Criticality of $C^1$-topology minimax levels for the Tonelli action and integer-coefficient multiplicity for superlinear Hamiltonians with conormal boundary conditions

Two explicitly stated open questions from the 2007 paper of Abbondandolo and Figalli on high-action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces are the subject of this problem. (i) For a Tonelli Lagrangian $L$ (smooth, fiberwise strictly convex, and superlinear in the velocities) on a compact configuration manifold, the minimax levels $c'_\alpha(L)$ — defined via supports of singular cycles in the space of $C^1$ curves endowed with the $C^1$ topology — are known to dominate the action of some Euler–Lagrange orbit, and are known to be critical values when $L$ has at most quadratic growth in the velocities; whether this persists for every Tonelli Lagrangian is not known. (ii) The source's multiplicity theorem for superlinear Hamiltonians with coercive action integrand and conormal boundary conditions was proved with $\mathbb{Z}_2$-coefficients via mod-2 degree theory for Fredholm maps, and the extension to integer coefficients via coherent orientations was suggested but not carried out; an integer-coefficient Floer homology with conormal boundary conditions is known only under quadratic-type growth assumptions strictly stronger than the superlinear class of the source theorem. The objective is to resolve both questions, with no growth restriction beyond superlinearity.

Background

Let $M$ be a smooth closed (compact, without boundary) manifold, the configuration space of a mechanical system, and let $L:[0,1]\times TM\to\mathbb{R}$ be a $C^2$ function on the tangent bundle $TM$, whose points are pairs $(q,v)$ of a position and a velocity. Such an $L$ is called a Tonelli Lagrangian when it is strictly convex in the velocity variable (the Hessian $\partial_{vv}L$ is positive definite) and superlinear ($L(t,q,v)\ge K|v|-C(K)$ for every constant $K>0$, so it grows faster than any linear function of the speed). These are the standard hypotheses under which the Legendre transform $(q,v)\mapsto(q,\partial_vL)$ is a diffeomorphism onto the cotangent bundle $T^*M$ and under which the action functional

$$\mathbb{A}_L(\gamma)=\int_0^1 L\bigl(t,\gamma(t),\gamma'(t)\bigr)\,dt$$

attains minima on suitable classes of curves. Critical points of $\mathbb{A}_L$ are exactly the solutions of the Euler–Lagrange equation $\frac{d}{dt}\partial_vL(t,\gamma,\gamma')=\partial_qL(t,\gamma,\gamma')$, the classical equations of motion. Abbondandolo and Figalli (2007) fix a closed submanifold $Q\subset M\times M$ and impose the non-local boundary condition $(\gamma(0),\gamma(1))\in Q$ together with $\partial_vL(0,\gamma(0),\gamma'(0))[\xi_0]=\partial_vL(1,\gamma(1),\gamma'(1))[\xi_1]$ for every tangent vector $(\xi_0,\xi_1)\in TQ$: the endpoints are coupled by $Q$ and the endpoint momenta are conormal to $Q$ (orthogonal to $Q$ under the pairing of velocities and covectors). Taking $Q$ to be the diagonal gives periodic orbits; a single point gives the fixed-endpoint (Dirichlet) problem; $Q=M\times M$ gives a Neumann-type problem.

For Lagrangians with at most quadratic growth in $v$ and suitable uniform convexity, Benci (1986) showed that $\mathbb{A}_L$ is $C^1$ on the Hilbert manifold of $W^{1,2}$ curves (curves with square-integrable derivative, endowed with the $W^{1,2}$ Sobolev topology) and satisfies the Palais–Smale condition — every sequence with bounded action and differential tending to zero has a convergent subsequence — so classical Lusternik–Schnirelmann minimax theory applies: the number of critical points is at least the cuplength of the path space plus one, where the cuplength of a topological space is the largest number of cohomology classes of positive degree whose cup product is nonzero. Benci also obtained Morse-index estimates (the Morse index of a solution counting the independent directions in which the second variation of the action is negative definite) and, when infinitely many homology groups of the path space are nonzero, sequences of solutions with diverging action. Cieliebak (1994) proved the Hamiltonian analogue for Hamiltonians on $T^*M$ that are quadratic at infinity in the momenta, using Floer-theoretic methods (analysis of a Cauchy–Riemann-type equation whose one-dimensional solutions are gradient-flow lines of the action).

