Extension of germs of holomorphic foliations

We consider the problem of extending germs of plane holomorphic foliations to foliations of compact surfaces. We show that the germs that become regular after a single blow up and admit meromorphic first integrals can be extended, after local changes of coordinates, to foliations of compact surfaces. We also show that the simplest elements in this class can be defined by polynomial equations. On the other hand we prove that, in the absence of meromorphic first integrals there are uncountably many elements without polynomial representations.Comment: 17 page

Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design

We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enneper surfaces.publishedVersio

Detecting Symmetries of Rational Plane and Space Curves

This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.Comment: 19 page

On curves with Poritsky property

For a given closed convex planar curve γ with smooth boundary and a given p > 0, the string construction is obtained by putting a string surrounding γ of length p + |γ| to the plane. Then we pull some point of the string "outwards from γ" until its final position A, when the string becomes stretched completely. The set of all the points A thus obtained is a planar convex curve Γ p. The billiard reflection T p from the curve Γ p acts on oriented lines, and γ is a caustic for Γ p : that is, the family of lines tangent to γ is T p-invariant. The action of the reflection T p on the tangent lines to γ ≃ S 1 induces its action on the tangency points: a circle diffeomorphism T p : γ → γ. We say that γ has string Poritsky property, if it admits a parameter t (called Poritsky-Lazutkin string length) in which all the transformations T p are translations t → t + c p. These definitions also make sense for germs of curves γ. Poritsky property is closely related to the famous Birkhoff Conjecture. It is classically known that each conic has string Poritsky property. In 1950 H.Poritsky proved the converse: each germ of planar curve with Poritsky property is a conic. In the present paper we extend this Poritsky's result to germs of curves to all the simply connected complete Riemannian surfaces of constant curvature and to outer billiards on all these surfaces. We also consider the general case of curves with Poritsky property on any two-dimensional surface with Riemannian metric and prove a formula for the derivative of the Poritsky-Lazutkin length as a function of the natural length parameter. In this general setting we also prove the following uniqueness result: a germ of curve with Poritsky property is uniquely determined by its 4-th jet. In the Euclidean case this statement follows from the above-mentioned Poritsky’s result