CONVEXITY PROPERTIES AND COMPLETE HYPERBOLICITY OF LEMPERT’S ELLIPTIC TUBES

  International Journal of Mathematics Vol. 22, No. 5 (2011) 603 c World Scientific Publishing Company

DOI: 10.1142/S0129167X11006921

  CONVEXITY PROPERTIES AND COMPLETE HYPERBOLICITY OF LEMPERT’S ELLIPTIC TUBES

  

DANIELE ALESSANDRINI

Institut de Recherche Math´ ematique Avanc´ ee, CNRS and UDS

7 rue Ren´ e Descartes, F-67084 Strasbourg Cedex, France

daniele.alessandrini@gmail.com

  

ALBERTO SARACCO

Dipartimento di Matematica, Universit` a di Parma

Parco Area delle Science 53/A, I-43124 Parma, Italy

alberto.saracco@unipr.it

  

Received 17 February 2010

Revised 17 June 2010

_n are C-convex and We prove that elliptic tubes over properly convex domains D ⊂ RP

complete Kobayashi-hyperbolic. We also study a natural construction of complexification

of convex real projective manifolds.

  

Keywords: Elliptic tubes; C-convexity; Kobayashi hyperbolicity; convex real projective

manifolds. Mathematics Subject Classification 2010: 32F17, 32Q45, 57M50

  1. Introduction _n The notion of elliptic tube of a subset of R was defined and studied by Lempert _n in

  where we deal with subsets of RP to emphasize the projective invariance of the construction. _{n e n} If D ⊂ RP , the elliptic tube D is a subset of CP , a “complexification” of D. _e

  Some properties of the set D transfer automatically to D , for example openness, _e closedness, connectedness and boundedness of D imply the same for D .

  The same is not true for convexity; indeed Lempert had shown that the elliptic

  2

  2

  tube over a triangle in R is not a convex subset of C . This is not too surprising, _n as the property of convexity in C is not projectively invariant. In

  , Lempert was

  particularly interested in the case where the base domain D is a properly convex _e domain. He showed that in this case the elliptic tube D is linearly convex, hence pseudoconvex. He also proved that the Hilbert distance on D is the restriction of _e the Kobayashi distance on D . In _e a Monge–Amp`ere maximal problem is studied

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D. Alessandrini & A. Saracco

  _e

  In this paper we show that if D is a properly convex domain, the elliptic tube D _e is C-convex, and that the Kobayashi distance on D is complete. Then we use the projective invariance of the elliptic tubes to construct a natural complexification of convex real projective manifolds. In this paper, by a convex real projective mani- fold we mean a manifold carrying a convex real projective structure, a geometric structure in the sense of Klein (see

  for details).

  Section

   introduces the different notions of convexity needed in the sequel,

  namely linear convexity and C-convexity, and the necessary instruments for the proofs in the following sections, among which an important one is the notion of projective dual complement.

  In Sec.

   the definition and the first properties of elliptic tubes are given. We also _n

  prove that if D ⊂ RP is an open convex set, then the tube over the dual comple- ment of D coincide with the dual complement of the tube over D. A consequence _e is that D is linearly convex. As a corollary, elliptic tubes over convex domains are pseudoconvex, holomorphically convex, domains of holomorphy, polynomially convex, Runge domains, and are convex with respect to the linear fractions.

  In Sec. _{k e} we show some regularity properties of elliptic tubes. Namely, if ∂D is of class C then also ∂D \∂D is. Unless D is very special (projectively equivalent to the ball), there is no regularity at the real points of the boundary of the elliptic tube.

  This prevents the use of the equivalence between linearly convexity and C-convexity

  1

  which holds for connected C -smooth domains. Then we prove that elliptic tubes over properly convex domains are C-convex. Using a characterization of Kobayashi- complete hyperbolicity of C-convex domains given in

  , this shows that elliptic

  tubes over properly convex domains are complete Kobayashi-hyperbolic, and that they are taut and hyperconvex.

  In Sec.

  we apply our results to real and complex projective manifolds (in the

  sense of Klein). To every convex real projective manifold we associate in a natural way a complex projective manifold, that we call its complexification. We show that this complex projective manifold is complete Kobayashi-hyperbolic and that it is homeomorphic in a natural way to the tangent bundle of the original convex real projective manifold.

  2. Linearly Convex and C-convex Sets In this section, we will recall some results about linear convexity and C-convexity that we will need in the following. We will also recall the definition of convexity for a subset of the real projective space.

  Let K be a field, V be a finite-dimensional vector space over K, P = P(V ) be

  ∗ ∗ ∗

  the corresponding projective space, V be the dual of V and P = P(V ) be the projective dual.

