by Kai Cieliebak
1. Some complex analysis
Holomorphic convexity and Levi problem
Stein manifolds
Embeddings into \( \mathbb{C}^N \)
Plurisubharmonicity
Rational and polynomial convexity
2. The topology of Stein manifolds
The first condition (arising from the complex structure) is the existence of an almost complex structure \( J \), that is, an endomorphism of the tangent bundle with \( J^2=-\operatorname{id} \). Given such \( J \), we associate to a function \( \phi:V\to\mathbb{R} \) the 1-form \( d^\mathbb{C}\phi=d\phi\circ J \). We call the function \( J \)-convex if \( -dd^\mathbb{C}\phi(v,Jv) > 0 \) for each nonzero tangent vector \( v \). If \( J \) is integrable (i.e., induced by a complex structure), then \( J \)-convexity is equivalent to strict plurisubharmonicity.
The second condition arises from the existence of an exhausting \( J \)-convex function \( \phi:V\to\mathbb{R} \). After a \( C^2 \)-small perturbation, we may assume that \( \phi \) is Morse. Then all its critical points have Morse index \( \leq n \) (since the restriction of \( \phi \) to each holomorphic curve is strictly subharmonic, hence has no local maximum and therefore index at most 1). By Morse theory, this implies that \( V \) has a handles of index at most \( n \).
In dimension \( \neq 4 \) these two conditions are also sufficient:
See [7] for a more detailed proof and [9] for a less detailed account of the same.
3. Stein versus Weinstein
Liouville and Weinstein domains. In this article, a domain will always mean a compact manifold with smooth boundary.6 Fix a domain \( W \) of real dimension \( 2n \). A Liouville form on \( W \) is a 1-form \( \lambda \) for which \( \omega=d\lambda \) is symplectic and \( \lambda|_{\partial W} \) is a contact form inducing the boundary orientation. The pair \( (W,\lambda) \) is called a Liouville domain. Via \( \lambda=i_X\omega \), a Liouville form is equivalent to a pair \( (\omega,X) \) consisting of a symplectic form \( \omega \) and a Liouville field \( X \), that is, a vector field which points outward along \( \partial W \) and satisfies \( L_X\omega=\omega \).
A defining function for \( W \) is a smooth function \( \phi:W\to(-\infty,c] \) with regular level set \( \partial W=\phi^{-1}(c) \). Let us call a vector field \( X \) gradient-like for \( \phi \) if \( d\phi=g(X,\cdot\,) \) for some positive smooth \( (2,0) \) tensor field \( g \), where “positive” means \( g(v,v) > 0 \) for all \( v\neq 0 \). It is shown in [e54] that this agrees with the notion of “gradient-like” from [7] if \( \phi \) is Morse or generalized Morse, but it is defined without any nondegeneracy assumption on \( \phi \). A Weinstein structure on \( W \) is a pair \( (\lambda,\phi) \) consisting of a Liouville form \( \lambda \) and a defining function \( \phi \) for which the Liouville field \( X \) is gradient-like. The triple \( (W,\lambda,\phi) \) is called a Weinstein domain.
Let us denote by \( \mathfrak{Stein} \), \( \mathfrak{Weinstein} \) and \( \mathfrak{Liouville} \) the spaces of Stein, Weinstein and Liouville structures on \( W \), respectively. We have canonical maps \[ \mathfrak{Stein} \xrightarrow{\ \mathfrak{W}\ \ } \mathfrak{Weinstein} \xrightarrow{\ \mathfrak{L}\ \ } \mathfrak{Liouville} \] given by \( \mathfrak{W}(J,\phi)=(-d^\mathbb{C}\phi,\phi) \) and \( \mathfrak{L}(\lambda,\phi)=\lambda \). Concerning the first map, we have the following theorem and conjecture from [7] (see [e54] for their adaptation to the above notion of “gradient-like”).
Thus every Weinstein structure on \( W \) is Weinstein homotopic to a Stein structure (in particular, the topological conditions for the existence of a Stein or Weinstein structure are the same), and any two Stein structures that are Weinstein homotopic are Stein homotopic.
A first step towards this conjecture would be showing that the local singularity theories of Stein and Weinstein structures agree. For example, the following question has a positive answer for critical points of Morse or birth-death type (see [7]) but is open in general.
