Local Lipschitz geometry of weighted homogeneous surfaces

LOCAL LIPSCHITZ GEOMETR Y OF WEIGHTED HOMOGENE OUS SURF A CES L. BIRBRAIR AND A. FERNANDES Abstract. W e compute Hoelder Complexes,i.e. the complete bi-Lipsc hitz inv ariants, for germs of real weighe d homogeneous algebraic or semi algebraic surfaces. 1. Introduction A basic question of Metric Theory of Sing ularities is Lipsch itz Classifica tio n of Singular Sets. Some recent results of sev eral authors are devoted t o Lipschitz inv ariants of semialgebraic or alg e braic sets with singularities. (See, for example, [1],[3],[4],[5],[7]). H¨ older Complex es, constructed in [1], are complete bi-Lipschitz inv ariants for germs of semia lg ebraic surfaces. Lˆ e Dung T ra ng asked the following natural ques - tion: what is the r e lation b etw een Lipschitz inv aria nt s and the algebra ic nature of the semia lg ebraic s ets? In this pap er w e give a complete answer to this question for weight ed homo geneous surfac e s in R n , i.e. we compute the ex p o nen ts in H¨ older Complexes of these se ts . In order to compute these exp o nen ts we consider weigh ted homogeneous singu- lar foliations in R n and prov e that the cor resp onding H¨ older E xp onents can b e computed in terms o f order s of contact of leav es of such a foliation. W e would like to tha nk Profes s or Lˆ e Dung T ra ng fo r a very interesting ques tion and very imp ortant comments on a preliminar y version of this pap er and P rofessor F uensant a Aro ca for interesting discus s ions. 2. Preliminaries and main resul ts W e are going to recall a definition of Cano nic a l H¨ older Complex presented in [1]. Date : August 30, 2021. 1 2 L. BIRBR AIR AND A. FERNANDES An A bstr act H¨ older Complex is a pair ( Γ , β ), where Γ is a finite graph, E Γ is the set o f edges o f Γ a nd β : E Γ → Q is a rational v alued function such that for each g ∈ E Γ , w e have β ( g ) ≥ 1. A v ertex a ∈ V Γ is ca lled smo oth or artificial if a is connec ted with exac tly tw o edge s and these edges connect a with exactly tw o vertices of Γ. A vertex a is called a lo op ve rtex if a is co nnected with exactly tw o edges and these edg es connect a with the s ame vertex of Γ. An Abstr act H¨ older Complex (Γ , β ) is called Canonic al or Simplifie d if (1) Γ has no a rtificial vertices; (2) for an y lo op v ertex connected with tw o edges g 1 and g 2 we hav e β ( g 1 ) = β ( g 2 ). The Standar d H¨ older T riangle T β is a semialgebraic subset of R 2 defined as follows: T β = { ( x, y ) ∈ R 2 : 0 ≤ y ≤ x β , 0 ≤ x ≤ 1 } , where β ≥ 1 is a rational n umber. A semialgebra ic set X ⊂ R n is called a Ge ometric H¨ older Complex asso ciate d t o (Γ , β ) if (1) there exis ts a homeo morphism Φ : cone (Γ) → X , where cone (Γ) is a co ne ov er Γ; (2) for any edge g ∈ E Γ , the image of the set cone ( g ) ⊂ cone (Γ) b y the map Φ is semialgebra ically bi-Lipschitz equiv alent, with resp ect to the inner metric, to T