Carlos Gutierrez

A solution to the bidimensional Global Asymptotic Stability Conjecture

If Y : ℜ2 → ℜ2 is а С1 vector field such that Y(0) = 0 and, for all q ∈ ℜ2, all the eigenvalues of DX(q) have negative real part, then the stable manifold of 0 is ℜ2. n nLet ρ ∈ [0, ∞) and Y : ℜ2 → ℜ2 be a C1 map such that, for all q ∈ ℜ2, the determinant of DY (q) is positive and moreover, for all p ∈ ℜ2, with |p| ≥ ρ, the spectrum of DY (p) is disjoint of the non-negative real half axis. Then Y is injective.

Asymptotic stability at infinity of planar vector fields

Let ρ>0 andX be aC1 vector field on the plane such that: (i) for allq∈ℝ2, Det(DX(q))>0; and (ii) for allp∈ℝ2, with ‖p‖≥ρ, Trace(D(X(p))<0. IfX has a singularity and ∫ℝ2 Trace(DX)dx⋏dy is less than 0 (resp. greater or equal than 0), then the point at infinity of the Riemann sphere ℝ2∪{∞} is a repellor (resp. an attractor) ofX.

Lines of curvature on surfaces immersed in ℝ4

The differential equation of thelines of curvature for immersions of surfaces into ℝ4 is established. It is shown that, for a class of generic immersions of a surface into ℝ4 in theCr-topology,r≥4, all of the umbilic points are locally topologically stable. This type of umbilic points is described.

On C r -closing for flows on 2-manifolds

For some full measure subset ℬ of the set of interval exchange transformations (IETs) the following is satisfied. Let X be a Cr, 1 ≤ r ≤ ∞, vector field, with finitely many singularities, on a compact orientable surface M. Given a non-trivial recurrent point pM of X, the holonomy map around p is semi-conjugate to an IET E : [0,1)→[0,1). If Eℬ then there exists a Cr vector field Y, arbitrarily close to X, in the Cr-topology, such that Y has a closed trajectory passing through p.

On a Conjecture of Carathéodoryr Analyticity Versus Smoothness

TheclassicalCarathx13eodoryConjecturestatesthateverysmo othconvexemb eddingofa2-sphereinR3musthaveatleasttwoumbilics.Aellknownapproachtotheproblemisbasedonasemi-lo calargument.ForanysurfaceinR3,theeigenspacesofthesecondfundamentalformde netwoorthog-onalline elds(principaldirections)whosesingu-laritiesareexactlytheumbilics.Toeachisolatedumbilicwecanattachtheindexofeitheronetwo elds,whichishalfofaninteger,andthesumofthoseindexesistheEulercharacteristicsurface,ifthesurfaceiscompactandallumbilicsareisolated.So,ifanemb eddedspherehasonlyoneumbilic,thismusthaveindextwo.Wjustob-servethat,uptoaninversionR3,wcanalwayssupp osethatthecurvatureatagivenumbilicisp ositive,andthereforetheconvexityhyp othesisisnotrelevantforthisargument.Examplesofumbilicsindexjareknownforalljx141.Alo calconjecturestrongerthanCarathx13eo-dorys,knownastheLo ewnerconjecture,statesthattherearenoumbilicsofindexgreaterthenone.Thisconjecturehasb eenassertedtob etrueforanalyticsurfacesbyseveralauthors[Hamburger1940{1941;Bol1943{1944;Klotz1959;Titus1973],implyingthereforeCarathx13eodorysConjectureforcAKPeters,Ltd.1058-6458/96

Dissipative vector fields on the plane with infinitely many attracting hyperbolic singularities

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Lines of principal curvature for mappings with Whitney umbrella singularities

For anyb>0, there are dissipative analytic vector fields on ℝ2 which, when restricted to ℝ×(−b,b), have positive Jacobian and infinitely many (attracting) singularities.