Contact geometry in the restricted three-body problem: a survey

These are expanded notes for an online mini-course taught for postgraduate students at UDELAR, Montevideo, Uruguay, in November 2020, remotely from the Mittag-Leffler Institute in Djursholm, Sweden. Lectures were recorded and are available on YouTube on the author’s personal channel (∼ 8hs of material in three lectures).

[1]  Richard Siefring Intersection theory of punctured pseudoholomorphic curves , 2009, 0907.0470.

[2]  Wilhelm Klingenberg,et al.  Lectures on closed geodesics , 1978 .

[3]  G. Hill On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon , 1886 .

[4]  H. Poincaré,et al.  Les méthodes nouvelles de la mécanique céleste , 1899 .

[5]  Raoul Bott,et al.  On the iteration of closed geodesics and the sturm intersection theory , 1956 .

[6]  C. Conley,et al.  On Some New Long Periodic Solutions of The Plane Restricted Three Body Problem , 1963 .

[7]  K. Wysocki,et al.  Genus zero global surfaces of section for Reeb flows and a result of Birkhoff , 2019, Journal of the European Mathematical Society.

[8]  V. Arnold Some remarks on symplectic monodromy of Milnor fibrations , 1995 .

[9]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[10]  E. Zehnder,et al.  The dynamics on three-dimensional strictly convex energy surfaces , 1998 .

[11]  A. Oancea Morse theory, closed geodesics, and the homology of free loop spaces , 2014, 1406.3107.

[12]  W. Neumann Generalizations of the Poincaré Birkhoff fixed point theorem , 1977, Bulletin of the Australian Mathematical Society.

[13]  KEITH BURNS,et al.  Open problems and questions about geodesics , 2013, Ergodic Theory and Dynamical Systems.

[14]  Wolfgang Meyer,et al.  Periodic geodesics on compact riemannian manifolds , 1969 .

[15]  Onathan Exactly fillable contact structures without Stein fillings , 2012 .

[16]  A. Oancea A survey of Floer homology for manifolds with contact type boundary or symplectic homology , 2004, Ensaios Matemáticos.

[17]  Bahar Acu The Weinstein conjecture for iterated planar contact structures , 2017, 1710.07724.

[18]  H. Geiges,et al.  Diffeomorphism type of symplectic fillings of unit cotangent bundles , 2019, Journal of Topology and Analysis.

[19]  Nancy Hingston,et al.  Equivariant Morse theory and closed geodesics , 1984 .

[20]  William Dymock,et al.  1892 , 1893, The Indian medical gazette.

[21]  D. Mcduff Symplectic manifolds with contact type boundaries , 1991 .

[22]  C. Croke Poincaré's problem and the length of the shortest closed geodesic on a convex hypersurface , 1982 .

[23]  Christophe Golé,et al.  Poincaré's proof of Poincaré's last geometric theorem , 1990 .

[24]  H. Hofer,et al.  The Weinstein conjecture in cotangent bundles and related results , 1988 .

[25]  A. Katok ERGODIC PERTURBATIONS OF DEGENERATE INTEGRABLE HAMILTONIAN SYSTEMS , 1973 .

[26]  C. Wendl Strongly fillable contact manifolds and $J$-holomorphic foliations , 2008, 0806.3193.

[27]  K. Cieliebak,et al.  From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds , 2012 .

[28]  Otto van Koert,et al.  Global Surfaces of Section in the Planar Restricted 3-Body Problem , 2011, Archive for Rational Mechanics and Analysis.

[29]  M. Gromov Pseudo holomorphic curves in symplectic manifolds , 1985 .

[30]  Umberto L. Hryniewicz Fast finite-energy planes in symplectizations and applications , 2008, 0812.4076.

[31]  Otto van Koert,et al.  Global hypersurfaces of section in the spatial restricted three-body problem , 2020, Nonlinearity.

[32]  F. Bourgeois,et al.  Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces , 2007, 0704.1039.

[33]  M. Hutchings,et al.  Torsion contact forms in three dimensions have two or infinitely many Reeb orbits , 2017, Geometry & Topology.

[34]  C. Viterbo A proof of Weinstein’s conjecture in ℝ 2n , 1987 .

[35]  Hansjörg Geiges,et al.  An introduction to contact topology , 2008 .

[36]  H. Geiges,et al.  The diffeomorphism type of symplectic fillings , 2016, Journal of Symplectic Geometry.

[37]  Başak Z. Gürel,et al.  The Conley Conjecture and Beyond , 2014, 1411.7723.

[38]  J. Milnor Singular points of complex hypersurfaces , 1968 .

[39]  B. Ozbagci,et al.  Fillings of unit cotangent bundles of nonorientable surfaces , 2016, 1609.01891.

[40]  On the Floer homology of cotangent bundles , 2004, math/0408280.

[41]  K. Cieliebak,et al.  Symplectic homology and the Eilenberg-Steenrod axioms , 2015, 1511.00485.

[42]  Basak Z. Gurel,et al.  Conley Conjecture Revisited , 2016, 1609.05592.

[43]  R. S. Ward,et al.  Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces , 1999 .

[44]  P. Steerenberg,et al.  Targeting pathophysiological rhythms: prednisone chronotherapy shows sustained efficacy in rheumatoid arthritis. , 2010, Annals of the rheumatic diseases.

[45]  Henri Poincaré,et al.  Sur un théorème de géométrie , 1912 .