For a general Tonelli Lagrangian, which may grow superquadratically in $v$, no functional setting is known in which the action is simultaneously $C^1$ and Palais–Smale: on $W^{1,2}$ curves the action need not be differentiable, and in the $C^1$ topology its sublevel sets are non-compact. Abbondandolo and Figalli bypassed this obstruction by a modification technique: replace $L$ outside a ball $\{|v|\le R\}$ in each fiber by a convex quadratic $R$-modification $L_0$ (equal to $L$ for $|v|\le R$, uniformly convex and quadratically growing for $|v|$ large), run the quadratic-growth theory for $\mathbb{A}_{L_0}$ on $W^{1,2}$, and use an a priori estimate — extremals of bounded action have bounded $C^2$ norm, and critical points of $\mathbb{A}_{L_0}$ with action below a given bound actually satisfy $\|\gamma'\|_\infty\le R$ once $R$ is large — to conclude that the critical points so produced are genuine Euler–Lagrange orbits of $L$. This yielded multiplicity theorems for arbitrary Tonelli Lagrangians and, on the Hamiltonian side, for superlinear Hamiltonians.

Two limitations of the modification argument remained, and the source paper states both explicitly. First, on the Lagrangian side, fix a nonzero homology class $\alpha\in H_k(C_Q([0,1],M))$, where $C_Q([0,1],M)$ is the space of continuous curves $\gamma$ with $(\gamma(0),\gamma(1))\in Q$ and $\alpha$ is represented by a singular cycle (a finite formal sum of continuous maps of simplices). Let $\mathscr{K}'_\alpha$ be the family of supports of singular cycles representing $\alpha$ within the space of $C^1$ curves carrying the $C^1$ topology, and define the minimax level

$$c'_\alpha(L)=\inf_{K\in\mathscr{K}'_\alpha}\ \max_{\gamma\in K}\mathbb{A}_L(\gamma).$$

The modification argument produces, for every $\varepsilon>0$, a critical point of $\mathbb{A}_L$ with action at most $c'_\alpha(L)+\varepsilon$, hence by the a priori bounds a critical point $\gamma$ with $\mathbb{A}_L(\gamma)\le c'_\alpha(L)$. Under an additional at-most-quadratic growth assumption on $L$ one can choose modifications with $L_0\ge L$ and reverse the inequality, proving that $c'_\alpha(L)$ is a critical value; the authors state that it is not clear to them whether this persists for every Tonelli Lagrangian. Second, on the Hamiltonian side, the multiplicity theorem covers Hamiltonians $H:[0,1]\times T^*M\to\mathbb{R}$ that are superlinear in the momenta, whose action integrand $DH[Y]-H$ is coercive (here $Y$ is the Liouville vector field $\sum p_i\partial_{p_i}$ and $DH[Y]$ the directional derivative; for convex $H$ this quantity equals the Legendre–Fenchel dual Lagrangian of $H$ in the momenta, evaluated along the orbit, so its coercivity is the Hamiltonian counterpart of superlinear growth), and whose flow is complete; the boundary condition is the conormal one, $(x(0),-x(1))\in N^*Q$, the conormal bundle of $Q$. The theorem gives at least $\operatorname{cuplength}_{\mathbb{Z}_2}(C_Q([0,1],M))+1$ solutions, and infinitely many solutions with diverging action when infinitely many $\mathbb{Z}_2$-homology groups of the path space are nonzero. The proof uses the Smale $\mathbb{Z}_2$-degree for Fredholm sections of Hilbert bundles — a generalization of Brouwer degree to infinite dimensions, where the linearization at a zero is a Fredholm operator (an operator with finite-dimensional kernel and closed range of finite codimension); working mod 2 avoids the orientation issue for Fredholm maps. The authors remark that the extension to integer-valued cohomology could probably be obtained through the determinant bundle over the space of Fredholm operators — the coherent-orientation machinery introduced by Floer and Hofer (1993), which consistently orients the determinants of the linearizations so that counts with signs are well defined — but they do not carry this out. The numerical bound $\operatorname{cuplength}_{\mathbb{Z}}+1$ is never larger than the $\mathbb{Z}_2$-bound already proved, and nonvanishing integral homology in infinitely many degrees implies the corresponding mod-2 condition.