Concerning the map \( \mathfrak{L} \), there exist Liouville domains \( W \) which cannot be Weinstein because they have homology above half their dimension. The first such examples were constructed by McDuff [e17]; further examples arise from Lie groups [e19], foliations [6], and Anosov flows [e21], [e51]. However, the following question is open.
Evidence for a positive answer arises from recent work by
Honda
and
Huang
on convex hypersurfaces in contact manifolds
[e49].
An interesting test case
is the Anosov Liouville domains
from
[e17],
[e21],
[e51].
These have the form \( W=[-1,1]\times M \)
for a closed oriented 3-manifold \( M \), so their stabilization
\( W\times B^2 \) (with its canonical Liouville structure after smoothing
the corners) satisfies the hypothesis in Question 3.4.
J. Breen
and
A. Christian
have recently posted a proof that \( W\times
B^2 \) is Liouville homotopic to a Weinstein structure if \( M \) is a torus
bundle with its suspension Anosov flow.
Liouville and Weinstein manifolds
Evidence for a positive answer arises from the following theorem.
In particular, this theorem applies to the Anosov Liouville manifolds \( V=\mathbb{R}\times M \) arising from [e17], [e21], [e51]. It improves an earlier result by Eliashberg and Gromov [2], using the theory of flexible Weinstein manifolds discussed in the next section.
4. Flexible Weinstein manifolds
Loose Legendrians
A formal Legendrian embedding \( (f,F^s) \) is a smooth embedding \( f:\Lambda\hookrightarrow M \) covered by a homotopy of monomorphisms \( F^s:T\Lambda\to TM \), \( s\in[0,1] \), such that \( F^0=df \) and \( F^1:T\Lambda\to\xi \) is isotropic. Let us denote by \( \mathfrak{Leg} \) and \( \mathfrak{FormalLeg} \) the spaces of Legendrian embeddings and formal Legendrian embeddings \( \Lambda\hookrightarrow M \), respectively. Associating to a Legendrian embedding \( f:\Lambda\hookrightarrow M \) the formal Legendrian embedding \( (f,F^s) \) with \( F^s=df \) for all \( s\in[0,1] \) defines a canonical map \[ \mathfrak{F}:\mathfrak{Leg}\to\mathfrak{FormalLeg}. \]
Flexible Weinstein domains and manifolds
For example, consider the case \( V=\mathbb{R}^{2n} \) with \( n\geq 3 \). Then the space \( \Omega^2_{\operatorname{nondeg}} \) is contractible, so Theorem 4.3 asserts that \( \mathfrak{Weinstein}_{\operatorname{flex}} \) is path connected and Conjecture 4.4 asks that \( \mathfrak{Weinstein}_{\operatorname{flex}} \) is contractible. On the other hand, M. McLean [e35] has constructed infinitely many pairwise non-symplectomorphic (in particular, nonhomotopic) Weinstein structures on \( \mathbb{R}^{2n} \) (for \( n\geq 4 \), extended to \( n=3 \) by M. Abouzaid and P. Seidel [e37]), so Theorem 4.3 becomes false if we replace \( \mathfrak{Weinstein}_{\operatorname{flex}} \) by \( \mathfrak{Weinstein} \).
Another property of flexible Weinstein structures is that, after a Weinstein homotopy, one can arbitrarily prescribe the Morse function:
Subcritical Weinstein domains and manifolds
Subflexible Weinstein domains
A Weinstein domain is called subflexible if it is deformation equivalent to a sublevel set of a flexible Weinstein manifold. It follows that each subflexible Weinstein domain has vanishing symplectic homology [e35]. Nonflexibility of a subflexible Weinstein domain is detected in [e47] by nonvanishing of a suitably twisted version of symplectic homology.
Flexible Weinstein subdomains
It follows that \( C:=\{\widetilde\phi\geq 0\} \) is diffeomorphic to \( [0,1]\times\partial W \) and \( \{\widetilde\phi\leq 0\} \) is the flexibilization \( \operatorname{Flex}(W) \) of \( W=(W,\lambda,\phi) \) (i.e., the flexible Weinstein structure on \( W \) formally homotopic to \( (W,d\lambda) \)), so we can state Theorem 4.9 concisely as \[ W\sim\operatorname{Flex}(W)\cup C. \] Here the flexible subdomain \( \operatorname{Flex}(W) \) carries all the smooth topology, whereas the topologically trivial cobordism \( C \) carries all the symplectic information.