  A subset E ⊂ P is said to be linearly convex if the complement of E is a union of projective hyperplanes or, in other words, if every point in the complement of E

  Lempert’s Elliptic Tubes 605

  The intersection of linearly convex sets is linearly convex, and the space P is linearly convex. If E ⊂ P is any set, the linearly convex hull of E is the intersection of all the linearly convex sets containing E. This is just the complement of the

  ∗

  union of all the projective hyperplanes disjoint from E. Every element ξ ∈ P is a projective class of linear functionals, hence it has a well defined kernel denoted by ker(ξ), which is a projective hyperplane of P. If x ∈ P, the set

  ∗

  Ann(x) = {ξ ∈ P | x ∈ ker(ξ)}

  ∗ ∗∗

  is called the annihilator of x, and it is a projective hyperplane of P . If φ : P → P is the canonical identification, then Ann(x) = ker(φ(x)).

  Let E ⊂ P be a set. Then the dual complement of E is the set

  ∗ ∗

  E = {ξ ∈ P | ker(ξ) ∩ E = ∅}, i.e. it is the set of all projective hyperplanes disjoint from E. Note that if E ⊂ E ,

  1

  2 ∗ ∗

  then E ⊂ E .

  2

1 Lemma 2.1. If

  E ⊂ P, then

  ∗ ∗ E = P \ Ann(x). _x

∈E

∗

  In particular is linearly convex.

  E

  ∗ Proof. Let ξ ∈ E . If x ∈ P is such that ξ ∈ Ann(x), then x ∈ ker(ξ), hence x ∈ E. ∗ Therefore ξ ∈ P \ Ann(x). _{x ∈E}

  ∗ Let ξ ∈ E , then there is x ∈ E ∩ ker(ξ). Hence ξ ∈ Ann(x), with x ∈ E. ∗ ∗ ∗∗ −1 ∗

  If F ⊂ P , we will identify F ⊂ P with φ (F ) ⊂ P, and we will write

  ∗ −1 ∗

  F = φ (F ) ⊂ P. More explicitly we have

  ∗ F = {x ∈ P | Ann(x) ∩ F = ∅}.

  By previous lemma, we have the alternative description

  ∗ F = P\ ker(ξ). _ξ

∈F

∗∗

  ∗∗

  Lemma 2.2. E is the linearly convex hull of

  E. In particular E ⊂ E , and if

  ∗∗ E is linearly convex, then E = E . ∗∗ ∗ Proof.

  x ∈ E iff Ann(x) is disjoint from E iff every projective hyperplane con-

  ∗∗

  taining x intersects E. Hence the complement of E is the union of the projective hyperplanes not intersecting E.

  ∗

  Let K = R or K = C. It is easy to see that if E is open, then E is compact,

  ∗ ∗

  and if E is compact then E is open. E is bounded in some affine patch if and

  606

D. Alessandrini & A. Saracco

  Let E ⊂ P. A projective hyperplane is said to be tangent to E if it intersects the boundary ∂E but it does not intersects the interior part int(E).

  ∗ ∗

  Proposition 2.1. Let E ⊂ P be open. Then (E) = int(E ). Moreover we have

  ∗ ∗ ∀ ξ ∈ P ξ ∈ ∂E ⇔ ker(ξ) is tangent to E.

  ∗ ∗

  Let E ⊂ P be compact. Then (int(E)) ⊃ E . Moreover we have

  ∗ ∗ ∀ ξ ∈ P ξ ∈ ∂E ⇒ ker(ξ) is tangent to E.

  Proof.

  See Proposition 2.5.1]. If K = R a set E ⊂ P is said to be convex if it does not contain projective lines and if the intersection with every projective line is connected. An open or compact convex set is homeomorphic to an open or closed ball. Theorem 2.1. The connected components of a linearly convex set are convex. Open or compact convex sets are linearly convex. If

  E is connected and open or compact, then E is convex iff it is linearly convex. Moreover, the dual complement of a convex set is convex.

  Proof.

  See Proposition. 1.3.4, Theorems 1.3.6 and 1.3.11]. _n A convex set E ⊂ RP is said to be properly convex if it is not contained in a projective hyperplane and it does not contain an affine line. _n

  Theorem 2.2. Let E ⊂ RP be a properly convex open set. Then : _n \H. (1) E is relatively compact in an affine patch RP (2) int(E) and E are also properly convex, and int(E) = int(E) and int(E) = E.

  ∗ ∗ ∗ ∗

  (3) (E) = int(E ) and (int(E)) = E . Proof. See Proposition 1.3.1 and Theorem 1.3.14]. _n

  Proposition 2.2. Let E ⊂ RP be a compact convex set. Then E has a basis of properly convex neighborhoods. Proof.

  A linearly convex set has a basis of linearly convex neighborhoods (see

  

  p. 17]) (U α ). As E is connected, it is always contained in the interior of a connected component of U α that is convex. _n Proposition 2.3. Let be a convex open set not containing straight lines.