[46]  Doris Hein The Conley conjecture for the cotangent bundle , 2010, 1006.0372.

[47]  H. Hofer,et al.  Symplectic homology I open sets in ℂn , 1994 .

[48]  A. Sorrentino Lecture notes on Mather's theory for Lagrangian systems , 2010, 1011.0590.

[49]  H. Rademacher On the average indices of closed geodesics , 1989 .

[50]  G. Hill Researches in the Lunar Theory , 1878 .

[51]  A. Fet,et al.  Variational problems on closed manifolds , 1953 .

[52]  Geonwoo Kim,et al.  The contact geometry of the spatial circular restricted 3-body problem , 2020, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg.

[53]  Systems of global surfaces of section for dynamically convex Reeb flows on the 3-sphere , 2011, 1105.2077.

[54]  M. Gromov Homotopical effects of dilatation , 1978 .

[55]  F. Takens,et al.  Generic properties of geodesic flows , 1972 .

[56]  John Franks,et al.  Geodesics onS2 and periodic points of annulus homeomorphisms , 1992 .

[57]  Alexander F. Ritter,et al.  Invariance of symplectic cohomology and twisted cotangent bundles over surfaces , 2018, 1807.02086.

[58]  Umberto L. Hryniewicz,et al.  On the existence of disk-like global sections for Reeb flows on the tight $3$-sphere , 2010, 1006.0049.

[59]  Vinicius G. B. Ramos,et al.  The asymptotics of ECH capacities , 2012, 1210.2167.

[60]  Y. Eliashberg Unique holomorphically fillable contact structure on the 3-torus , 1996 .

[61]  Joel W. Robbin,et al.  The Maslov index for paths , 1993 .

[62]  The Conley Conjecture , 2006, math/0610956.

[63]  U. Frauenfelder,et al.  The Conley–Zehnder indices of the rotating Kepler problem , 2012, Mathematical Proceedings of the Cambridge Philosophical Society.

[64]  C. Wendl,et al.  Weak and strong fillability of higher dimensional contact manifolds , 2011, 1111.6008.

[65]  Kei Irie Dense existence of periodic Reeb orbits and ECH spectral invariants , 2015, 1508.07542.

[66]  W. Ziller Geometry of the Katok examples , 1983, Ergodic Theory and Dynamical Systems.

[67]  A. Stipsicz,et al.  The topology of Stein fillable manifolds in high dimensions I , 2013, 1306.2746.

[68]  Ana Cannas da Silva,et al.  Lectures on Symplectic Geometry , 2008 .

[69]  H. Geiges A brief history of contact geometry and topology , 2001 .

[70]  T. Ōba Lefschetz–Bott Fibrations on Line Bundles Over Symplectic Manifolds , 2019, International Mathematics Research Notices.

[71]  E. Crouser,et al.  54 , 2018, The Devil's Fork.

[72]  A. Chenciner Poincaré and the Three-Body Problem , 2015 .

[73]  H. Hofer Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three , 1993 .

[74]  Nancy Hingston,et al.  On the growth of the number of closed geodesics on the two-sphere , 1993 .

[75]  Otto van Koert,et al.  Brieskorn manifolds in contact topology , 2013, 1310.0343.

[76]  109 , 2019, Medicine & Science in Sports & Exercise.

[77]  H. Hofer,et al.  On the Weinstein conjecture in higher dimensions , 2007, 0705.3953.

[78]  H. Geiges,et al.  Reeb dynamics inspired by Katok’s example in Finsler geometry , 2017, Mathematische Annalen.

[79]  Otto van Koert,et al.  The Restricted Three-Body Problem and Holomorphic Curves , 2018 .

[80]  I. Batalin,et al.  Gauge theory , 2001 .

[81]  V. Bangert,et al.  ON THE EXISTENCE OF CLOSED GEODESICS ON TWO-SPHERES , 1993 .

[82]  J. Zukas Introduction to the Modern Theory of Dynamical Systems , 1998 .

[83]  D. Sullivan,et al.  The homology theory of the closed geodesic problem , 1976 .

[84]  J. Moser A fixed point theorem in symplectic geometry , 1978 .

[85]  W. Ziller The free loop space of globally symmetric spaces , 1977 .

[86]  On the lengths of closed geodesics on a two-sphere , 1997 .

[87]  59 , 2019, Critical Care Medicine.

[88]  Agustin Moreno,et al.  Bourgeois contact structures: Tightness, fillability and applications , 2019, Inventiones mathematicae.

[89]  Tian-Jun Li,et al.  Uniruled Caps and Calabi-Yau Caps , 2014 .

[90]  Henri Poincaré,et al.  Sur les lignes géodésiques des surfaces convexes , 1905 .

[91]  Umberto L. Hryniewicz,et al.  GLOBAL SURFACES OF SECTION FOR REEB FLOWS IN DIMENSION THREE AND BEYOND , 2017, Proceedings of the International Congress of Mathematicians (ICM 2018).

[92]  C. Viterbo Functors and Computations in Floer Homology with Applications, I , 1999 .

[93]  Kei Irie EQUIDISTRIBUTED PERIODIC ORBITS OF C∞-GENERIC THREE-DIMENSIONAL REEB FLOWS , 2021 .

[94]  H. Hofer,et al.  The weinstein conjecture in the presence of holomorphic spheres , 1992 .