Theorem 4.9 has a surprising consequence for the critical points of a Weinstein domain \( W=(W,\lambda,\phi) \) of dimension \( 2n \). Following [e52], call \( W \) smoothly subcritical if it admits a defining Morse function without critical points of index \( \geq n \), and smoothly critical otherwise. Call \( W \) Weinstein subcritical if there exists a Weinstein structure homotopic to \( (\lambda,\phi) \) whose Morse function has no critical points of index \( \geq n \), and Weinstein critical otherwise. Let \( \operatorname{Crit}(W) \) be the minimal number of critical points of a Morse function on \( W \), and \( \operatorname{WCrit}(W) \) the minimal number of critical points of a Morse function appearing in a Weinstein structure homotopic to \( (\lambda,\phi) \).
The first case in the following theorem follows from
Theorem 4.6 (more generally for flexible \( W \)) and the third
case from Theorem 4.9
(the second case
requires additional arguments).
5. The topology of rationally and polynomially convex subsets
Domains
- \( W \) is smoothly isotopic to a rationally convex domain if and only if it admits a defining Morse function without critical points of index \( > n \).
- \( W \) is smoothly isotopic to a polynomially convex domain if and only if
it satisfies, in addition, the following topological condition:
- (T) The inclusion homomorphism \( H_n(W;G)\to H_n(V;G) \) is injective for every abelian group \( G \).
For \( V=\mathbb{C}^n \), condition (T) in Theorem 5.1(b)
reads \( H_n(W;G)=0 \) for every abelian group \( G \). By the universal
coefficient theorem, this is equivalent to \( H_n(W;\mathbb{Z})=0 \) and
\( H_{n-1}(W;\mathbb{Z}) \) having no torsion.
For example, this shows that for each \( k\in\mathbb{N} \) there exists some
disjoint union of \( k \) balls in \( \mathbb{C}^n \) which is polynomially convex.
On the other hand, the combination of Theorem 4.8,
Criterion 5.2(b)
below,
and the Stein–Weinstein correspondence yields
polynomially convex domains in \( \mathbb{C}^n \) which are nonflexible, in
particular they admit no defining Morse functions without critical
points of index \( \geq n \).
The proof of Theorem 5.1 is based on the following criterion for rational convexity due to S. Nemirovski [e34] (deduced from ([e20], Theorem 1.1), cf. ([11], Criterion 3.1)), and for polynomial convexity due to K. Oka [e4] (cf. ([e31], Theorem 1.3.8)). Here a domain \( W \) in a Stein manifold \( (V,J) \) is called \( J \)-convex if its boundary is a \( J \)-convex hypersurface, that is, a regular level set of a \( J \)-convex function defined on its neighbourhood.
- \( W \) is rationally convex if and only if there exists a defining \( J \)-convex function \( \phi:W\to(-\infty,0] \) such that the 2-form \( -dd^\mathbb{C}\phi \) on \( W \) extends to an exact Kähler form \( \omega \) on the whole \( (V,J) \).
- \( W \) is polynomially convex if and only if there exists an exhausting \( J \)-convex function \( \phi:V\to\mathbb{R} \) with regular sublevel set \( W=\{\phi\leq 0\} \).
Note that an immediate consequence of part (a) is the rational convexity of a disjoint union \( B=B_1\amalg\cdots\amalg B_k\subset\mathbb{C}^n \) of finitely many balls [e34]: if \( B_j \) has center \( a_j \) and radius \( r_j \), then \( \phi(z)=|z-a_j|^2-r_j^2 \) for \( z\in B_j \) is a defining \( i \)-convex function for \( B \) with \( -dd^\mathbb{C}\phi \) the standard Kähler form on \( \mathbb{C}^n \).
Sketch of proof of Theorem 5.1. (a) The “only if” follows immediately from Criterion 5.2(a). By Morse theory, the “if” comes down to the following inductive statement: Let \( W,\phi,\omega \) be as in Criterion 5.2(a), and \( \Delta\subset V\setminus{\operatorname{Int}}\,W \) a transversely attached \( k \)-disk, \( k\leq n \). Then \( \Delta \) is isotopic through transversely attached \( k \)-disks to \( \widetilde\Delta \) such that \( W\cup\widetilde\Delta \) is contained in an arbitrarily small \( J \)-convex domain \( \widetilde W \). This is proved in three steps.