β , where β = β ( g ). If Ψ : Φ( cone ( g )) → T β is the corres p o nding bi- Lipschitz map, then Ψ( x 0 ) = 0 , where x 0 ∈ X is the image of the vertex of cone (Γ) by the ma p Φ. Theorem 2.1. [1] L et X ⊂ R n b e a close d semialgebr aic set of di mension 2 and let x 0 ∈ X . Then ther e exist s a unique ( up to isomorphism) Canonic al H¨ older Complex (Γ , β ) such that, for sufficiently smal l ǫ > 0 , X ∩ B ( x 0 , ǫ ) is a Ge ometric H¨ older Complex, asso ciate d to (Γ , β ) . Let a 1 ≥ a 2 ≥ · · · ≥ a n be a finite se q uence o f p ositive integer num bers . A Weighte d homo gene ous foliation F ( a 1 ,...,a n ) in R n with the weigh ts a 1 ≥ a 2 ≥ · · · ≥ a n is a singula r fo lia tion defined as a family of curv es ( t a 1 x 1 , . . . , t a n x n ), where x = ( x 1 , . . . , x n ) ∈ R n and t ∈ (0 , + ∞ ). The S tandar d N ewton Simplex , a sso ciated LOCAL LIPSCHITZ GEOMETR Y OF WEIGHTED HOM OGENEOUS SURF ACES 3 to a w eighted homogeneo us folia tio n F ( a 1 ,...,a n ) is the conv ex hull of the p oints ( a 1 , 0 . . . , 0) , (0 , a 2 , 0 , . . . , 0) , . . . , (0 , . . . , 0 , a n ). All the 1-dimensio nal fa c e s of the Standard Newton Simplex b elong to subspac e s R 2 ij = span { (0 , . . . , a i , 0 , . . . , 0) , (0 , . . . , a j , 0 , . . . , 0) } . The quotient a i a j ( i < j ) is called the dir e ction of ( i, j )-1-dimensiona l face o f a Standard Newton Simplex. Let x 6 = 0 b e a point in R n . Denote by γ x the clos ure of the leaf o f F ( a 1 ,...,a n ) passing thr ough the p o int x . A set X ⊂ R n is called ( a 1 , . . . , a n ) -weighte d homo ge- ne ous if, for all x ∈ X , we hav e γ x ⊂ X . Example 2. 2. Let f : R 3 → R b e a weigh ted ho mo geneous p oly nomial w ith re s pe c t to weight s a ≥ b ≥ c . Then X = f − 1 (0) is an ( a, b, c )- weigh ted ho mogeneous algebraic set. Example 2.3. Let F : R m → R n be a p o ly nomial map with co ordinate functions F = ( f 1 , . . . , f n ) such that f 1 , . . . , f n are weigh ted homogene o us p olyno mials with degrees ( d 1 , . . . , d n ). Then F ( R m ) is a ( d 1 , . . . , d n )-weigh ted ho mogeneous semial- gebraic s et. Prop ositio n 2. 4. L et X ⊂ R n b e a semi algebr aic ( a 1 , . . . , a n ) -weighte d homo ge- ne ous subset. Then S ing ( X ) is also an ( a 1 , . . . , a n ) -weighte d homo gene ous set. Pr o of. Let us c onsider a ma p ϕ t : R n → R n defined as fo llows: ϕ t ( x 1 , . . . , x n ) = ( t a 1 x 1 , . . . , t a n x n ) . Note that a r estriction of this ma p to R n − { 0 } is a diffeomor phism, for all t > 0. If X ⊂ R n is a semialg ebraic ( a 1 , . . . , a n )-weigh ted ho mo geneous subset, th en, for all t > 0, ϕ t ( X ) = X . Thu s, ϕ t ( S ing ( X )) = S ing ( X ) and hence S ing ( X ) is ( a 1 , . . . , a n )-weigh ted homogeneous set.  