  D ⊂ R

  1

  1

  2 Then for all

  ε > 0, there are two strictly convex domains D = D (ε) and D =

  2

  1

  2

  2

  1

  ⊂ D ⊂ D \D is contained D (ε) with real analytic boundary, such that D , and D in the ^{2 1}

  ε-neighborhood of ∂D (i.e. sup d(x, ∂D) < ε, where d makes reference _{n x ∈D \D} to the Euclidean distance on R ).

  _n Lempert’s Elliptic Tubes 607

  Corollary 2.1. Let be a properly convex open set. Then there is an D ⊂ RP exhaustion of k with real analytic boundary.

  D made by strictly convex open sets D Proof. Since D is properly convex and open, it is starred with respect to all of its points. Consider an affine patch containing D such that D is convex and 0 ∈ D. Fix a sequence 1 > δ k > 0 strictly decreasing to 0. We define E = (1 − δ k )D ⊂ D.

  1−δ k

  Suppose we have defined D k ⊂ E , strictly convex and with real analytic _⋐ 1−δ k boundary (and D j D l , for j < l < ¯ k), for all k < ¯ k. We need to construct D ¯ ⊂ _k E . Fix ε ¯ < min{δ ¯ , d(∂D ¯ , ∂E )}. E is convex and bounded, so by

  1−δ ¯ k k k k 1−δ k ¯ 1−δ k ¯ −1

  the previous proposition we can find D ¯ = D ¯ (ε ¯ ) ⊂ E strictly convex with _{k k k 1−δ ¯ k} real analytic boundary such that E \D ¯ is contained in the ε ¯ -neighborhood of _⋑ 1−δ k ¯ k k E 1−δ . So D ¯ D ¯ . _{¯ k k k −1} The defined sets D k have the required regularity properties and form an exhaustion of D.

  If K = C a set E ⊂ P is C-convex if it does not contain projective lines and if the intersection with every projective line is connected and simply connected. An open C-convex set is homeomorphic to an open ball (see Theorem 2.4.2]).

  Let E P be an open connected set. If a ∈ ∂E, we denote by Γ(a) the set of

  ∗ all tangent hyperplanes to E at a. We have Γ(a) = Ann(a) ∩ E . _n

  Theorem 2.3. If is an open connected set E CP , with n > 1, E is C-convex iff for all a ∈ ∂E, Γ(a) is nonempty and connected.

  Proof. See Theorem 2.5.2].

  3. Elliptic Tubes _{n n}

• 1

  \{0} → RP We denote by π : R the natural projection.

  1 Let I ⊂ RP be an interval, with extremes a and a 1 . Then there are two vectors

  2

  v , v ∈ R such that π(v ) = a , π(v ) = a and

  1

  1

  1

  int(I) = π({c v + c v | c , c ∈ R, c c > 0}),

  1

  

1

  1

  1 I = π({c v + c v | c , c ∈ R, c c ≥ 0}).

  1

  

1

  1

  1 ∗

  1

2 Consider the dual basis v , v ∈ (R ) . Now we have

  1

  1

  int(I) = {x ∈ RP | v (x)v (x) > 0},

  1

  1 I = {x ∈ RP | v (x)v (x) ≥ 0}.

  1

  1

  ⊂ CP Consider the natural inclusion RP . Consider the sets _e

  1

  1

  int(I) = π({c v + c v | c , c ∈ C, ℜ(c c ) > 0}) = {z ∈ CP | ℜ(v (z)v (z)) > 0}, _e

  1

  1

  1

  

1

  1

  1 I = π({c v + c v | c , c ∈ C, ℜ(c c ) ≥ 0}) = {z ∈ CP | ℜ(v (z)v (z)) ≥ 0}.

  1

  1

  1

  

1

  608

  D. Alessandrini & A. Saracco _{e e}

  It is easy to see that the sets int(I) and I are just the open and closed circle _{e e}

  1

  in CP with diameter I. The circles int(I) and I are well defined up to projective

  1 changes of coordinates in RP . _n

  In the same way, if I ⊂ RP is a segment with extremes a and a there are two _{n +1}

  1

  vectors v , v ∈ R such that π(v ) = a , π(v ) = a and

  1

  1

  1

  int(I) = π({c v + c v | c , c ∈ R, c c > 0}),

  1

  

1

  1

  1 I = π({c v + c v | c , c ∈ R, c c ≥ 0}).

  1

  

1

  1

  1 _C

  If L is the real projective line containing I, we denote by L the complexification _{e C e} of L. Then int(I) and I are the circles in L with diameter I, well defined as in the previous section: _e int(I) = π({c v + c v | c , c ∈ C, ℜ(c c ) > 0}), _e

  1

  1

  1

  1 I = π({c v + c v | c , c ∈ C, ℜ(c c ) ≥ 0}). _n

  1

  1

  1

  1 Let D ⊂ RP be a set. The elliptic tube with base D is defined (as in ) as

  the set _{e e n} | I ⊂ D is a closed segment } ⊂ CP D = ∪{I . The definition is projectively invariant. If D is open, then this definition is equivalent to _{e e n} D = ∪{I | I ⊂ D is an open segment } ⊂ CP . _{e e}

