The gives Theorem 4. To make the proof more concise, we prove a new theorem from which Theorems 2 and3 follows readily. We introduce the conception of non-landslide at first.A Beltrami differential µ (not necessarily extremal) in Bel ( S ) is said to be landslide if there exists a non-empty open subset G ⊂ S such thatesssup z ∈ G | µ ( z ) | < k µ k ∞ ;otherwise, µ is said to be of non-landslide .Throughout the section, we denote by ∆( ζ, r ) the round disk { z : | z − ζ | < r } ( r >
0) and let U λ = { α ∈ [ µ ] Z : k α k ∞ ≤ λ } . Theorem 5.
Suppose [ µ ] Z ∈ Z (∆) and λ ≥ k [ µ ] Z k . If U λ contains a landslide Beltramidifferential, then U λ contains infinitely many weakly non-decreasable dilatations.Proof. If [ µ ] Z = [0] Z and λ = 0, nothing needs to prove. Now let λ > α be a landslide dilatation in U λ . Then there is λ ′ ∈ (0 , λ ) and a sub-domain G ⊂ ∆ such that | α ( z ) | ≤ λ ′ on G . Applying Lemma 2.1 on G , we can find a Beltramidifferential χ ∈ U λ such that χ ( z ) = 0 on some small disk ∆( ζ, ρ ) ⊂ G . GUOWU YAO
By Lemma 2.2, we can find a weakly non-decreasable dilatation ν ∈ U λ such that | ν ( z ) | ≤ | χ ( z ) | a.e. on ∆. It is obvious that ν ( z ) = 0 on ∆( ζ, ρ ).Let D = ∆( ξ, r ) be a small round disk in ∆( ζ, ρ ), r ∈ (0 , ρ ). We regard [0 | D ] Z as the basepoint in the infinitesimal Teichm¨uller space Z ( D ). By Theorem 1, we maychoose a pseudo non-decreasable dilatation γ = 0 | D in [0 | D ] Z whose support set in D has empty interior such that k γ k ∞ ≤ λ . Put β ( z ) = ( ν ( z ) , z ∈ ∆ \ D,γ ( z ) , z ∈ D. If β is a weakly non-decreasable dilatation in U λ , then let ν D = β . In fact, ν D isnecessarily a pseudo non-decreasable dilatation since β | D = γ is decreasble but is notstrongly decreasable in [0 | D ] Z .Otherwise, β is not a weakly non-decreasable dilatation in U λ , and then by theproof of Lemma 2.2 there is a weakly non-decreasable dilatation β ′ in U λ such that(1) | β ′ ( z ) | ≤ | β ( z ) | ,(2) there exists a small round disk ∆( z ′ , r ′ ) ⊂ ∆ and a positive number δ > | β ′ ( z ) | ≤ | β ( z ) | − δ, for almost all z ∈ ∆( z ′ , r ′ ) . Since the support set of β on ∆( ζ, ρ ) has empty interior, it forces that ∆( z ′ , r ′ ) ⊂ ∆ \ ∆( ζ, ρ ). Let ν ′ D = β ′ . Claim. β ′ | D [0 | D ] Z where β ′ | D is the restriction of β ′ on D .Suppose to the contrary. Then β ′ | D ∈ [0 | D ] Z . Let e β ′ ( z ) = ( β ′ ( z ) , z ∈ ∆ \ D, , z ∈ D. It is clear that e β ′ ∈ U λ . Since β ( z ) = β ′ ( z ) = 0 on ∆( ζ, ρ ) \ D , it is easy to verify that(1) e β ′ ( z ) = ν ( z ) = 0 on ∆( ζ, ρ ),(2) | e β ′ ( z ) | ≤ | ν ( z ) | ,(3) | e β ′ ( z ) | < | ν ( z ) | − δ on ∆( z ′ , r ′ ).Thus, ν is strongly decreasable on ∆, a contradiction. The claim is proved.For convenience, let e ν D denote either ν D or ν ′ D .Now, let { D n = ∆( z n , r n ) } be a sequence of round disks in ∆( ζ, ρ ) which aremutually disjoint and { γ n ∈ [0 | D n ] Z : γ n = 0 | D n } be a sequence of Beltrami differentialswhose support sets have empty interior in D n respectively. By the previous analysis,we get either a pseudo non-decreasable dilatation ν D n or a weakly non-decreasabledilatation ν ′ D n in U λ . It is easy to check that whenever n = m , it holds that e ν D n = e ν D m .Thus, we get infinitely many weakly non-decreasable dilatations in U λ . Note.
If the weakly non-decreasable dilatation in U is unique, then it is necessarily anon-decreasable dilatation in [ µ ] Z . Proof of Theorems 2 and 3 . At first, let λ > k [ µ ] Z k . It is evident that [ µ ] Z contains a landslide Beltrami differential. Then by Theorem 5, there are infinitely manyweakly non-decreasable dilatations in U λ . The gives Theorem 2. on-uniqueness of infinitesimally weakly non-decreasable extremal dilatations λ = k [ µ ] Z k . Then U λ just contains all extremal dilatations in [ µ ] Z .Now assume U λ >
1. Then [ µ ] Z contains infinitely many extremal dilatations. Case 1.
Each extremal dilatation in [ µ ] Z is non-landslide.By the way, it is an open problem whether there exists [ µ ] Z such that the extremalin [ µ ] Z is not unique and each extremal in [ µ ] Z is non-landslide. Anyway, by definitioneach extremal in [ µ ] Z is weakly non-decreasable, and hence [ µ ] Z contains either infinitelymany non-decreasable extremal dilatations or infinitely many pseudo non-decreasableextremal dilatations. Case 2. [ µ ] Z contains a landslide extremal dilatation.Then by Theorem 5, there are infinitely many weakly non-decreasable dilatationsin U λ .Theorem 3 now follows. R EFERENCES [1] E. Reich,
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