Step 1. By Eliashberg–Murphy’s h-principle for Lagrangian caps ([8], Theorem 2.2), \( \Delta \) is isotopic through transversely attached \( k \)-disks to \( \widetilde\Delta \) which is \( \omega \)-isotropic and attached along an isotropic sphere \( \partial\widetilde\Delta\subset\partial W \) (which is loose if \( k=n \)). Moreover, we can arrange that \( \widetilde\Delta \) is real analytic and \( J \)-orthogonally attached, and \( \omega \) is real analytic near \( \widetilde\Delta \).
Step 2. The next ingredient is the following easy consequence of the \( \partial\overline\partial \)-lemma ([11], Proposition 3.6): Let \( L \) be a real analytic Lagrangian submanifold (possibly noncompact and/or with boundary) in a Kähler manifold \( (V,J,\omega) \) with real analytic Kähler form \( \omega \), and let \( \rho:L\to\mathbb{R} \) be any real analytic function. Then there exists a unique real analytic function \( \psi \) on a neighborhood of \( L \) satisfying \[ -dd^\mathbb{C}\psi=\omega,\qquad \psi|_{L}=\rho, \qquad (d^\mathbb{C}\psi)|_{L}=0. \]
Step 3. We extend \( \widetilde\Delta \) from Step 1 to a real analytic Lagrangian \( L\cong D^k\times D^{n-k} \), and apply Step 2 with \( \rho \) having a unique index \( k \) critical point on \( \widetilde\Delta \). Using \( J \)-orthogonality, we can adjust \( \psi \) to fit together with \( \phi \) near \( \partial\widetilde\Delta \) to a \( J \)-convex function \( \widehat\phi \) on a neighbourhood \( U \) of \( W\cup\widetilde\Delta \) with \( -dd^\mathbb{C}\widehat\phi=\omega \) near \( \partial U \). Finally, we use \( J \)-convex surroundings to modify \( \widehat\phi \) on \( U \) to a \( J \)-convex function \( \widetilde\phi \), still satisfying \( -dd^\mathbb{C}\widetilde\phi=\omega \) near \( \partial U \), with a regular sublevel set \( \widetilde W=\{\widehat\phi\leq c\} \) containing \( W\cup\widetilde\Delta \). By Criterion 5.2(a), \( \widetilde W \) is rationally convex.
(b) The “only if” follows from Criterion 5.2(b) which implies \( H_{n+1}(V,W;G)=0 \). For the “if”, suppose first that \( (V,J) \) is flexible. The hypothesis of part (a) together with condition (T) imply the existence of an exhausting Morse function \( \psi:V\to\mathbb{R} \) without critical points of index \( > n \) and regular sublevel set \( W=\{\psi\leq 0\} \) ([11], Lemma 2.1). By Theorem 4.6, the flexible Weinstein structure \( \mathfrak{W}(J,\phi) \) on \( V \) is homotopic to a Weinstein structure \( (\lambda,\psi) \) with the given function \( \psi \). Thus \( W=\{\psi\leq 0\} \) is a Weinstein subdomain of \( (V,\lambda,\psi) \) and the result follows from the Stein–Weinstein correspondence ([12], Theorem 1.7(c)). If \( (V,J) \) is not flexible we use the splitting \( V=\operatorname{Flex}(V)\cup C \) from Theorem 4.9 (transferred to the Stein setting) and apply the previous argument to the flexible Stein manifold \( \operatorname{Flex}(V) \). This completes the proof. □
In complex dimension 2, S. Nemirovski and K. Siegel [e44] have answered the question on rational convexity in \( \mathbb{C}^2 \) for a special class of domains, disk bundles over surfaces. In contrast to the case of complex dimension \( n\geq 3 \), there exist \( i \)-convex domains in \( \mathbb{C}^2 \) (e.g., the unit disk cotangent bundle of the Klein bottle) that cannot be realized as rationally convex domains. Beyond this, the following question remains wide open.
Totally real submanifolds
Proof. The proof is a slight adaptation of the proof of ([e53], Proposition 1.2).
Assume first that \( p\notin H(K) \). Then there exists a holomorphic \( g:V\to\mathbb{C} \) with \( g(p)=0 \) and \( g|_K\neq 0 \). Then the meromorphic function \( h=f/g \) with \( f\equiv 1 \) satisfies \( g|_K\neq 0 \), \( f(p)\neq 0 \), and \( |h(p)|=\infty > \max_K|h| \), hence \( p\notin\widehat K_\mathcal{R} \).
