The geometry of the critical set of nonlinear periodic Sturm–Liouville operators

Burghelea, Dan; Saldanha, Nicolau C.; Tomei, Carlos

doi:10.1016/j.jde.2008.10.021

Cited by 9 publications

(10 citation statements)

References 22 publications

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“…We prove in Lemma 7.1 that, given Q, the closed subset M k ⊂ L Q of multiconvex curves of multiplicity k is either empty or a contractible submanifold 6 The homotopy type of L ±1 -October 13, 2018 Figure 3: Two multiconvex curves of multiplicity 3 of codimension 2k − 2 with trivial normal bundle. Assuming −z convex, we next construct in Lemma 7.3 maps h 2k−2 : S 2k−2 → L (−1) k z which intersect M k transversally and exactly once and are homotopic to a constant as maps S 2k−2 → I (−1) k z (details of the construction of the path in Section 5 are used here to verify that h 2k−2 has the desired properties).…”

Section: Introductionmentioning

confidence: 98%

“…This can occur in five different ways corresponding to five Bruhat cells to which Γ(t 0 ; t + 0 ) may belong. The two generic cases are when γ is about to leave the hemisphere defined by Γ(t 0 ) (but not at the point γ(t 0 )) or, conversely, when the geodesic defined by Γ(t) passes through γ(t 0 ) (but not aligned with γ ′ (t 0 )) so that γ is about to enter its own convex hull: these correspond to P (123); 6 and P (132);0 , in this order, and to the first two diagrams in Figure 5: Notice that these matrices have two inversions and therefore their Bruhat cells have dimension 2. Two more exceptional cases correspond to the matrices P (23);2 and P (12);4 which, having one inversion, correspond to Bruhat cells of dimension 1.…”

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confidence: 99%

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The homotopy type of spaces of locally convex curves in the sphere

Saldanha

2015

Geom. Topol.

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A smooth curve γ : [0, 1] → S 2 is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally convex curves γ with γ(0) = γ(1) = e 1 and γThe space L −1,c is known to be contractible. We prove that L +1 and L −1,n are homotopy equivalent to (ΩS 3 ) ∨ S 2 ∨ S 6 ∨ S 10 ∨ · · · and (ΩS 3 ) ∨ S 4 ∨ S 8 ∨ S 12 ∨ · · · , respectively. As a corollary, we deduce the homotopy type of the components of the space Free(S 1 , S 2 ) of free curves γ : S 1 → S 2 (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces Free([0, 1], S 2 ) with fixed initial and final frames.

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Section: Introductionmentioning

confidence: 98%

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confidence: 99%

The homotopy type of spaces of locally convex curves in the sphere

Saldanha

2015

Geom. Topol.

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“…This point of view was the original motivation of V. Arnold, B. Khesin, V. Ovsienko, B. Shapiro and M. Shapiro for considering this class of questions in the early nineties [16,17,18,28]. The second author was first led to consider this subject while studying the critical sets of nonlinear differential operators with periodic coefficients, in a series of works with D. Burghelea and C. Tomei [3,4,5,26].…”

Section: Introductionmentioning

confidence: 99%

A CW complex homotopy equivalent to spaces of locally convex curves

Goulart¹,

Saldanha²

2021

Preprint

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Locally convex (or nondegenerate) curves in the sphere S n (or the projective space) have been studied for several reasons, including the study of linear ordinary differential equations of order n + 1. Taking Frenet frames allows us to obtain corresponding curves Γ in the group Spin n+1 ; recall that Π : Spin n+1 → Flag n+1 is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves Γ with prescribed initial and final points appears to be a hard problem. Due to known results, we may focus on L n , the space of (sufficiently smooth) locally convex curves Γ : [0, 1] → Spin n+1 with Γ(0) = 1 and Π(Γ(1)) = Π(1). Convex curves form a contractible connected component of L n ; there are 2 n+1 other components, corresponding to non convex curves, one for each endpoint. The homotopy type of L n has so far been determined only for n = 2 (the case n = 1 is trivial). This paper is a step towards solving the problem for larger values of n.The itinerary of a locally convex curve Γ : [0, 1] → Spin n+1 belongs to W n , the set of finite words in the alphabet S n+1 {e}. The itinerary of a curve lists the non open Bruhat cells crossed by the curve. Itineraries yield a stratification of the space L n . We construct a CW complex D n which is a kind of dual of L n under this stratification: the construction is similar to Poincaré duality. The CW complex D n is homotopy equivalent to L n . The cells of D n are naturally labeled by words in W n so that D n is infinite but locally finite. Explicit glueing instructions are described for lower dimensions.As an application, we describe an open subset Y n ⊂ L n , a union of strata of L n . In each non convex component of L n , the intersection with Y n is connected and dense. Most connected components of L n are contained in Y n . For n > 3, in the other components the complement of Y n has codimension at least 2. We prove that Y n is homotopically equivalent to the disjoint union of 2 n+1 copies of Ω Spin n+1 . In particular, for all n ≥ 2, all connected components of L n are simply connected.

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“…This point of view was the original motivation of B. Khesin and B. Shapiro for considering the problem in the early nineties [32,33,34,61]. The second named author was first led to consider this problem while studying the topology and geometry of critical sets of nonlinear differential operators with periodic coefficients, in a series of works with D. Burghelea and C. Tomei [13,14,15,55] In Section 2 we review some basics of the symmetric group: useful representations of a permutation and the poset structures given by the strong and weak Bruhat orders. We also introduce the multiplicities of a permutation.…”

Section: Introductionmentioning

confidence: 99%

Combinatorialization of spaces of nondegenerate spherical curves

Goulart¹,

Saldanha²

2018

Preprint

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A parametric curve γ of class C n on the n-sphere is said to be nondegenerate (or locally convex) when det γ(t), γ (t), • • • , γ (n) (t) > 0 for all values of the parameter t. We orthogonalize this ordered basis to obtain the Frenet frame Fγ of γ assuming values in the orthogonal group SO n+1 (or its universal double cover, Spin n+1 ), which we decompose into Schubert or Bruhat cells. To each nondegenerate curve γ we assign its itinerary: a word w in the alphabet S n+1 {e} that encodes the succession of non open Schubert cells pierced by the complete flag of R n+1 spanned by the columns of Fγ . Without loss of generality, we can focus on nondegenerate curves with initial and final flags both fixed at the (non oriented) standard complete flag. For such curves, given a word w, the subspace of curves following the itinerary w is a contractible globally collared topological submanifold of finite codimension. By a construction reminiscent of Poincaré duality, we define abstract cell complexes mapped into the original space of curves by weak homotopy equivalences. The gluing instructions come from a partial order in the set of words. The main aim of this construction is to attempt to determine the homotopy type of spaces of nondegenerate curves for n > 2. The reader may want to contrast the present paper's combinatorial approach with the geometry-flavoured methods of previous works.

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The geometry of the critical set of nonlinear periodic Sturm–Liouville operators

Cited by 9 publications

References 22 publications

The homotopy type of spaces of locally convex curves in the sphere

The homotopy type of spaces of locally convex curves in the sphere

A CW complex homotopy equivalent to spaces of locally convex curves

Combinatorialization of spaces of nondegenerate spherical curves

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