Mathematische Annalen

, Volume 373, Issue 3–4, pp 1299–1327 | Cite as

Surfaces containing two circles through each point

  • M. SkopenkovEmail author
  • R. Krasauskas


We find all analytic surfaces in 3-dimensional Euclidean space such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface (and analytically depending on the point). The search for such surfaces traces back to the works of Darboux from XIXth century. We prove that such a surface is an image of a subset of one of the following sets under some composition of inversions:
  • the set \(\{p+q:p\in \alpha ,q\in \beta \}\), where \(\alpha ,\beta \) are two circles in \(\mathbb {R}^3\);

  • the set \(\{2\frac{p \times q^{}}{|p+q|^2}:p\in \alpha ,q\in \beta ,p+q\ne 0\}\), where \(\alpha ,\beta \) are circles in the unit sphere \({S}^2\);

  • the set \(\{(x,y,z): Q(x,y,z,x^2+y^2+z^2)=0\}\), where \(Q\in \mathbb {R}[x,y,z,t]\) has degree 2 or 1.

    The proof uses a new factorization technique for quaternionic polynomials.

Mathematics Subject Classification

51B10 13F15 16H05 



This results have been presented at Moscow Mathematical Society seminar, SFB “Discretization in geometry and dynamics” colloquim in Berlin, Ya. Sinai—G. Margulis conference, and the conference “Perspectives in real geometry” in Luminy. The authors are especially grateful to A. Pakharev for joint numerical experiments, related studies [26], and for pointing out that numerous surfaces we tried to invent all have form (4). The authors are grateful to N. Lubbes and L. Shi for parts of Fig. 1, to A. Gaifullin and S. Ivanov for finding gaps in earlier versions of the proofs, to A. Bobenko, J. Capco, S. Galkin, A. Kanunnikov, O. Karpenkov, A. Klyachko, J. Kollár, W. Kühnel, A. Kuznetsov, N. Moshchevitin, S. Orevkov, F. Petrov, R. Pignatelli, F. Polizzi, H. Pottmann, G. Robinson, I. Sabitov, J. Schicho, K. Shramov, S. Tikhomirov, V. Timorin, M. Verbitsky, E. Vinberg, J. Zahl, S. Zubė for useful discussions. The first author is grateful to King Abdullah University of Science and Technology for hosting him during the start of the work over the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Research University Higher School of Economics (Faculty of Mathematics) and Institute for Information Transmission Problems of the Russian Academy of SciencesMoscowRussia
  2. 2.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania

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