A Viscosity method for the min-max construction of closed geodesics

Michelat, Alexis; Rivière, Tristan

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1282-1324.

We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. The existence is proved in the case of surfaces, and reduced to a topological condition in general. We also construct counter-examples in dimension $1$ and $2$ to the $\epsilon$-regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index of our sequence of critical points converging towards a closed non-trivial geodesic.

Classification : 49J35, 58B20, 58E10
Mots clés : Geodesics, minimax problems, Finsler geometry

W.K. Allard and F.J. Almgren, The structure of stationary one dimensional varifolds with positive density. Invent. Math. Springer-Verlag (1976). | Zbl

N. Anantharaman, L’héritage scientifique de Poincaré. In Chap. 7. Belin (2006).

T. Aubin, Some Nonlinear Problems in Riemannian Geometry. Springer-Verlag (1998). | Zbl

W. Ballmann, G. Thorbergsson and W. Ziller, Closed geodesics and the fundamental group. Duke Math. J. 48 (1981). | DOI | Zbl

V. Bangert, On the existence of closed geodesics on two-spheres. Int. J. Math. 4 (1993) 1–10. | DOI | Zbl

V. Benci and F. Giannoni, On the existence of closed geodesics on noncompact riemannian manifolds. Duke Math. J. 68 (1992). | DOI | Zbl

F. Bethuel, H. Brezis and F. Hélein, Ginzburg−Landau Vortices. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Boston (1994). | Zbl

D.G. Birkhoff, Dynamical systems with two degrees of freedom. Amer. Math. Soc. (1917). | JFM

G.D. Birkhoff, Dynamical systems. American Mathematical Society, New York (1927). | JFM

R. Bott, Lectures on Morse theory, old and new. Bull. Amer. Math. Soc. (New Series) 7 (1982) 331–358. | Zbl

K. Burns and V.S. Matveev, Open problems and questions about geodesics. Preprint (2014). | arXiv

P. Buser and H. Parlier, The distribution of simple closed geodesics on a Riemann surface. American Mathematical Society (2004). | Zbl

P.F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Enginners and Scientists. 2nd edition, Revised. Springer-Verlag (1971). | Zbl

E. Calabi and J. Cao. Simple closed geodesics on convex surfaces. J. Differ. Geom. 36 (1992) 517–549. | DOI | MR | Zbl

T.H. Colding and C.De Lellis, The min-max construction of minimal surfaces. Surveys of Differential Geometry, IX. International Press (2003). | MR | Zbl

T.H. Colding and W.P. Minicozzi, A Course in Minimal Surfaces. Amer. Math. Soc. 121 (2011). | MR | Zbl

F. Da Lio and T. Riviére, A viscosity approach to the min-max construction of free boundary discs. In preparation (2015).

H. Duan and Y. Long, Multiple closed geodesics on $3$-spheres. Adv. Math. 221 (2009) 1757–1803. | DOI | MR | Zbl

H. Federer, Geometric Measure Theory. Springer-Verlag (1969). | MR | Zbl

W. Fenchel, Über krümmung und Windung geschlossener Raumkurven. Math. Ann. 101 (1929) 238–252. | DOI | JFM | MR

J. Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms. Invent. Math. 108 (1992) 403–418. | DOI | MR | Zbl

J.S. Hadamard, Les surfaces à courbure opposées et leurs lignes géodésiques. J. Math. Pures Appl. 4 (1898) 27–74. | JFM | Numdam

R. Hardt and L. Simon, Seminar on Geometric Measure Theory. Birkhaüser (1986). | MR | Zbl

M.W. Hirsch, Differential Topology. Grad. Texts Math. Springer-Verlag New-York (1976). | MR | Zbl

F. Hélein, Applications harmoniques, lois de conservation, et repères mobiles. Diderot éditeur, Sciences et Arts (1996).

J. Jost. Riemannian Geometry and Geometric Analysis, sixth edition. Springer (2011). | MR | Zbl

W. Klingenberg, Lectures on Closed Geodesics. Springer-Verlag (1977). | MR | Zbl

S.G. Krantz, Geometric Integration Theory. Cornerstones (2008). | MR | Zbl

T. Lamm, Fourth order approximation of harmonic maps from surfaces. Calc. Var. 27 (2006) 125–157. | DOI | MR | Zbl

T. Lamm, Energy identity for approximations of harmonic maps from surfaces. Trans. Amer. Math. Soc. 362 (2010) 4077–4097. | DOI | MR | Zbl