Problem Statement

Both questions concern the completion of the variational framework of Abbondandolo and Figalli (2007) for superlinear systems on compact configuration spaces — the first on the Lagrangian side, the second on the Hamiltonian side. The common objective is to bring the minimax theory of the Tonelli action and its Hamiltonian counterpart to the same level of completeness as the classical quadratic-growth theory, with no growth restriction beyond superlinearity. The two parts are technically independent; a solution of either part alone is a legitimate partial answer.

Part (i). Let $M$ be a smooth closed manifold, $Q\subset M\times M$ a closed submanifold, and $L:[0,1]\times TM\to\mathbb{R}$ a $C^2$ Tonelli Lagrangian whose associated Hamiltonian flow is complete (automatic, for instance, when $L$ is autonomous — by conservation of energy, the coercivity of the Hamiltonian, and the compactness of $M$ — and guaranteed by the growth condition $-\partial_tL(t,q,v)\le c(1+\partial_vL(t,q,v)[v]-L(t,q,v))$ on the time derivative). For a nonzero class $\alpha\in H_k(C_Q([0,1],M))$, let $\mathscr{K}'_\alpha$ be the family of supports of singular cycles representing $\alpha$ in the space $C^1_Q([0,1],M)$ of $C^1$ curves with $(\gamma(0),\gamma(1))\in Q$, equipped with the $C^1$ topology, and let $c'_\alpha(L)=\inf_{K\in\mathscr{K}'_\alpha}\max_{\gamma\in K}\mathbb{A}_L(\gamma)$. Determine whether $c'_\alpha(L)$ is necessarily a critical value of the Tonelli action — that is, whether there exists an extremal curve $\gamma$ (a solution of the Euler–Lagrange equation with the $Q$-boundary condition) with $\mathbb{A}_L(\gamma)=c'_\alpha(L)$ — for every Tonelli Lagrangian, with no growth-rate restriction beyond superlinearity. What is known: a critical point with $\mathbb{A}_L\le c'_\alpha(L)$ always exists, and equality holds when $L$ has at most quadratic growth in the velocities. A negative answer requires an explicit Tonelli Lagrangian and class $\alpha$ for which $c'_\alpha(L)$ is computed and shown not to be attained.

Part (ii). Let $M$ be a smooth closed manifold and $Q\subset M\times M$ a closed submanifold, as in part (i), and let $N^*Q\subset T^*(M\times M)$ denote the conormal bundle of $Q$ (the set of covectors annihilating the tangent spaces of $Q$). Let $H:[0,1]\times T^*M\to\mathbb{R}$ be $C^2$ with coercive action integrand $DH(t,q,p)[Y(q,p)]-H(t,q,p)\ge a(|p|)$ for some function $a(s)\to+\infty$ (where $Y$ is the Liouville vector field), superlinear growth $H(t,q,p)\ge h(|p|)$ with $h(s)/s\to+\infty$ as $s\to+\infty$, and complete Hamiltonian flow. Prove, with integer coefficients, the multiplicity conclusions that Abbondandolo and Figalli established with $\mathbb{Z}_2$-coefficients: the boundary value problem $x'=X_H(t,x)$, $(x(0),-x(1))\in N^*Q$ (where $X_H$ is the Hamiltonian vector field, given in local coordinates $(q,p)$ on $T^*M$ by $X_H=(\partial_pH,-\partial_qH)$) has at least $\operatorname{cuplength}_{\mathbb{Z}}(C_Q([0,1],M))+1$ solutions, and infinitely many solutions with diverging action whenever $H_k(C_Q([0,1],M);\mathbb{Z})\neq0$ for infinitely many $k$; in the nondegenerate case, establish the corresponding homology-graded counts — here nondegenerate means that at every solution the linearized boundary value problem is invertible, and the graded count asserts that for each degree $k$ the number of solutions of Morse-type index $k$ is at least the rank of $H_k(C_Q([0,1],M);\mathbb{Z})$.

