Bulging deformations of convex R P 2 -manifolds
aa r X i v : . [ m a t h . DG ] F e b BULGING DEFORMATIONS OF CONVEX RP -MANIFOLDS WILLIAM M. GOLDMAN
Abstract.
We define deformations of convex RP -surfaces. A convex RP -manifold is a representation of a surface S as a quo-tient Ω / Γ, where Ω ⊂ RP is a convex domain and Γ ⊂ SL (3 , R ) is adiscrete group of collineations acting properly on Ω. We shall describea construction of deformations of such structures based on Thurston’searthquake deformations for hyperbolic surfaces and quakebend defor-mations for CP -manifolds.In general if Ω / Γ is a convex RP -manifold which is a closed sur-face S with χ ( S ), then either ∂ Ω is a conic, or ∂ Ω is a C convexcurve (Benz´ecri [1]) which is not C (Kuiper [5]). In fact its derivativeis H¨older continuous with H¨older exponent strictly between 1 and 2.Figure 1 depicts such a domain tiled by the (3 , , Figure 1.
A convex domain tiled by triangles
Date : April 3, 2019.1991
Mathematics Subject Classification.
Key words and phrases. surface, RP -structure, measured lamination, earth-quake, Teichm¨uller space.The author gratefully acknowledges partial support from National Science Foun-dation grant . This drawing actually arises from Lie algebras (see Kac-Vinberg [4]).Namely the Cartan matrix C = − − − − − − determines a group of reflections as follows. For i = 1 , , E ii denote the elementary matrix having entry 1 in the i -th diagonal slot.Then, for i = 1 , ,
3, the reflections ρ i = I − E ii C generate a discrete subgroup of SL (3 , Z ) which acts properly on theconvex domain depicted in Figure 1 (and appears on the cover of theNovember 2002 Notices of the American Mathematical Society).. Thisgroup is the Weyl group of a hyperbolic Kac-Moody Lie algebra.We describe here a general construction of such convex domains aslimits of piecewise conic curves.If Ω / Γ is a convex RP -manifold homeomorphic to a closed esurface S with χ ( S ) <
0, then every element γ ∈ Γ is positive hyperbolic, thatis, conjugate in SL (3 , R ) to a diagonal matrix of the form δ = e s e t
00 0 e − s − t . where s > t > − s − t . Its centralizer is the maximal R -split torus A consisting of all diagonal matrices in SL (3 , R ). It is isomorphic to aCartesian product R ∗ × R ∗ and has four connected components. Itsidentity component A + consists of diagonal matrices with positive en-tries.The roots are linear functionals on its Lie algebra a , the Cartansubalgebra.
Namely, a consists of diagonal matrices(0.0.1) a = a a
00 0 a . where a + a + a = 0. The roots are the six linear functionals on a defined by a α ij a i − a j where 1 ≤ i = j ≤
3. Evidently α ji = − α ij .Writing a ( s, t ) for the diagonal matrix (0.0.1) with a = s, a = 2 , a = − s − t, ULGING DEFORMATIONS OF CONVEX RP -MANIFOLDS 3 the roots are the linear functionals defined by a ( s, t ) α s − ta ( s, t ) α t − sa ( s, t ) α t − ( − s − t ) = s + 2 ta ( s, t ) α ( − s − t ) − t = − s − ta ( s, t ) α ( − s − t ) − s = − s − ta ( s, t ) α s − ( − s − t ) = 2 s + t which we write as α = (cid:2) − (cid:3) α = (cid:2) − (cid:3) α = (cid:2) (cid:3) α = (cid:2) − − (cid:3) α = (cid:2) − − (cid:3) α = (cid:2) (cid:3) The
Weyl group is generated by reflections in the roots and in thiscase is just the symmetric group, consisting of permutations of thethree variables a , a , a in a (as in (0.0.1)). A fundamental domain isthe Weyl chamber consisting of all a satisfying α > α > α > α are the positive simple roots. In other words, the roots are totally ordered by:the rule α > α > α > > α > α > α > α . In terms of the parametrization of a by a ( s, t ), the Weyl chamber equals { a ( s, t ) | s ≥ t ≥ − s } . The trace form on sl (3 , R ) defines the inner product h , i with associ-ated quadratic form tr (cid:0) a ( s, t ) (cid:1) = 2( s + st + t ) = 2 | s + ωt | where ω = + √− = e πi/ is the primitive sixth root of 1. WILLIAM M. GOLDMAN
The elements of sl (3 , R ) which dual to the roots (via the inner prod-uct h , i ) are the root vectors: h = a (1 , − ,h = a ( − , ,h = a (0 , ,h = a (0 , − ,h = a ( − , ,h = a (1 , a ( s, λs ) = s λs
00 0 − (1 + λ ) s where 1 ≥ λ ≥ − . Its boundary consists of the rays generated by the singular elements a (1 ,
1) = h + h = h + 2 h and a (2 , −
1) = h + h = 2 h + h . The sum of the simple positive roots is the element a (1 ,
0) = h = h + h which generates the one-parameter subgroup H t := exp (cid:0) a ( t, (cid:1) = e t e − t . The orbits of A + on RP are the four open 2-simplices defined by thehomogeneous coordinates, their (six) edges and their (three) vertices.The orbits of H t are arcs of conics depicted in Figure 2.Associated to any measured geodesic lamination λ on a hyperbolicsurface S is bulging deformation as an RP -surface. Namely, one appliesa one-parameter group of collineations e t
00 0 1 ULGING DEFORMATIONS OF CONVEX RP -MANIFOLDS 5 Figure 2.
Conics tangent to a triangleto the coordinates on either side of a leaf. This extends Thurston’s earthquake deformations (the analog of
Fenchel-Nielsen twist deforma-tions along possibly infinite geodesic laminations), and the bendingdeformations in PSL (2 , C ).In general, if S is a convex RP -manifold, then deformations are de-termined by a geodesic lamination with a transverse measure takingvalues in the Weyl chamber of SL (3 , R ). When S is itself a hyper-bolic surface, all the deformations in the singular directions becomeearthquakes and deform ∂ ˜ S trivially (just as in PSL (2 , C ). Figure 3.
Deforming a conic
WILLIAM M. GOLDMAN
Figure 4.
A piecewise conic
Figure 5.
Bulging data
Figure 6.
The deformed conic
ULGING DEFORMATIONS OF CONVEX RP -MANIFOLDS 7 Figure 7.
The conic with its deformation
References
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Bull. Soc.Math. France (1960), 229–3322. Casselman, W., “An exotic Coxeter complex,” Cover picture and p.1274, No-tices of the American Mathematical Society, 49 (10), November 2002.3. Goldman, W., Convex real projective structures on compact surfaces,
J. Diff.Geo. (1990), 791–845.4. Kac, V. and Vinberg, E. B. Quasi-homogeneous cones,
Math. Notes (1967),231–235, (translated from Math. Zametki (1967), 347-354)5. Kuiper, N., On convex locally projective spaces,
Convegno Int. Geometria Diff.Italy (1954), 200–213
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