  Note that if D is open also D is open, and if D is closed also D is closed. If _{e e} D is connected also D is connected. If D is contained in an affine patch also D is _e contained in the same affine patch, and if D is bounded in an affine patch, also D is bounded in the same affine patch. Moreover, if D is contained in an affine patch, _e then D is a deformation retract of D . If D is open (not necessarily affine) there is _{e e} a projectively invariant deformation retraction from D to D. In these cases D is homotopically equivalent to D. _{n −1 ∗ n +1}

  Let D ⊂ RP be an open convex set. Then π (D ) ⊂ R is the union of two disjoint open convex cones. We choose one of these convex cones, and we denote it by ˜ D.

  ∗ ∗ ∗ n −1 n +1

  ⊂ (RP Let D ) be the dual complement of D. Then π (D) ⊂ (R ) is the union of two disjoint open convex cones. Exactly one of these two convex cones

  ∗

  contains only linear functionals that are positive on ˜ D; we denote this cone by ( ˜ _{n ∗} D) .

  ∗

  Let F ⊂ (RP ) be a compact set such that F = D. In other words F is a

  ∗

  compact set such that the linearly convex hull of F is D . If ξ ∈ F, we denote by

  ∗

  ˜ ξ an element of ( ˜

  D) such that π( ˜ ξ) = ξ. Then _n | ∀ f, g ∈ F : ˜ D = {x ∈ RP f (x)˜ g(x) > 0}.

  Theorem 3.1. Let _{e n} D and F as above. Then D = {z ∈ CP | ∀ f, g ∈ F : ℜ( ˜ f (z)˜ g(z)) > 0}.

  Lempert’s Elliptic Tubes 609

  Theorem 3.2. Let D and F as above. Then _{e ∗ e} (F ) = D . _{e e e e} ∗ Proof. _e (F ) ⊃ D : We have to show that if ξ ∈ F , then ker(ξ) ∩ D = ∅. If

  ˜ ξ ∈ F , then by definition of elliptic tubes there are f, g ∈ F such that ˜ ξ = c f +c

_n

1 g ˜

  ˜ ∈ C, ℜ(c with c , c

  1 c 1 ) ≥ 0. Let z ∈ CP be such that ˜ ξ(z) = c f (z) + c 1 g(z) = 0. ˜

  Then ˜ f (z) c |˜ g(z)|

  1

  ℜ( ˜ f (z)˜ g(z)) = |˜ g(z)|ℜ = −|˜ g(z)|ℜ = − ℜ(c c 1 ) ≤ 0.

  |c | _e g(z) ˜ c Hence z ∈ D . _{e e e e} ∗ (F ) ⊂ D : We have to show that if z ∈ D there exists an element ξ ∈ F _e such that z ∈ ker(ξ). If z ∈ D there are f, g ∈ F such that ℜ( ˜ f (z)˜ g(z)) ≤ 0. Now _e if ξ is such that ˜ ξ = ˜ g(z) ˜ f − ˜ f (z)˜ _{n e} g, then ξ ∈ F , and z ∈ ker(ξ). Corollary 3.1. If is an open convex set is linearly convex.

  D ⊂ RP , then D Proof. A dual complement is always linearly convex. Note that this statement also follows from the first part of _n Proof of Theorem 2.2].

  Lemma 3.1. Let D ⊂ RP be a compact convex set , and let (U k ) be a family of _{e e} open convex sets such that U k ⊂ U k and U k = D. Then D = U . _{e e e e} +1 _k Proof. _{e e n} D ⊂ U : For all k we have D ⊂ U k , hence D ⊂ U . _{k k} D ⊃ U : We only have to prove that if L ⊂ RP is a real projective line, _{C C C C k e e e} and L is its complexification, then D ∩ L ⊃ L ∩ ( U ) = (U ∩ L ). This _{C C k k e} is easy because L ∩ D and L ∩ U k are just intervals, and L ∩ D and L ∩ U k are just circles. _{n e} Corollary 3.2. If is a compact convex set is linearly convex.

  D ⊂ RP , then D Proof. By Proposition

  we can always construct a family of open properly _{e e} convex sets such that U k ⊂ U k and U k = D. By the previous lemma D = U .

_{e e e}

+1 _k

  If z ∈ D , we can find a k such that z ∈ U , and by previous corollary, U is linearly _{k k e} convex, then there is a projective hyperplane H containing z and disjoint from U . _{e k} Hence D is linearly convex.

  Corollary 3.3. Let D be an open or compact properly convex set. Then _{e ∗ e}

  ∗ (D ) = (D ) . _{e e} ∗ ∗

  Proof.