A clos ed semialg e braic set X ⊂ R n is called a semialgebr aic su rfac e if dim X = 2. Theorem 2.5. L et X ⊂ R n b e a semialgebr aic ( a 1 , . . . , a n ) -weighte d homo gene ous surfac e. L et (Γ , β ) b e the Canonic al H¨ o lder Complex of X at 0 . Then, for e ach g ∈ E Γ , β ( g ) = 1 or is one of the ( i, j ) -dir e ctions of t he Standar d Newton Simplex asso ciate d to F ( a 1 ,...,a n ) . 4 L. BIRBRAIR AND A. FERNANDES F or the case of weigh ted ho mogeneous sur faces in R 3 , we will prove the following result. Theorem 2.6. L et X ⊂ R 3 b e a semialgebr aic ( a 1 , a 2 , a 3 ) -weighte d homo gene ous surfac e. If 0 ∈ X is an isolate d singular p oint and the lo c al link of X at 0 is c onne cte d, then t he germ of X at 0 is bi-Lipschitz e quivalent, with r esp e ct t o the inner metric, to a germ at 0 of a β - horn, i.e. a surfac e define d as fol lows: H β = { ( x 1 , x 2 , y ) ∈ R 3 : ( x 2 1 + x 2 2 ) = y 2 β } , wher e β is e qu al to 1 or to a 2 a 3 . 3. Order cont a ct of semialgebraic arcs. Weighted ho mogeneous f olia tion s Recall that a semialg ebraic ar c γ at a p o int x 0 ∈ R n is image of a s emiagebraic map ¯ γ : [0 , ǫ ) → R n such that ¯ γ (0) = x 0 and ¯ γ ( s ) 6 = 0, for s 6 = 0. Let γ 1 , γ 2 be tw o semialgebra ic arcs at x 0 . These ar cs ca n b e r eparametriz e d near x 0 in the following form: γ i ( t ) = { x ∈ γ i : k x − x 0 k = t } ; i = 1 , 2 . Let ρ ( t ) = k γ 1 ( t ) − γ 2 ( t ) k . Since ρ is a semialg ebraic function, we hav e ρ ( t ) = at λ + o ( t λ ) , where λ is a rational n umber bigger or equal to 1 and a > 0. The num ber λ is called the or der of c ontact of γ 1 and γ 2 . W e use the no tation λ ( γ 1 , γ 2 ). Set e γ i ( t ) = { x ∈ γ i : k x − x 0 k max = t } ; i = 1 , 2 . Let e ρ ( t ) = k e γ 1 ( t ) − e γ 2 ( t ) k max . Recall that k x k max = max {| x 1 | , . . . , | x n |} . Since e ρ is als o a semialg ebraic function, we hav e e ρ ( t ) = e at e λ + o ( t e λ ) , where ˜ λ is a rationa l num be r bigg er or equal to 1 and ˜ a > 0 . Prop ositio n 3 .1. The n u mb er e λ , define d ab ove, is e qu al to λ ( γ 1 , γ 2 ) . W e ar e g o ing to prove this prop osition in Section 6. LOCAL LIPSCHITZ GEOMETR Y OF WEIGHTED HOM OGENEOUS SURF ACES 5 Prop ositio n 3 .2. [2] L et γ 1 , γ 2 and γ 3 b e thr e e s emialgebr aic ar cs at x 0 ∈ R n . L et λ ( γ 1 , γ 2 ) ≥ λ ( γ 2 , γ 3 ) ≥ λ ( γ 1 , γ 3 ) , then λ ( γ 2 , γ 3 ) = λ ( γ 1 , γ 3 ) . The main r e s ult o f this section is the following. Theorem 3.3. L et F ( a 1 ,...,a n ) b e a weighte d homo gene ous foliatio n in R n . F or al l x 6 = y in R n , t he or der c ontact λ ( γ x , γ y ) is e qual t o 1 or to a dir e ction of a 1-dimensional fac e of t he Standar d N ewton Simplex asso ciate d to F ( a 1 ,...,a n ) . Pr o of. Let us pro ceed by induction on n . First, w e consider the ca se n = 2 . In this case, all the leaves of this foliation can b e presented in o ne of the following for ms: (1) x 1 ≥ 0, x 2 = ax α 1 , where a ∈ R and α = a 1 a 2 ; (2) x 1 ≤ 0, x 2 = a | x 1 | α , where a ∈ R and α = a 1 a 2 ; (3) x 1 = 0, x 2 ≤ 0; (4) x 1 = 0, x 2 ≥ 0. Using Prop o s