Singular Monge-Ampere foliations (original) (raw)
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Singular Monge-Amp�re foliations
Mathematische Annalen, 2003
This paper generalizes results of Lempert and Szöke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C 3 everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampère equation. These results are proved using techniques from contact geometry.
Foliations and complex monge-ampère equations
Communications on Pure and Applied Mathematics, 1977
Here we shall study some nonlinear partial differential equations involving the complex hessian cfic) = (d2f/dz, a&) which are related to the theory of functions of several complex variables. Since 8 and a are defined independently of the choice of local coordinates, the forms in the equations are invariant under holomorphic mappings. Certain properties of the forms on the left-hand side of (l.l), (1.2), and (1.3) were discussed by Chern, Levine, and Nirenberg [4] for the case where f is a bounded, real, plurisubharmonic function. The approach here is to drop the assumption that f is real and plurisubharmonic and require instead that f is Y 3 . In C", the (1, 1)-form %f may be identified with the complex hessian (hi) so that (l.l), (1.2), and ( are easily seen to be equivalent (respectively) to the following three conditions (1.1y rank c f i~) 5 p , A familiar example of a foliation which is defined by equations of complex Monge-Amphre type is the Levi foliation of a real submanifold S of C". It is shown that if the Levi foliation Y of S is nontrivial, then S is a complex analogue of a developable surface. This applies to an example of Sommer, which is the complex analogue of a ruled, but not developable, surface.
Contact geometry of one dimensional holomorphic foliations
Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let \theta be a contact form for (V,HT(V)), \xi_0 the Reeb vector field determined by \theta and assume that \xi_0 is of class C^k. In this paper we prove the following theorem (cf. Theorem 4.1): if the integral curves of \xi_0 are real analytic then there exist an open neighbourhood N\subset M of V and a solution u\in C^k(N) of the complex Monge-Amp\`ere equation (dd^c u)^(n+1)=0 on N which is a defining equation for V. Moreover, the Monge-Amp\`ere foliation associated to u induces on V that one associated to the Reeb vector field. The converse is also true. The result is obtained solving a Cauchy problem for infinitesimal symmetries of CR distributions of codimension one which is of independent interest (cf. Theorem 3.1). Comment: 15 pages
Monge-Ampère foliations with singularities at the boundary of strongly convex domains
Mathematische Annalen, 2005
Let D ⊂ C N be a bounded strongly convex domain with smooth boundary. We consider a Monge-Ampère type equation in D with a simple pole at the boundary. Using the Lempert foliation of D in extremal discs, we construct a solution u whose level sets are boundaries of horospheres. Among other things, we show that the biholomorphisms between strongly convex domains are exactly those maps which preserves our solution.
Annali di Matematica Pura ed Applicata, 2019
In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if F is a singular Finsler foliation with closed leaves on a Randers manifold (M, Z) with Zermelo data (h, W), then F is a singular Riemannian foliation on the Riemannian manifold (M, h). As a direct consequence, we infer that the regular leaves are equifocal submanifolds (a generalization of isoparametric submanifolds) when the wind W is an infinitesimal homothety of h (e.g., when W is a Killing vector field or M has constant Finsler curvature). We also present a slice theorem that locally relates singular Finsler foliations on Finsler manifolds with singular Finsler foliations on Minkowski spaces.
Harmonic forms and near-minimal singular foliations
Commentarii Mathematici Helvetici, 2002
For a closed 1-form ω with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which ω is harmonic. For a codimension 1 foliation F , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of F are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form ω has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, ω is harmonic and the associated foliation F ω is comprised of minimal leaves. However, when ω has singularities, the foliation Fω is not necessarily minimal. We show that the Calabi condition enables one to find a metric in which ω is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the ω-singular set. In fact, we prove the best possible result of this type: we construct families of metrics in which, as U shrinks to the singular set, the taut geometry of the foliation F ω outside U remains stable. Furthermore, all compact leaves missing U are volume minimizing cycles in their homology classes. Their volumes are controlled explicitly.
Singular holomorphic foliations
1984
A generalized Nash Blow-up M ′ with respect to coherent subsheaves of locally free sheaves is defined for complex spaces. It is shown that M ′ is locally isomorphic to a monoidal transformation and hence is analytic. Examples of M ′ are given. Applications are given to Serre’s extension problem and reductive group actions. A C∗-action on Grassmannians is defined, fixed point sets and Bialynicki-Birula decomposition are described. This action is generalized to Grassmann bundles. The Grassmann graph construction is defined for the analytic case and it is shown that for a compact Kaehler manifold the cycle at infinity is an analytic cycle. A calculation involving the localized classes of graph construction is given. Nash residue for singular holomorphic foliations is defined and it is shown that the residue of Baum-Bott and the Nash residue differ by a term that comes from the Grassmann graph construction of the singular foliation. As an application conclusions are drawn about the rati...
Holomorphic foliations and Kupka singular sets
Communications in Analysis and Geometry, 1999
Let T be a nondicritical codimension one holomorphic foliation on the fc-dimensional complex projective space M = CP(fc), k > 3. Assume that the closure K^) of the Kupka singular set is a complete intersection projective variety and that the points in K(!F)-K^) exhibit slightly generic holomorphic integrating factors. Then the Kupka components have normalizable transverse type and therefore the foliation F is given by a closed rational 1form on CP(fc). In general (for M a complex manifold and K^) not necessarily complete intersection), we prove the existence of an affine or projective transverse structure in a neighborhood of the Kupka singular set. 1. Introduction. 1.1. Singular holomorphic foliations. According to Probenius Theorem [23] a (nonsingular) codimension one holomorphic foliation T on a complex manifold M is given by an open covering M = (J Uj of M such that in each Uj we have a nonsingular holomorphic 1-form Uj that satisfies the integrability condition ujj A dujj = 0, and such that for each Uj fl [/; ^ 0 we have UJ^JJ nU = dij-^jlu^jj. f or some holomorphic nonvanishing function gij in Ui D Uj. If M is a complex projective space M = CP(fc) then it is known that there are no such foliations without singularities. On the other hand we may consider singular holomorphic foliations defined as follows: A singular codimension one holomorphic foliation on M is defined by an open cover M = [jUj and integrable holo-jeJ morphic 1-forms as above, but that may have singularities. The singular 1 Scardua was supported by CNPq-Brasil and Min Affaires Etrangeres-France.
Schlesinger foliation for deformations of foliations
arXiv (Cornell University), 2016
In this article, we show that for any deformation of analytic foliations, there exists a maximal analytic singular foliation on the space of parameters along the leaves of which the deformation is integrable.