  If D is open, then by the previous theorem we know that D = ((D ) ) . _{e e e e ∗ ∗ ∗} ∗ ∗∗

  610

  D. Alessandrini & A. Saracco _{e ∗ e e} ∗ ∗ ∗∗ If D is compact, then D is open, hence we have ((D ) ) = (D ) = D . _{e e ∗}

  ∗ Hence (D ) = (D ) . _e

  Corollary 3.4. Let is pseudoconvex D be an open properly convex set. Then D , holomorphically convex

  , a domain of holomorphy, polynomially convex, a Runge domain , and it is convex with respect to the linear fractions. Let D be a compact _e properly convex set. Then D is polynomially convex. _e Proof.

  This follows from the fact that D is linearly convex and with connected dual complement. See

  Propositions 2.1.8, 2.1.9 and 2.1.11].

  4. Regularity and C-convexity of Elliptic Tubes _n Let D ⊂ RP be an open properly convex set, let h D be the Hilbert distance on D _e (see

  Sec. 3] for all the definitions needed here), and let k D be the Kobayashi _{e e}

  distance on D . Note that as D is bounded in some affine patch, also D is bounded _{e e} in the same affine patch, hence D is Kobayashi hyperbolic, i.e. k D is a non- degenerate distance. _n Theorem 4.1. If D ⊂ RP is an open properly convex set , then _e (1) For all p, q ∈ D, h D (p, q) = k D (p, q). _{n e C e} is a real projective line is an open disk (2) If L ⊂ RP , then L ∩ D , and k D _{C e} restricted to is the Poincar´ e distance on the disk. _e L ∩ D _{C C} contains is just (3) If z ∈ D \D, let L be the unique real line such that L z (L the line containing z and ¯ z). Then there exists x ∈ L such that _{e e} k D (z, x) = min k D (z, p). _p

  ∈D

  Proof. Note that the first statement is

  Theorem 3.1]. The other two can be proved in the same way.

  Consider the functions: _{e e} d : D ∋ z → min k D (z, p) ∈ R ≥0 , _{e p ∈D} φ : D ∋ z → 2 arctan(tanh(d(z))) ∈ [0, π/2).

  Choose a real projective hyperplane H such that D is bounded in the affine _{n n e n n C} patch R = RP \H. Hence D is bounded in the affine patch C = CP \H . With _{e n} these affine coordinates, D ⊂ D ⊕ iR . Consider the function _{n −1} p : D ⊕ iR ∋ z = x + iy → inf{t > 0 | x + t y ∈ D}.

  _{n n C} Lempert’s Elliptic Tubes 611

  Lemma 4.1. Let L ⊂ R be a real line. Then L ∩ (D ⊕ iR ) is a strip S = (a, b) ⊕ iR and _{  if} y _ y ≥ 0, b − x p| S (z) = p| S (x + iy) = _{ } y if y < 0. a − x

  Moreover , _e {p(z)p(¯ \∂D. z) = 1} = ∂D Proof.

  It is a simple computation. We can now define _e p(z) + p(¯ z) u : D ∋ z → arctan ∈ [0, ∞).

  1 − p(z)p(¯ z) Proposition 4.1. _{n e} u = φ. _C Proof.

  Choose a real line L ⊂ R , and compute explicitly u on L ∩D . Also φ can _{C e} ∩ D be computed explicitly on L using the previous theorem. Choose coordinates _{C e}

  ∩ D on L such that it is the unit disk in C: then on this disk both functions are equal to

  2|ℑ(z)| 1 + z arctan = arg .

  

2

  1 − z 1 − |z| _k Proposition 4.2. Suppose that

  D has a boundary of class C (with k ∈ N ∪ {∞} ∪ _k on their domains. {ω}). Then the functions p and u are of class C _n

  −1 Proof.

  Let z = x + iy ∈ D ⊕ iR . The half-line {x + t y | t > 0} cuts ∂D in a _n single point h ∈ ∂D. By hypothesis, there exists a neighborhood U of h in C and

  −1

  a function f : U → R such that ∂D ∩ U = f (0) and ∀ ζ ∈ ∂D : df ζ = 0. Then _{n ′ ′ ′} there exists a neighborhood V of z in C , such that for all z = x + iy ∈ V , p(w)

  ′ −1 ′

  is precisely the unique value of t such that f (x + t y ) = 0. Observe that ∂

  1

  −1 −1

  f (x + t y) = − ∇f (x + t y), y ,

  

2

  ∂t t which on ∂D is non-vanishing since the direction y is transversal to ∂D. Thus, by _k the implicit function theorem p is of class C near z. _k

  612

  D. Alessandrini & A. Saracco _k

  Proposition 4.3. Suppose that D has a boundary of class C (with k ∈ N ∪ {∞} ∪ _{e k} {ω}). Then ∂D \∂D is of class C . _{e n} Proof. _e ∂D \∂D ⊂ D ⊕ iR , and it is precisely the set where p(z)p(¯ z) = 1. Fix _C ζ ∈ ∂D \∂D. Let L be the complex line containing ζ and ¯ ζ. Then

  2 _C

  y p(z)p(¯ z)| L = − _C (b − x)(a − x) where x + iy is the coordinate on L corresponding to z. We have ∂ _{C e}

  2 (p(z)p(¯ z))| = = 0. _L

  ∩∂D

  ∂y y _{e k} Hence by the implicit function theorem ∂D \∂D is of class C . _n Theorem 4.2. Suppose that is a properly convex open set with

  D ⊂ RP ∂D of

  1 e class . Then is C-convex.