ition 3.1 o ne can show that λ ( γ x , γ y ) is equal to 1 or to α . Suppo se that the statement is true fo r all the weighted homo geneous foliatio ns F ( a 1 ,...,a k ) in R k for k < n . Consider a foliation F ( a 1 ,...,a n ) . Note that the restric- tion of F ( a 1 ,...,a n ) to the h yp erplane x n = 0 is a weighted homogeneo us folia tion F ( a 1 ,...,a n − 1 ) in R n − 1 . Thus, fo r any t w o curves γ y , γ z belo ngs to the h yp erplane, the statement is true, by the induction hypotheses . Thus, we ca n suppo se that γ y , γ z are chosen in such a w ay that z = ( z 1 , . . . , z n ) and z n 6 = 0. Let y = ( y 1 , . . . , y n ). If y n = 0 , then the unit tangent vector at zero to γ y belo ng to the hyperplane x n = 0 and the unit ta ngent vector to γ z do es not b elong to this hyperplane. Thus, λ ( γ y , γ z ) = 1. If y and z belo ng to different sides of the hyperpla ne x n = 0 , then their unit ta ngent vectors at zero cannot co incide. Again, in this c a se, λ ( γ y , γ z ) = 1 . Now, we suppos e that z n > 0 and y n > 0 (the case z n < 0 and y n < 0 can be trea ted in the same wa y). Cons ider the parametriza tion ¯ γ y ( t ) of γ y and ¯ γ z ( t ) of γ z defined in the b eginning of this section. W e hav e ¯ γ y ( t ) = ( t a 1 a n ¯ y 1 , . . . , t a n − 1 a n ¯ y n − 1 , t ) and ¯ γ z ( t ) = ( t a 1 a n ¯ z 1 , . . . , t a n − 1 a n ¯ z n − 1 , t ) 6 L. BIRBRAIR AND A. FERNANDES where ¯ y = ( ¯ y 1 , . . . , ¯ y n − 1 , 1 ) a nd ¯ z = ( ¯ z 1 , . . . , ¯ z n − 1 , 1 ) a re the intersections of γ y and γ z , resp ectively , with the hyper plane x n = 1. W e obtain e ρ ( t ) = m ax { t a 1 a n | ¯ y 1 − ¯ z 1 | , . . . , t a n − 1 a n | ¯ y n − 1 − ¯ z n − 1 |} . Hence, λ ( γ y , γ z ) c an b e equa l to a 1 a n , . . . , a n − 1 a n . The theorem is prov ed.  R emark 3.4 . By the constructio n, it is clear t hat, for all the pairs ( i > j ), there exists a pair of curves γ y and γ z such tha t λ ( γ y , γ z ) = a i a j . 4. H ¨ older exponents. Horn exponents. A semialg e braic surface X ⊂ R n is ca lle d a β -H¨ older T riangle at x 0 ∈ X if the ger m o f X at x 0 is s e mia lgebraica lly bi-Lipschitz equiv alent to a g erm of the standard β -H¨ older T riangle T β ⊂ R 2 , with resp ect to the inner metric, and the image of the p oint x 0 by the corresp onding bi-Lipschit z map is the p oint (0 , 0) ∈ R 2 . The inv erse images of the bo undary curves of T β , containing (0 , 0) ar e called sides of the β -H¨ older T riangle X . The num b er β is called the H ¨ older Exp onent o f X at x 0 . W e use the no tation β ( X , x 0 ). A semia lgebraic surface X ⊂ R n is called a β -Horn a t a p o int x 0 ∈ X if the germ of X at x 0 is semia lgebraica lly bi-Lipschitz e quiv a lent to the germ a t (0 , 0 , 0) ∈ R 3 of the standa r d β -Horn, i.e. a semialgebra ic set defined as follows: H β = { ( x 1 , x 2 , y ) ∈ R 3 : ( x 2 1 + x 2 2 ) = y 2 β } , with resp ect to the inner metric, and the image of