References

1. Alberto Abbondandolo and Alessio Figalli, High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces, Journal of Differential Equations 234 (2007), 626–653. DOI: 10.1016/j.jde.2006.10.015; arXiv:math/0609633. https://doi.org/10.1016/j.jde.2006.10.015
2. Vieri Benci, Periodic solutions of Lagrangian systems on a compact manifold, Journal of Differential Equations 63 (1986), 135–161. DOI: 10.1016/0022-0396(86)90045-8. https://doi.org/10.1016/0022-0396(86)90045-8
3. Kai Cieliebak, Pseudo-holomorphic curves and periodic orbits on cotangent bundles, Journal de Mathématiques Pures et Appliquées (9) 73 (1994), no. 3, 251–278. zbMATH: 0837.58010. https://zbmath.org/811443
4. Andreas Floer and Helmut Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Mathematische Zeitschrift 212 (1993), 13–38. DOI: 10.1007/BF02571639. https://doi.org/10.1007/BF02571639
5. Alberto Abbondandolo, Alessandro Portaluri and Matthias Schwarz, The homology of path spaces and Floer homology with conormal boundary conditions, Journal of Fixed Point Theory and Applications 4 (2008), 263–293. DOI: 10.1007/s11784-008-0097-y; arXiv:0810.1977. https://doi.org/10.1007/s11784-008-0097-y
6. Alberto Abbondandolo and Matthias Schwarz, A smooth pseudo-gradient for the Lagrangian action functional, Advanced Nonlinear Studies 9 (2009), 597–623. DOI: 10.1515/ans-2009-0402; arXiv:0812.4364. https://doi.org/10.1515/ans-2009-0402
7. Marco Mazzucchelli, The Lagrangian Conley conjecture, Commentarii Mathematici Helvetici 86 (2011), 189–246. DOI: 10.4171/CMH/222; arXiv:0810.2108. https://doi.org/10.4171/CMH/222
8. Guangcun Lu, The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems, Journal of Functional Analysis 256 (2009), 2967–3034. DOI: 10.1016/j.jfa.2009.01.001; arXiv:0806.0425. https://doi.org/10.1016/j.jfa.2009.01.001
9. Guangcun Lu, Corrigendum to 'The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems' [J. Funct. Anal. 256 (2009) 2967–3034], Journal of Functional Analysis 261 (2011), 542–589. DOI: 10.1016/j.jfa.2011.02.027. https://doi.org/10.1016/j.jfa.2011.02.027
10. M. Arcostanzo, M.-C. Arnaud, P. Bolle and M. Zavidovique, Tonelli Hamiltonians without conjugate points and $C^0$-integrability, Mathematische Zeitschrift 280 (2015), 165–194. DOI: 10.1007/s00209-015-1417-8; arXiv:1309.6076. https://doi.org/10.1007/s00209-015-1417-8
11. Luca Asselle, On the existence of Euler–Lagrange orbits satisfying the conormal boundary conditions, Journal of Functional Analysis 271 (2016), 3513–3553. DOI: 10.1016/j.jfa.2016.08.023; arXiv:1507.05883. https://doi.org/10.1016/j.jfa.2016.08.023
12. Duanzhi Zhang and Zhihao Zhao, Infinitely many brake orbits of Tonelli Hamiltonian systems on the cotangent bundle, preprint (2023). arXiv:2302.09472. https://arxiv.org/abs/2302.09472
13. Miguel Paternain, Periodic orbits of Tonelli Lagrangians with prescribed action or period, Proceedings of the American Mathematical Society 152 (2024), 209–222. DOI: 10.1090/proc/16421. https://doi.org/10.1090/proc/16421
14. Filip Broćić and Dylan Cant, Bordism classes of loops and Floer's equation in cotangent bundles, Journal of Fixed Point Theory and Applications 26 (2024), article 26. DOI: 10.1007/s11784-024-01114-x; arXiv:2305.11783. https://doi.org/10.1007/s11784-024-01114-x
15. Guangcun Lu, Bifurcations in Lagrangian systems and geodesics I, Calculus of Variations and Partial Differential Equations 65 (2026), article 159. DOI: 10.1007/s00526-026-03318-0. https://doi.org/10.1007/s00526-026-03318-0