  C D _{e e} Proof. _e Let a ∈ ∂D . Since D is linearly convex, Γ(a) = ∅. If a ∈ ∂D, we know that ∂D is smooth at a, hence it has only one tangent real hyperplane H at a. Note that H has real dimension 2n − 1. It contains only one complex hyperplane, hence Γ(a) is a single point, hence it is connected. If a ∈ ∂D, then a is a real point. Γ(a) = _{e ∗ e}

  ∗

  Ann(a) ∩ (D ) = Ann(a) ∩ (D ) . By the second part of Proposition _n

  Ann(a) ∩

  R is a real hyperplane (of dimension n − 1) tangent to D, hence Ann(a) intersects _e

  ∗ (D ) in a single point, and also Γ(a) is a single point hence it is connected. _{n e}

  Corollary 4.1. Suppose that D ⊂ RP is a properly convex open set. Then D is C-convex. Proof. By Corollary

  there is an exhaustion of D made by (strictly) convex _e domains D k whose boundary is real analytic. By Theorem 5.1, each D is C-convex. _k

  From the definition of the elliptic tubes is obvious that _{e e e} D = ∪D k ⇒ D = ∪D . _k Hence D is an increasing union of C-convex open sets, thus C-convex (see

   Proposition 2.2.2]). _{n e}

  Corollary 4.2. Suppose D ⊂ RP is a properly convex open set. Then D is complete Kobayashi hyperbolic , taut, and hyperconvex. _e Proof. By previous corollary, D is a bounded C-convex open set. Hence the state- ment follows from

  Theorem 1]. _{n e}

  Lempert’s Elliptic Tubes 613 Proof.

  If D is bounded, it is Corollary

  If D is unbounded, it is an increasing _e

  union of bounded convex open sets, hence D is an increasing union of C-convex open sets, and hence C-convex. _{n e} Corollary 4.4. Suppose that is a compact convex set. Then is

  D ⊂ RP D C-convex. _e

  ∗ ∗ Proof.

  If D is compact, then D is open, hence (D ) is C-convex. By

  

  Theorem 2.3.9], the dual complement of an open C-convex set is C-convex, hence _e D is C-convex.

  5. Complexification of Convex Real Projective Manifolds The projective invariance of the elliptic tubes may be used to construct a com- plexification of real manifolds with a suitable structure, namely with a structure of convex real projective manifold. The complexifications will have a structure of com- plex projective manifolds. These structures are geometric structures in the sense of Klein (see

  for a survey paper). Here we recall the definitions we need in the following.

  Let K be R or C, and let M be a manifold of dimension n if K = R, and _n of dimension 2n if K = C. A KP -structure on M is given by a maximal atlas {(U i , φ i )}, where the sets U i form an open covering of M and the charts φ i : U i → _n KP are projectively compatible, i.e. the transition maps

  −1

  φ i,j = φ i | U ◦ φ | φ : φ j (U i ∩ U j ) → φ i (U i ∩ U j ) _{i ∩U j j (U i ∩U j ) j} ∩ have the property that for every connected component C of the intersection U i _{j n i,j | C C}

  U there exists a projective map A ∈ P GL +1 (K) such that φ = A| . A _{n n} KP

• manifold is a manifold with a KP -structure. They are called real projective manifolds if K = R and complex projective manifolds if K = C. _{n n n}
• For example every open subset of KP (KP itself included) has a natural KP structure given by the inclusion map and all the charts that are compatible with the inclusion map. More interesting examples can be constructed by taking an _n open subset Ω of KP and a subgroup Γ ⊂ P GL n (K) acting freely and properly
• 1

  discontinuously on Ω. Then the quotient space Ω/Γ is a manifold and it inherits a _{n n} KP -structure from Ω. The KP -manifolds we will consider here are of this form. _n The most interesting case is when K = R and Ω ⊂ RP is an open prop- erly convex set. Real projective manifolds of the form Ω/Γ are called convex real projective manifolds. It is possible to construct many interesting manifolds of this form. For example, according to the Klein model of hyperbolic space, the hyper- _{n n} bolic space is identified with an ellipsoid H ⊂ RP , and the group of hyperbolic isometries is identified with the group of projective transformations of the ellipsoid,

• O (1, n) ⊂ P GL n (R). By this identification, every complete hyperbolic m
• 1

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D. Alessandrini & A. Saracco

  examples of convex real projective manifolds. It is also possible to construct many interesting examples when Ω is not an ellipsoid. _n Let M = Ω/Γ be a convex real projective manifold, in particular Ω ⊂ RP is an open properly convex set and Γ ⊂ P GL n (R) acts freely and properly _e +1 discontinuously on Ω. Consider the elliptic tube Ω . By the projective invariance of _e the elliptic tube construction, the group Γ also acts on Ω . _e Proposition 5.1. The action of Γ on Ω is free and properly discontinuous. _e Proof.