J.M. Lee, Riemannian Manifolds, An Introduction to Curvature. Springer-Verlag, New York, Inc. (1997). | MR | Zbl

Y. Liokumovich, Sweepouts of Riemannian surfaces. Ph.D. thesis (2015). | MR

J. Langer and D.A. Singer, The total squared curvature of closed curves. J. Differ. Geom. 20 (1984) 1–22. | DOI | MR | Zbl

J. Langer and D.A. Singer, Curve straightening and a minimax argument for closed elastic curves. Topology 24 (1985) 75–88. | DOI | MR | Zbl

J. Milnor, Morse Theory. Ann. Math. Stud. Princeton University Press (1963). | MR | Zbl

J.F. Nash, The imbedding problem for Riemannian Manifolds. Ann. Math. 63 (1956) 20–63. | DOI | MR | Zbl

R.S. Palais, Morse theory on Hilbert manifolds. Topology 2 (1963) 299–340. | DOI | MR | Zbl

R.S. Palais, Homotopy theory of infinite dimensional manifolds. Topology 5 (1966) 1–16. | DOI | MR | Zbl

R.S. Palais, Critical points Theory and the Minimax Principle. Proc. Symp. Pure Math., Volume XV, Global Analysis (1970). | MR | Zbl

F. Paulin. Groupes et géométries. Lecture notes (2014).

U. Pinkall, Hopf tori in $S^3$. Invent. Math. 81 (1985) 379–389. | DOI | MR | Zbl

J.T. Pitts, Regularity and singulaity of one dimensional stationary integral varifolds on manifolds arising from variational methods in the large. Convegno sulla teoria geometrica dell’ integrazione e varietàminimali, Instituto Nazionale di Alta Matematica, CittàUiversitaria, Roma, May 1973, Symposia Mathematica, Volume XIV (1974). | MR | Zbl

H. Poincaré, Sur les lignes géodésiques des surfaces convexes. Trans. Amer. Math. Soc. 17 (1905). | JFM | MR

H.-B. Rademacher, On the average indices of closed geodesics. J. Differ. Geom. 29 (1989) 65–83. | MR | Zbl

T. Rivière, A Viscosity Method in the Min-max Theory of Minimal Surfaces. Preprint (2015). | arXiv | MR

M. Struwe, Positive solutions of critical semilinear elliptic equations on non-contractible planar domains. J. Eur. Math. Soc. 2 (2000) 329–388. | DOI | MR | Zbl

M. Struwe, Variational methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, fourth edition. Springer-Verlag (2008). | MR | Zbl

J. Sacks and K. Uhlenbeck, The Existence of Minimal Immersions of $2$-Spheres. Ann. Math. (1981). | MR | Zbl

D. Sullivan and M. Vigué-Poirrier, The homology theory of the closed geodesic problem. J. Differ. Geom. 11 (1976) 633–644. | MR | Zbl

I.A. Taimanov, On the existence of three nonselintersecting closed geodesics on manifolds homeomorphic to the $2$-sphere. Russ. Acad. Sci. Izv. Math. 40 (1992). | MR | Zbl

I.A. Taimanov, The type numbers of closed geodesics. Preprint (2010). | arXiv | MR | Zbl

L. Tartar, Compensated compactness and applications to partial differential equations. Lecture notes (1979). | MR | Zbl

L. Tartar, From Hyperbolic Systems to Kinetic theory, A Personalized Quest. Springer-Verlag, Berlin, Heidelberg (2008). | MR | Zbl

G. Tian, M.-C. Hong and H. Yin, The Yang-Mills $\alpha$-flow in Vector Bundles over Four Manifolds and its Applications. Comment. Math. Helv. 90 (2015) 75–120. | DOI | MR | Zbl

H. Whitney, Geometric Integration Theory. Princeton University Press (1957). | MR | Zbl