Conversely, assume that \( p\notin\widehat K_\mathcal{R} \). Then there exists a meromorphic function \( h=f/g \) with \( g|_K\neq 0 \), \( f(p),g(p) \) not both 0, and \( |h(p)| > \max_K|h| \). Set \( z:=f(p)/g(p)\in\mathbb{C}\cup\{\infty\} \). If \( z=\infty \), then \( g \) is holomorphic with \( g(p)=0 \) and \( g|_K\neq 0 \), hence \( p\notin H(K) \). In the case \( z\neq\infty \) consider the holomorphic function \( \widetilde f:=f-zg \) satisfying \( \widetilde f(p)=0 \). If \( \widetilde f(q)=0 \) for some \( q\in K \), then in view of \( g(q)\neq 0 \) we would obtain the contradiction \( |h(q)|=|z|=|h(p)| \). Thus \( \widetilde f|_K\neq 0 \) and we again conclude \( p\notin H(K) \). □
By ([e53], Proposition 1.2), this implies that our notion of rational convexity agrees with the notion of strong meromorphic convexity in [e53].
Criterion 5.2 has the following analogue for totally real submanifolds.7
- \( L \) is rationally convex if and only if there exists an exhausting \( J \)-convex function \( \phi:V\to\mathbb{R} \) such that \( -dd^\mathbb{C}\phi|_L=0 \).
- \( L \) is polynomially convex if and only if there exists an exhausting \( J \)-convex function \( \phi:V\to[0,\infty) \) such that \( L=\phi^{-1}(0) \).
Proof. In view of the preceding discussion, part (a) is just ([e53], Theorem 3.2). In (b) the “if” part follows from ([e16], Theorem 5.2.8), and the “only if” from ([e16], Theorem 5.1.6). □
This criterion implies the following necessary conditions in symplectic terms. Here \( L \) is called exact rationally convex if there exists an exhausting \( J \)-convex function \( \phi:V\to\mathbb{R} \) such that \( -d^\mathbb{C}\phi|_L \) is exact, so we have the implications \[ \text{polynomially convex} \quad\implies\quad \text{exact rationally convex} \quad\implies\quad \text{rationally convex}. \]
For \( (\mathbb{C}^n,i) \) with \( \phi(z)=|z|^2/4 \), so that \( -d^\mathbb{C}\phi \) is the standard Liouville form, this becomes the well-studied question of (exact) Lagrangian submanifolds of \( \mathbb{C}^n \). For example, the nonexistence of exact Lagrangian submanifolds [e14] and the classification of Lagrangian submanifolds of \( \mathbb{C}^2 \) (see [e36]) yield the following.
- An \( n \)-dimensional totally real closed submanifold of \( \mathbb{C}^n \) is never exact rationally convex (in particular not polynomially convex).
- The rationally convex totally real closed surfaces in \( \mathbb{C}^2 \) are precisely the 2-torus and the nonorientable surfaces with negative Euler characteristic divisible by 4.
Consider now a closed manifold \( M \) of dimension \( n \). Following [12], its Grauert tube is the cotangent bundle \( T^*M \) with its canonical (up to Stein homotopy) Stein structure (carrying a \( J \)-convex function with a Morse–Bott minimum along the zero section and no other critical points). In this setting, the Nearby Lagrangian Conjecture from symplectic topology takes the following form (cf. [12]).
While this conjecture is in general open, here are some partial results (cf. [12]).
- The Nearby Lagrangian Conjecture holds for \( M=S^2 \) (R. Hind [e40]) and \( M=T^2 \) (G. Dimitroglou Rizell, E. Goodman and A. Ivrii [e43]).
- For every exact rationally convex totally real closed \( n \)-dimensional submanifold \( L\subset T^*M \) of a Grauert tube, the restriction \( \pi|_L:L\to M \) of the cotangent bundle projection is a simple homotopy equivalence (M. Abouzaid and T. Kragh [e46]).
- Every exact rationally convex totally real closed \( n \)-dimensional submanifold \( L \) in the Grauert tube of \( S^n \) is homeomorphic to \( S^n \) and bounds a parallelizable \( (n+1) \)-manifold (M. Abouzaid [e41], T. Ekholm, T. Kragh and I. Smith [e45]).
