Weil-Petersson Metric Geometry Quick Overview

WP Metric Geometry Quick View SA W July 3, 2007 Let T be the T eichm¨ uller space for marked genus g , n punctured Riemann surfaces R of negative Euler characteristic. By Uniformization a co nformal structure determines a unique complete co mpatible h yp er b olic metr ic ds 2 . T is a complex manifold of dim C =3 g − 3 + n with the cotangent space at R repres ent ed by Q ( R ), the space of holomorphic quadr a tic differentials on R with at most simple po les at the puncture s. The W eil-Petersson Hermitian cometr ic is h ϕ, ψ i = Z R ϕ ψ ( ds 2 ) − 1 . The WP metric is in v ariant under the ac tion of the mapping class gr oup MCG, [Iv a02]. The metric is K¨ ahler, non complete with negative sectional curv ature with sup T = 0 (except for dim T = 1) and inf T = −∞ . In pr actic e and exp erienc e the WP ge ometry of T c orr esp onds to the hyp erb olic ge ometry of surfac es. The augmen ted T eic hm ¨ uller space, [Abi77, B er74]. The augmented T e- ichm¨ uller space T is a MCG in v ariant b ordifica tion (a partial compactifica tion) following the approach for Baily-B orel bo rdifications. ( T is an analog o f the S L (2; Z ) inv ariant b ordification Q for H provided with the Sa take horo cycle top ology .) The b ordificatio n is a lso describ ed as the Chabauty topolo gy clo- sure of the faithful cofinite representations of π 1 ( R ) in to P S L (2; R ). T / M C G is homeomorphic to the Deligne-Mumford stable cur ve compactification of the mo duli space of Riema nn surfac e s . The elements of T − T are marked deg ener- ate hyperb olic s tr uctures for which certain simple non periphera l closed curves are r epr esent e d by pair s of cusps. CA T(0) geometry . Imp ortant is tha t T is WP complete with ( T , d W P ) a C AT (0) metric space (a generalize d complete, simply co nnected, non p ositively curved space.) The lar ge-scale geometry is describ ed by T b eing quasi isometr ic to P G ( R ) the p ants gr aph of R with unit-leng th edge metric, [Br o03]. T is a stratified space with each op en strata characterized a s the union of a ll geo desic s (distance r ealizing paths) containing a g iven point as an interior p oint. T itself is characterized as the close d WP conv ex hull of the maximally degener ate hyperb olic str uctures (the unions of thrice punctured spher es.) An application of the C AT (0) geometry and the imp ortant rigidity of the c omplex of curves C ( R ), [Iv a02, Chap. 3 ] is that the is o metry gro up Isom W P coincides with MCG. Geo desi c-length functions. Asso cia ted to each non trivial, non p eriph- eral fr ee ho motopy class on R is the leng th of the unique ds 2 geo desic in the homotopy class. Geo desic- le ngths ℓ α ( R ) on T are elements of the WP ge- ometry . The F enc hel-Nielsen t wist (r ig ht earthquake) deformation abo ut α is describ ed by 2 t α = J grad ℓ α for J the T eic hm¨ uller a lmost complex s tructure. The WP Hermitian pair ing is describ ed in terms of the hyperb olic tr igonometry for R = H / Γ with h grad ℓ α , J grad ℓ β i = 4 ω W P ( t α , t β ) = − 2 X p ∈ α ∩ β cos θ p 1 for the geo des ic s α, β and int ersection ang les θ ∗ on R a nd in upp e r half plane h grad ℓ α , grad ℓ β i = 2 π  ℓ α δ αβ + X ′ h A i\ Γ / h B i ( u log u + 1 u − 1 − 2  for the Kronecker delta δ ∗ , where for C ∈ h A i\ Γ / h B i then u = u ( ˜ α, C ( ˜ β )) is the cosine of the intersection angle if the lifts ˜ α and C ( ˜ β ) int ersect and is o therwise cosh d ( ˜ α, C ( ˜ β )). WP conv exit y and curv ature. Informa tion o n geodesic s is provided by the strict conv exity of geo desic-lengths (more generally convexit y of the total length of a me asur e d ge o desic lamina tion ) along WP geo desics (square ro o ts are also conv ex.) F o r surfaces with cusps the distances along g eo desics between horo cycles are a lso strictly conv ex . Geo des ic -length sublevel sets ar e conv e x. On T the WP Levi-Civita connection D is describ ed for s mall geo desic-length, λ = grad ℓ 1 / 2 α and tangent U by D U λ α = 3 ℓ − 1 / 2 α h J λ α , U i J λ α + O ( ℓ 3 / 2 α k U k ) . The remainder term constant is uniform for b ounded geo desic-leng th. Bo unds for the gradient and Hessian of geo desic-leng th give r ise to a pplications. The WP curv ature of the s pa n { t α , J t α } is O ( − ℓ − 1 α ). Similar ly for a pair of deformations suppo rted on different compo nen ts of R − { shor t g eodesics } the curv a tur e of the corr esp onding 2-plane is O ( − shor t separating l eng ths ). F enc hel-Niel sen co ordinates. Mar ked hyperb olic pair s of pan ts are de- termined by their thr ee b oundary g eo desic-lengths in R + . The FN twist-length co ordinates ( ℓ j , τ j ) 3 g − 3+ n j =1 for as semblin g hyperb olic pan ts pro vide global c o o r- dinates for T with expansions ω W P = 1 2 X dℓ j ∧ dτ j and on the Bers r egion { ℓ j < c 0 } the e x pansions h , i ≍ X ( dℓ 1 / 2 j ) 2 + ( dℓ 1 / 2 j ◦ J ) 2 ≍ X Hess ℓ j with comparability uniform in terms of c 0 . At the maximally degene r ate s tr uc- ture the WP metr ic has the expansio ns h , i = 2 π X ( dℓ 1 / 2 j ) 2 + ( dℓ 1 / 2 j ◦ J ) 2 + O ( X ℓ 3 j h , i ) = π 6 X Hess ℓ 2 j ℓ j + O ( X ℓ 2 j h , i ) . Alexandro v tangen t cone. F o r a pair of geo de s ics (distance realizing curves) fro m a common point in a C AT (0) metric spa ce there is a well-defined initial angle, [BH99, Cha p. I I .3 ]. A tangent cone is defined in terms of initial angle. At a p oint p of a stra ta T ( σ ) ⊂ T − T , σ ∈ C ( R ), the WP Alexandr ov tangent cone AC p is isometric to a pr o duct of a E uclidean or thant and the 2 tangent s pa ce T T ( σ ) with WP pair ing. The isometry is given in terms o f initial der iv atives of g eo desic-length functions. The dimension of the Euclidean orthant is the co unt of ge o desic-lengths trivial o n T ( σ ) (the count of no des.) A prop erty of T is tha t a WP geo desic tang e ntial to T ( σ ) at p actually lies in T ( σ ). Applications inc lude that g e o de s ics do not r efr act at the b ordificatio n and that the a ng le o f incidence and reflection co incide for certain limits of degenerating geo desics. A further application is for combinatorial harmonic maps. Certain groups a c ting o n E uclidean buildings and gro up extensions acting on Cayley graphs s a tisfying a Poincar ´ e type inequality for links of p oints will have a globa l fixed po in t for an action o n T . All important attributions and references are provided in the in tro ductio ns of [W ol0 3, W o l07]. References [Abi77] William Abikoff. Deg enerating families of Riemann surfaces. A nn. of Math. (2) , 1 05(1):29– 44, 1977 . [Ber74] Lipman Bers. Spaces of de g enerating Riemann surface s . In Disc on- tinuous gr oups and Riemann su rfac es (Pr o c. Conf., Univ. Maryland, Col le ge Park, Md., 1973) , pages 43 –55. Ann. of Math. Studies, No. 79. Princeton Univ. P ress, Princeton, N.J., 1974. [BH99] Martin R. Bridson and Andr´ e Haefliger . Metric sp ac es of non-p ositive curvatur e . Spr inger-V erlag, Berlin, 1 999. [Bro03 ] Jeffrey F. Bro ck. The Weil-Peter sson metr ic a nd volumes of 3- dimensional hyper bo lic conv ex cor es. J. Amer. Math. So c. , 16(3):495– 535 (electro nic), 2 003. [Iv a02] Nik o lai V. Iv anov. Mapping class groups. In H andb o ok of ge ometric top olo gy , pages 5 23–63 3. North-Holland, Amsterdam, 2002. [W ol0 3] Scott A. W olpert. Geometry of the Weil-Peters son completion of T e- ichm¨ uller spac e . In Surveys in Differ ential Ge ometry VIII: Pap ers in Honor of Calabi, L awson, Siu and Uhlenb e ck , pa ges 357– 393. Intl. Press, Ca mbridge, MA, 2003 . on Arxiv: math/0 50252 8 . [W ol0 7] Scott A. W olper t. Behavior of geo desic-length functions on Teichm¨ uller space. preprint on Arxiv: math/0 70155 6 , 2007. 3