  To see that the action of Γ on Ω is free, suppose, by contradiction, that an _e element γ ∈ Γ has a fixed point z ∈ Ω \Ω. As γ is a real matrix, also the conjugate point ¯ z is fixed by γ, and also the unique complex line containing z and ¯ z. This complex line is real (in the sense of conjugation-invariant) hence it intersects Ω in _e a segment and it intersects Ω in a disc. As γ does not fix any point of Ω, it acts on the segment as a nontrivial translation. This action extends to the disc without fixing any point of the disc, and this is a contradiction with the fact that z was a fixed point. _e

  Consider the group G of all bi-holomorphisms of Ω , equipped with the compact- _e open topology. As Ω is Kobayashi-hyperbolic, the group G is a Lie group and it _e acts properly on Ω (see

  Introduction], for a discussion of these properties).

  As Γ acts properly discontinuously on Ω, it is discrete for the topology it has as a subgroup of P GL n (R), that is the same it has as a subgroup of P GL n (C).

• 1
• 1

  The topology of this latter group is the compact-open topology for its action on _n CP (see

  Sec. 2.6]). Hence the group Γ is discrete even with the topology it has as a subgroup of G.

  This implies that Γ is closed in G. In fact, by

  Chapter 2, 1.8], a subgroup

  H of G is discrete if and only if there exists a neighborhood U of 1 in G such that H ∩ U = {1}. Now Γ is discrete, its closure Γ is again a subgroup, but Γ ∩ U = {1}, hence Γ is discrete itself, hence Γ = Γ. _e

  As Γ is closed in G and the action of G on Ω is proper, then also the action of _e Γ on Ω is proper. As Γ is discrete, the action is properly discontinuous. _e We have seen that Γ acts freely and properly discontinuously on Ω . Hence _{e e} the quotient M = Ω /Γ is a manifold and it has a natural complex projective _e structure. We call the complex projective manifold M the complexification of M . _e

  In the remaining part of this section we describe the manifold M . The inclusion _{e e e} Ω ⊂ Ω gives an inclusion M ⊂ M . The complex conjugation on Ω is compatible _e with the action of Γ, hence it induces an anti-holomorphic involution on M that _e has M as locus of fixed points. The Kobayashi metric of M is the quotient of the _{e e}

  Kobayashi metric on Ω by the action of Γ, hence M is a Kobayashi-hyperbolic complex manifold. As a corollary of the theorems in the first part we obtain: Theorem 5.1. Let

  M be a convex real projective manifold. Then the Kobayashi _e

  Lempert’s Elliptic Tubes 615 Proof.

  It follows from the observations made above and Corollary

_e

  Finally, we describe the topology of M , by showing that it is homeomorphic in a natural way with the tangent bundle to M . _e Theorem 5.2. Let be its complexifica-

  M be a convex real projective manifold, M tion and T M be the tangent bundle of M . Then there is a natural homeomorphism _e f : M →T M ˜ whose restriction to M is the zero section of T M . _n Proof.

  Let M = Ω/Γ, where Ω ⊂ RP is an open properly convex set and Γ ⊂ P GL n +1 (R) is a subgroup acting freely and properly discontinuously on Ω. We _e will give a natural homeomorphism f between Ω and the tangent space of Ω, _n T Ω = Ω × R , which is projectively equivariant, hence passes to the quotient. _e

  If x ∈ Ω, then define f (x) ∈ T Ω as (x, 0). If z ∈ Ω \Ω, consider the complex _e line L z through z and ¯ z. I z = L z ∩ Ω is a segment, and ∆ z = L z ∩ Ω is a disk with diameter I z . Consider now the geodesic for the Poincar´e metric γ z joining _e z and z in ∆ z . This is also a geodesic for the Kobayashi metric in Ω . Define x z = γ z ∩ Ω = γ z ∩ I z . We will define f (z) as a vector in the tangent space at x z , _{e e e} T x Ω. Since Ω is a complex manifold, it has a complex structure J : T Ω → T Ω . _{z e}

  Let us denote by v ∈ T x Ω the unitary tangent vector to γ at x z (considering γ _z as a curve from z to ¯ z). Since γ is a geodesic connecting two complex conjugate points, Jv ∈ T x z Ω. Note that Jv is tangent to I z , thus the complex structure allows _e us to choose a direction on I z at x z . Let l z = k (z, x z ) (the Kobayashi distance).