the p oint x 0 by the cor resp onding bi-Lipschitz ma p is the p oint (0 , 0 , 0) ∈ R 3 . The num ber β is called the Horn Exp onent of X at x 0 . W e a re g oing to use the same notation β ( X , x 0 ). The follo wing result is useful for calculations o f H¨ o lder Exp onents and Horn Exp onents. Theorem 4.1. L et X ⊂ R n b e a semialgebr aic s urfac e. L et x 0 ∈ X b e a p oint such that X is a β -H¨ older T riangle at x 0 or a β -Horn at x 0 . Then β ( X , x 0 ) = inf { λ ( γ 1 , γ 2 ) : γ 1 , γ 2 ar e semialgebr aic ar cs on X with γ 1 (0) = γ 2 (0) = x 0 } . Pr o of. W e are going to pr ov e the s tatement for a β -H¨ o lder T riangle. The pr o of for a β -Ho rn is the same. Let X ⊂ R n be a β - H¨ older T riangle at x 0 ∈ X . By the LOCAL LIPSCHITZ GEOMETR Y OF WEIGHTED HOM OGENEOUS SURF ACES 7 main re sult of [1] (se e also [6]), there exists a finite set of semialgebr aic arcs a t x 0 , { γ 1 , . . . , γ k } , γ i ⊂ X for all i = 1 , . . . k , such that (1) γ i , γ i +1 are sides o f a β i -H¨ older T riangle X i ⊂ X , where β i = λ ( γ i , γ i +1 ); (2) γ j ∩ X i = x 0 if j 6 = i and j 6 = i + 1; (3) X i ∩ B ( x 0 , ǫ ) is nor mally embedded in R n , for sufficiently small ǫ > 0. By the simplification theorem o f [1], β ( X , x 0 ) = min β i . Let α 1 , α 2 ⊂ X b e tw o semia lgebraic arcs at x 0 . Since α 1 and α 2 are semialg e- braic, then there exist tw o subsets X j 1 and X j 2 , defined a bove, such that α 1 ⊂ X j 1 and α 2 ⊂ X j 2 . W e c an supp os e that j 1 < j 2 . By P rop osition 3 .2, we obta in λ ( α 1 , α 2 ) = min { λ ( α 1 , γ j 1 +1 ) , λ ( γ j 1 +1 ,γ j 1 +2 ) , . . . , λ ( γ j 2 , α 2 ) } . By the sa me rea son, λ ( α 1 , γ j 1 +1 ) ≥ β j 1 and λ ( γ j 2 , α 2 ) ≥ β j 2 . By these thr ee inequa lities , we o btain λ ( α 1 , α 2 ) ≥ β ( X , x 0 ). On the other ha nd, there exists a pair γ i , γ i +1 , such that β ( X , x 0 ) = β i = λ ( γ i , γ i +1 ).  5. Canon ical H ¨ older Complex for weighted ho mogeneous surf ace s. This section is devoted to a pro of of Theorem 2.5. W e use induction on the dimension o f the a m bient space R n . Let X ⊂ R 2 be a closed semialgebra ic surface which is ( a 1 , a 2 )-weigh ted ho mo- geneous. If X 6 = R 2 , then X is a collec tio n of some H¨ older T ria ng les X 1 , . . . , X p such tha t X i ∩ X j = { 0 } if i 6 = j . By Pro po sition 2.4, we hav e tw o p ossibilities: (1) the boundar y curv es of X i belo ng to S ing ( X ) and, thus, the bo undary curves γ 1 and γ 2 are leav es of the weighted homogeneous foliation; (2) X i is a ” half ” o f a weigh ted homogeneo us β -Horn, in this case we can a ls o suppo se that the b oundar y cur ves of X i are leav es of this foliation. Thu s, X i is a ( a 1 , a 2 )-weigh ted homogeneous s et. If a β - H¨ o lder T r iangle X i int er- sects with the set x 2 = 0 o nly a t { 0 } , then β ( X i , 0 ) = a 1 a 2 . Otherwise, β ( X i , 0 ) = 1 . Let us obs e rve that β ( R 2 , 0 ) = 1 . The firs t step of induction