In view of these examples, we can ask a more general question.
Example 5.9(a) gives an affirmative answer to this question for \( V=T^*S^2 \) or \( T^*T^2 \), and Example 5.9(b) gives indivisibility of \( [L] \) for \( V=T^*M \). More generally, the Nearby Lagrangian Conjecture would imply an affirmative answer for \( V=T^*M \).
6. Holomorphic versus symplectic
Flexibility and subflexibility
Splitting and nonhyperbolicity
A necessary condition for this is the existence of a nonconstant holomorphic map \( \mathbb{C}\to V \) through each point of \( V \). In particular, \( V \) is nonhyperbolic in the sense of Kobayashi or Brody [e15], [e24]. It would be interesting to study the symplectic counterparts of these notions; see [e29] for some steps in this direction.
Boundary versus interior
- What does the boundary know about the interior?
- What does the interior know about the boundary?
For a Stein domain \( W \) both questions are well understood:
(1) The CR structure on the boundary (defined as the germ of a complex structure on a neighbourhood of \( \partial W \)) determines \( W \) uniquely up to biholomorphism. This was proved by H. Rossi [e10], by recovering \( W \) as the spectrum of the ring of holomorphic functions on \( \mathcal{O}p(\partial W) \).
(2) The biholomorphism type of the interior determines the boundary uniquely up to diffeomorphism. In fact, C. Fefferman [e12] proved that each biholomorphism between the interiors of Stein domains extends smoothly to the boundary, using that such a biholomorphism is an isometry for the Bergmann metrics. On the other hand, the Stein deformation type of the interior does not recover the boundary: S. Courte [e42] has constructed pairs of Stein domains whose interiors are deformation equivalent as Stein manifolds, but whose boundaries are nondiffeomorphic.
For a Weinstein domain \( W \) both questions are much more subtle:
(1) This question has several variants, depending on the structure assumed on the boundary and the equivalence relation for the interior. The strongest variant appears in a theorem of Gromov [e14]: Every Weinstein domain whose boundary is \( S^3 \) with its standard contact form is symplectomorphic to the standard ball \( B^4\subset\mathbb{R}^4 \). For the following variants, we assume that the boundary is given with its contact structure. Uniqueness of Weinstein fillings up to deformation equivalence is proved for the following manifolds with their standard contact structures: \( S^3 \), \( S^2\times S^1 \), the lens spaces \( L(p,1) \) for \( p\neq 4 \), and connected sums of these [5], [e28], [7]. Uniqueness up to symplectomorphism of the completion is proved for \( T^3 \) with its standard contact structure [e38]. In higher dimensions no uniqueness results up to deformation equivalence are known, but uniqueness up to diffeomorphism is proved for \( S^{2n-1} \) with its standard structure [e17], and more generally for contact manifolds admitting a subcritical Weinstein filling [e50]. On the other hand, in any dimension \( 4k-1\geq 3 \) there exist contact manifolds which admit infinitely many pairwise homotopy inequivalent Weinstein fillings [e26], [e30], [e48], and there exist contact 3-manifolds with infinitely many simply connected Weinstein fillings which are all homeomorphic but pairwise nondiffeomorphic [e33].
(2) This question also has two variants, depending on which symplectic structure we consider on the interior of \( W \). With the symplectic structure of its completion, we have the Weinstein analogue of S. Courte’s theorem [e42]: there exist pairs of Weinstein domains whose completions are symplectomorphic, but whose boundaries are nondiffeomorphic. With the (finite volume) symplectic structure induced from \( W \), we have some partial rigidity [e22]: if two Weinstein domains with nondegenerate periodic orbits on their boundaries have exact symplectomorphic interiors, then the action spectra of their boundaries agree.
Automorphisms
Convexity
In a Liouville manifold \( (V,\lambda) \), let us call a hypersurface \( S \) \( \lambda \)-convex if it admits a contact form \( \alpha \) such that \( \alpha-\lambda|_S \) is exact. The first obstructions to this property arise not near points but near closed characteristics. It is true but much less obvious that \( \lambda \)-convex hypersurfaces are not dense with respect to the Hausdorff metric on the space of closed hypersurfaces [e25].
Kai Cieliebak received his PhD from ETH Zürich in 1996. After postdocs at Harvard and Stanford and an associate professorship at Ludwig-Maximilians-Universität München, he is now full professor at Universität Augsburg.
