## Algebraic & Geometric Topology

### Helicity and the Mañé critical value

Gabriel P Paternain

#### Abstract

We establish a relationship between the helicity of a magnetic flow on a closed surface of genus $≥2$ and the Mañé critical value.

#### Article information

Source
Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1413-1422.

Dates
Accepted: 30 June 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513797034

Digital Object Identifier
doi:10.2140/agt.2009.9.1413

Mathematical Reviews number (MathSciNet)
MR2530122

Zentralblatt MATH identifier
1181.53035

#### Citation

Paternain, Gabriel P. Helicity and the Mañé critical value. Algebr. Geom. Topol. 9 (2009), no. 3, 1413--1422. doi:10.2140/agt.2009.9.1413. https://projecteuclid.org/euclid.agt/1513797034

#### References

• V I Arnold, On some problems in symplectic topology, from: “Topology and geometry–-Rohlin Seminar”, (O Y Viro, editor), Lecture Notes in Math. 1346, Springer, Berlin (1988) 1–5
• V I Arnold, B A Khesin, Topological methods in hydrodynamics, Applied Math. Sciences 125, Springer, New York (1998)
• K Burns, G P Paternain, Anosov magnetic flows, critical values and topological entropy, Nonlinearity 15 (2002) 281–314
• I Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Math. 115, Academic Press, Orlando, FL (1984) Including a chapter by B Randol, With an appendix by J Dodziuk
• K Cieliebak, U Frauenfelder, G P Paternain, Symplectic topology of Mañé's critical values
• G Contreras, L Macarini, G P Paternain, Periodic orbits for exact magnetic flows on surfaces, Int. Math. Res. Not. (2004) 361–387
• H Furstenberg, The unique ergodicity of the horocycle flow, from: “Recent advances in topological dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund)”, (A Beck, editor), Lecture Notes in Math. 318, Springer, Berlin (1973) 95–115
• V L Ginzburg, On closed trajectories of a charge in a magnetic field. An application of symplectic geometry, from: “Contact and symplectic geometry (Cambridge, 1994)”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 131–148
• V L Ginzburg, Comments to some of Arnold's problems (1981-9 and related problems and 1994-13), from: “Arnold's problems”, Springer-Verlag, Berlin (2004) 395–401, 557–558
• A Katok, Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynam. Systems 8$\sp *$ (1988) 139–152
• R Mañé, Lagrangian flows: the dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. $($N.S.$)$ 28 (1997) 141–153
• H K Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 106 (1969) 117–129
• S P Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37 (1982) 3–49, 248
• G P Paternain, Magnetic rigidity of horocycle flows, Pacific J. Math. 225 (2006) 301–323
• A Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. $($N.S.$)$ 20 (1956) 47–87
• C H Taubes, The Seiberg–Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007) 2117–2202
• A Verjovsky, R F Vila Freyer, The Jones–Witten invariant for flows on a $3$–dimensional manifold, Comm. Math. Phys. 163 (1994) 73–88