is done. Let X ⊂ R n be a se mialgebraic ( a 1 , . . . , a n )-weigh ted homogeneous sur fa ce. Let X i be a β - H¨ o lder T riangle cor resp onding to the Ca nonical Co mplex of X at 0. If 8 L. BIRBRAIR AND A. FERNANDES X i belo ngs to the hyperpla ne x n = 0, then the statement is true, by the induction hypothesis. Thus, let us s uppo se that X i do es not b elong to the hyp e rplane x n = 0, i.e. there ex is ts a curve γ z ⊂ X i such that γ z ∩ { x n = 0 } = { 0 } . No w, if X i ∩ { x n = 0 } 6 = { 0 } , then there exists a curve γ y ⊂ X i ∩ { x n = 0 } . The curves γ z and γ y hav e differen t unit ta ngent v ectors at 0 ∈ R n and, thus, β ( X i , 0 ) = 1. Now w e consider the ca se when X i ∩ { x n = 0 } = { 0 } . W e are going to show that, for any pair of semialg ebraic arcs α 1 , α 2 ⊂ X i with same initial po int 0 ∈ R n , there exists a pair o f leav es γ z 1 and γ z 2 such that λ ( α 1 , α 2 ) ≥ λ ( γ z 1 , γ z 2 ). In order to prov e this statement, we need the following lemma. Lemma 5.1. L et Y ⊂ R n b e a ( a 1 , . . . , a n ) weighte d ho mo gene ous surfac e, such that Y − { 0 } is c onne ct e d. L et Y ∩ { x n = 0 } = { 0 } . Supp ose t hat t he se ction Y ∩ { x n = 1 } is c ontaine d in the plane { x i = r } . Then, for every p ositive value ǫ , ther e exists a value r ( ǫ ) such that the se ction Y ∩ { x n = ǫ } is c ontaine d t o the plane x i = r ( ǫ ) . Pr o of. T ake r ( ǫ ) = r ǫ a i a n .  Let M = X i ∩ { x n = 1 } . Suppo se that there exists an index k such that M ⊂ { x k = r } . Let us consider a pr o jection P : R n → R n − 1 defined as fo llows: P ( x 1 , . . . , x n ) = ( x 1 , . . . , ˆ x k , . . . , x n ) . Observe that, for j 6 = k , P | X j is a bi-Lipschitz map and P ( X j ) is a ( a 1 , . . . , ˆ a k , . . . , a n )- weigh ted homogeneous subset in R n − 1 . Then we obtain our statemen t from the induction hypotheses. Now, let us supp ose that M 6⊂ { x n − 1 = r } , for a ll r . Let T : M × [0 , + ∞ ) → X i be a map defined as follows: T ( x 1 , . . . , x n , t ) = ( t a 1 a n x 1 , . . . , t a n − 1 a n x n − 1 , t ) . Clearly , the map T is se mia lgebraic, injective and sur jective, for t 6 = 0. Let α : [0 , ρ ] → X i be a semialgebraic arc in X i such that α (0) = 0, para meterized in suc h a wa y that α ′ (0) exists. Let ¯ α : [0 , ρ ] → M × [0 , 1] be a lifting of α , i.e. T ◦ ¯ α = α. This lifting is defined as follows: ¯ α ( s ) = T − 1 ( α ( s )), for s 6 = 0. Since ¯ α ( s ) is s emialgebraic , then lim s → 0 ¯ α ( s ) exists and b elo ng s to M × { 0 } . By the same reaso n, lim s → 0 ¯ α ′ ( s ) also exists. T he r efore, the a rc ¯ α near M × { 0 } can b e repara meterized LOCAL LIPSCHITZ GEOMETR Y OF WEIGHTED HOM OGENEOUS SURF ACES 9 in the following wa y: ¯ α ( t ) = ( x 1 ( t ) , . . . , x n − 1 ( t ) , t ). By definition of the map T , we hav e α ( t ) = ( t a 1 a n x 1 ( t ) , . . . , t a n − 1 a n x n − 1 ( t ) , t ) . Clearly , d dt | t =0 ( α ( t )) is