  Ω

  Then we can define · Jv) ∈ T Ω. f (z) = (x z , l z

  Note that the construction of f is projectively equivariant. We need to show that f is a homeomorphism. f is surjective. Let (x, w) ∈ T Ω. Consider the interval

  I = Ω ∩ {x + tw | t ∈ R},

  (x,w) _e

  and the elliptic tube over it, ∆ = I . Up to a projective transformation we

  (x,w)

(x,w)

  may suppose ∆ = ∆, the unit disk, and x = 0. Consider the two imaginary

  (x,w)

  points z , z such that k (x, z i ) = w . Then we have that f (z ) = (x, ±w) and

  1 2 ∆

  1 f (z ) is the opposite point.

  2 _e

  f is injective. Suppose z , z ∈ Ω are such that f (z ) = f (z ). Then the complex

  1

  2

  1

  2

  geodesics γ z and γ z must coincide, they are at the same distance from the real _{1 2} part Ω and on the same side (in the disk, where the real part disconnects), hence z = z .

  1

  2 _e

  f is continuous. Let us consider a sequence {z n } n ⊂ Ω with _e ∈N lim z n = z ∈ Ω . _n ∞ →∞

  616

D. Alessandrini & A. Saracco

  We have to show that lim f (z n ) = f (z ). _n ∞ →∞ _e Note that the points z n converge to z ∞ and the Kobayashi distances k Ω (z n , z n ) _e converge to k Ω (z ∞ , z ∞ ).

  ∈ Ω. Consider the points x ∩ Ω. First consider the case where z ∞ z n = γ z n _{e e}

  In this case we just need to estimate the distance k Ω (x z n , z ∞ ) ≤ k Ω (x z n , z n ) + _e k Ω (z n , z ∞ ) → 0. _e ∈ Ω \Ω. In this case we can suppose that all

  Then consider the case where z ∞ _{n \Ω, then the disks ∆ z e} the points z are in Ω n are well defined and they converge to _{z ∞ n} the disk ∆ . All we need to prove is that the part of the geodesics γ connecting z n and z n converge to the part of the geodesic γ connecting z and z . Consider

  ∞ ∞ ∞

  the limit set of the geodesics γ n , defined as ∈ γ }.

  Λ = {t ∈ T M | t = lim t k for some sequence t k n k Observe that Λ is connected since it is the limit set of connected sets in a finite- dimensional Euclidean space, that z , z ∈ Λ since they are the limit of the points

  ∞ ∞

  z k and z k respectively, and that Λ ⊂ ∆ z ∞ . We have to prove that Λ coincides with γ .

  ∞

  We first prove that Λ ⊂ γ . Arguing by contradiction, let us suppose that there

  ∞

  is a point ζ ∈ Λ\γ ∞ . By the unicity of the geodesic in the disk this means _e k (z , z ) = k ∞ (z , z ) < k ∞ (z , ζ) + k ∞ (ζ, z )

  

Ω ∞ ∞ ∆ ∞ ∞ ∆ ∞ ∆ ∞

_{e e} = k (z , ζ) + k (ζ, z ).

  Ω ∞ Ω ∞

  Since ζ ∈ Λ, this means that _{e e e e} lim k (z n , z n ) ≥ k (z , ζ) + k (ζ, z ) > k (z , z ), _n Ω Ω ∞ Ω ∞ Ω ∞ ∞ →∞ which is a contradiction. So Λ ⊂ γ .

  ∞

  Since Λ is a connected set contained into γ which is a topological closed interval

  ∞ and the endpoints of γ belong to Λ, indeed Λ = γ . ∞ ∞

  We have proved that f is a continuous bijective map between two domains of the same dimension. By Brouwer’s invariance of domain theorem, the map f is a homeomorphism.

  Acknowledgments We wish to thank Giuseppe Tomassini, Stefano Trapani and Sergio Venturini for helpful discussions.

  A. Saracco was partially supported by the MIUR project “Geometric Properties

  Lempert’s Elliptic Tubes 617

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[2] W. Goldman, Geometric structures and varieties of representations, Geometry of

Group Representations, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/Colo.

  1987, Contemp. Math., Vol. 74 (1988) 169–198.

[3] A. Isaev, Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds,

Lecture Notes in Mathematics, Vol. 1902 (Springer, Berlin, 2007).

  

[4] L. Lempert, Elliptic and hyperbolic tubes, Several complex variables, Stockholm,

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[5] N. Nikolov and A. Saracco, Hyperbolicity of C-convex domains, C. R. Acad. Bulgare

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[6] C. Parrini and G. Tomassini, ∂µ = f : Existence of bounded solutions in unbounded

domains, Boll. Un. Mat. Ital. B (7) 1 (1987) 1211–1226.

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₂₀ (2009) 1561–1581.

  

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