not co nt ained in the hyperplane x n = 0. Let α 1 , α 2 : [0 , ρ ] → X i be t wo arcs suc h that α 1 (0) = α 2 (0) = 0 . The ar c s α 1 and α 2 can b e par ameterized as follows: α 1 ( t ) = ( t a 1 a n y 1 ( t ) , . . . , t a 1 a n − 1 y n − 1 ( t ) , t ) and α 2 ( t ) = ( t a 1 a n z 1 ( t ) , . . . , t a 1 a n − 1 z n − 1 ( t ) , t ) . W e o btain: k α 1 ( t ) − α 2 ( t ) k max = max { t a i a n | y i ( t ) − z i ( t ) | ; i = 1 , . . . , n − 1 } . Since | y i ( t ) − z i ( t ) | is a b ounded function and a 1 ≥ · · · ≥ a n − 1 , we have k α 1 ( t ) − α 2 ( t ) k max = at λ + o ( t λ ) with λ ≥ a n − 1 a n . On the o ther ha nd, since M 6⊂ { x n − 1 = r } , there exist y = ( y 1 , . . . , y n − 1 , 1 ) , z = ( z 1 , . . . , z n − 1 , 1 ) ∈ M s uch that y n − 1 6 = z n − 1 . By Theorem 3.3, the lea ves γ y and γ z hav e the order o f c o ntact λ ( γ y , γ z ) = a n − 1 a n . The theorem is proved.  Pr o of of The or em 2 .6. Let X ⊂ R 3 be a semialg ebraic ( a 1 , a 2 , a 3 )-weigh ted ho mo- geneous surfa ce with a connected lo ca l link at 0. If X ∩ { x 3 = 0 } 6 = { 0 } , then, by the pro of o f the Theore m 2.5, we obtain tha t β ( X , 0) = 1. Note that β ( X , 0) can b e equal to a 1 a 3 only in the cas e that X ∩ { x 3 = ǫ } is totally included in a line x 2 = r ( ǫ ). But, since the lo cal link of X at 0 is connected, it implies that X ∩ { x 3 = ǫ } is the set defined by x 3 = ǫ , x 2 = r ( ǫ ). Then X is a union of standard leav es of F ( a 1 ,a 2 ,a 3 ) passing through the p oints b elong ing to the straight line x 3 = ǫ , x 2 = r ( ǫ ). Note, tha t if X ∩ { x 3 = 0 } = { 0 } then X cannot b e closed. The case β ( X , 0) = a 1 a 2 can o ccur if , and only if, X ⊂ { x 3 = 0 } . But, in this case, β ( X, 0) = 1.  10 L. BIRBRAIR AND A. FERNANDES 6. Order comp arison lemma Let K be a field of ger ms of s ubanalytic functions f : (0 , ǫ ) → R . Let ν : K → R be a canonica l v aluation on K . Namely , if f ( t ) = αt β + o ( t β ) with α 6 = 0 , we put ν ( f ) = β . Here we are going to prove a bit more genera l result such that Lemma 3.1 is a partial ca s e of it. Theorem 6.1 . L et k · k S b e a semialgebr aic norm on R n . L et γ 1 and γ 2 b e two semi- analytic ar cs such that γ 1 (0) = γ 2 (0) = x 0 ∈ R n . L et γ S i ( t ) b e a p ar ametrization of γ i such that k γ S i ( t ) − x 0 k S = t , i = 1 , 2 . L et λ S ( γ 1 , γ 2 ) = ν ( k γ S 1 ( t ) − γ S 2 ( t ) k S ) . Then λ S ( γ 1 , γ 2 ) = λ ( γ 1 , γ 2 ) . In order to pr ov e this theorem we nee d the following lemma. Lemma 6 .2. L et M ⊂ R n b e a semialgebr aic c onvex c omp act subset such that 0 ∈ I nt ( M ) . Then, for e ach smal l ǫ > 0 , ther e exists a n umb er δ > 0 such that, for e ach p air x , y ∈ ∂ M with k x − y k < ǫ , the angle b etwe en x and x − y satisfies the fol lowing ine quality δ < ∠ ( x, x − y ) < π − δ. Pr o of. Let T an g ( M ) b e a s ubset of R n × R P n − 1 of the pa irs ( x, l ) such that l is a stra ig ht line, l ∩ I nt ( M ) = ∅ and x ∈ ∂ M ∩ l . Clearly , T ang ( M ) is a compact semialgebra ic subs e t of R n × R P n − 1 . Let ang : R n × R P n − 1 → R b e a function defined as fo llows: ang ( x, l ) = sin( ∠ ( − → 0 x, l )) . Observe that sin( ∠ ( − → 0 x, l )) is a w ell defined function. Since T ang ( M ) is compact, then ther e exis ts ˜ δ > 0 such that, for all ( x, l ) ∈ T ang ( M ), we hav e ang ( x, l ) > ˜ δ . Let T ang ǫ ( M ) b e a set of pair s ( x, l ) where x ∈ ∂ M and l is a straight line passing through x and some y ∈ ∂ M s uch that k x − y k ≤ ǫ . Obs erve tha t T ang ǫ ( M ) is also a compa c t semialg ebraic se t. Since the Hausdorff limit lim ǫ → 0 T ang ǫ ( M ) b elongs to T ang ( M ), there exis ts ˜ ǫ > 0 such that ang ( x, l ) > ˜ δ 2 , for a ll ( x, l ) ∈ T ang ˜ ǫ ( M ). It pr ov es the lemma .  Pr o of of The or em 6 .1. Let x 0 and let γ 1 , γ 2 be arcs satisfying the co ndition of the theorem. L et us prov e that λ S ( γ 1 , γ 2 ) ≥ λ ( γ 1 , γ 2 ). Supp os e that λ S ( γ 1 , γ 2 ) < LOCAL LIPSCHITZ GEOMETR Y OF WEIGHTED HOM OGENEOUS SURF ACES 11 λ ( γ 1 , γ 2 ). Let γ 1 ( t ) a nd γ 2 ( t ) b e po ints such that k γ 1 ( t ) k = k γ 2 ( t ) k = t . Let τ = | γ 1 ( t ) k S . Let γ S 2 ( τ ) b e a p oint o n γ 2 such that k γ S 2 ( τ ) k S = τ . Thus, for small t , the angle at the vertex γ S 2 ( τ ) of the triangle γ 1 ( t ) , γ 2 ( t ) , γ S 2 ( τ ) must tend to zer o. The line defined by γ 2 ( t ) and γ S 2 ( τ ) tends to the tangent line of γ 2 at 0. Since the ball of r a dius τ , with resp ect to the norm k · k S , is a conv ex set a nd the o r igin belo ngs to this ball, we obtain a contradiction to Lemma 6 .2. Using the similar ar gument we ca n show that λ S ( γ 1 , γ 2 ) ≤ λ ( γ 1 , γ 2 ).  References [1] L. Birbrair, L o c al bi- L ipschitz classific ation of 2- dimensional semialgebr aic sets. Houston Journal of Mathematics, N3, vol.25, (1999), pp 453-472. [2] L. B i rbrair, A. F ernandes, Metric the ory of semialgebr aic curves. Revista M atem´ atica Com- plutense, vol.13, N 2, (2000), pp 369-382. [3] L. Birbrair , J.- P . Brasselet, Metric homolo gy . Comm. Pure Appl. Math. 53 (2000), no. 11, 1434–144 7. [4] A. F ernandes, T op olo gic al equival enc e of co mplex curves and bi- Lipschitz home omorp hisms. Michigan Math. J. 51 (2003), no. 3, 593–606 [5] S. Koik e, L. Pa unescu, The kissing dimension of sub analytic sets is pr eserve d by a b i -Lipschitz home omorphism . Prepri nt [6] K. Kurdyk a, On a sub analytic str atific ation satisfying a Whitne y pr op e rt y with exp onent 1. Real algebraic geometry (Rennes, 1991), 316–322 , Lecture Notes in Math., 1524, Springer, Berlin, 1992. [7] G. V alette, The link of the germ of a semi-algebr aic metric sp ac e . Preprint. Dep ar t amento de Ma tem ´ atica, Universidade Federal do Cear ´ a, A v. Mister Hull s/n,Campus do PICI, Bloco 914,CEP: 60 .455-760 - For t aleza - CE - Brasil. 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