Viterbo's conjecture on the spectral norm of Lagrangians in cotangent bundles and the Arnold-type principle for Hausdorff limits of the zero section

For a closed manifold $N$, the Lagrangian Arnold conjecture predicts that a Lagrangian submanifold of the cotangent bundle $T^*N$ that is Hamiltonian isotopic to the zero section $O_N$ must meet $O_N$ in at least $\mathrm{cl}(N)$ points, the cup length of $N$. Buhovsky, Humilière and Seyfaddini proved that an Arnold-type principle survives in continuous settings: for $C^0$-Lagrangians of the form $L=\phi(O_N)$ with $\phi$ a Hamiltonian homeomorphism, fewer spectral invariants than $\mathrm{cl}(N)$ force $L\cap O_N$ to be homologically non-trivial. They asked whether the hypothesis $L=\phi(O_N)$ can be weakened to $L$ being a Hausdorff limit of Lagrangians Hamiltonian isotopic to $O_N$, a question they connected to Viterbo's 2007 conjecture that $\gamma(L_i)\to 0$ whenever such Lagrangians Hausdorff-converge to $O_N$, equivalently a uniform bound $\gamma(L)\le C(N)$ on the spectral width inside the unit co-disc bundle. The conjecture is proven only for special bases (compact rank-one symmetric spaces, string point-invertible manifolds, homogeneous spaces) and under Riemannian bounds. The problem asks to settle Viterbo's conjecture for arbitrary closed $N$, and to determine whether the Arnold-type principle extends to Hausdorff limits.

Background

The Arnold conjecture in symplectic geometry predicts that a Hamiltonian diffeomorphism (the time-one map of a flow generated by a time-dependent Hamiltonian function) of a closed symplectic manifold has at least as many fixed points as the minimal number of critical points of a smooth function. Its homological version gives the sharper lower bound $\mathrm{cl}(M)$, the cup length of $M$: one plus the largest $k$ for which $k$ cohomology classes of strictly positive degree have a non-zero cup product. The conjecture admits a Lagrangian analogue: in the cotangent bundle $T^*N$ of a closed manifold $N$, with its canonical symplectic form $d\lambda$ ($\lambda$ the Liouville form whose fiber coordinates are momentum variables), a Lagrangian submanifold (an $n$-dimensional submanifold on which $d\lambda$ vanishes, e.g. the zero section $O_N$ of vectors with zero momentum) that is Hamiltonian isotopic to $O_N$ must intersect $O_N$ in at least $\mathrm{cl}(N)$ points; this is a theorem obtained by Floer-theoretic methods.

To such a Lagrangian $L$, Viterbo (1992) associated, via generating functions (families of functions parametrized quadratically at infinity whose critical points generate $L$) and equivalently via Floer theory, spectral invariants $\ell(a,L)$: minimax values of the action functional indexed by homology classes $a\in H_*(N)\setminus\{0\}$, lying in the action spectrum of $L$. The spectral width $\gamma(L)=\ell([N],L)-\ell([\mathrm{pt}],L)$ (where $[N]$ and $[\mathrm{pt}]$ are the fundamental and point classes) is finite, vanishes exactly when all spectral invariants of $L$ coincide, and induces Viterbo's non-degenerate distance on the Hamiltonian orbit of the zero section.

In $C^0$ symplectic topology one studies homeomorphisms that are uniform limits of symplectic or Hamiltonian diffeomorphisms; a Hamiltonian homeomorphism (hameo) is such a limit of Hamiltonian diffeomorphisms. The Arnold conjecture itself breaks down at this regularity: there exist Hamiltonian homeomorphisms with a single fixed point on closed symplectic manifolds of dimension at least 4, and (Proposition 1.2 of the source paper) Hamiltonian homeomorphisms of $T^*M$ for closed surfaces $M$ whose image of the zero section meets $O_M$ in a single point. Buhovsky, Humilière and Seyfaddini showed that a reformulation survives: if $X$ is a possibly non-smooth object carrying spectral invariants and the number of distinct spectral values of $X$ is smaller than $\mathrm{cl}(N)$, then the fixed/intersection set of $X$ is homologically non-trivial (for every open neighborhood $U$ of the set there is a $j>0$ such that the inclusion-induced map $H_j(U)\to H_j(M)$ into the ambient manifold $M$ is non-trivial), hence infinite. For cotangent bundles they made this principle effective for $C^0$-Lagrangians of the form $L=\phi(O_N)$, $\phi$ a compactly supported Hamiltonian homeomorphism, by proving that the smooth spectral invariants extend continuously to such $L$; for 1-jet bundles $J^1N=T^*N\times\mathbb{R}$ they proved the analogous statement directly for Hausdorff limits of Legendrians contact-isotopic to the zero section, the action values being readable as the $z$-coordinates of intersection points with the zero wall.

The gap between these two settings is the subject of this problem. For Lagrangians in $T^*N$, no geometric reading of action values is available, and the paper's Remark 1.4 asks whether the hypothesis $L=\phi(O_N)$ can be weakened to $L$ being a Hausdorff limit of Lagrangians Hamiltonian isotopic to $O_N$. The paper's Remark 4.4 connects this to a conjecture of Viterbo (2007): if Lagrangians $L_i\subset T^*N$, each Hamiltonian isotopic to $O_N$, Hausdorff-converge to $O_N$, then $\gamma(L_i)\to 0$. By the fiber-dilation map $(q,p)\mapsto(q,\lambda p)$, which multiplies the Liouville form by $\lambda$ and scales all spectral invariants linearly, this is equivalent to: there exists a constant $C(N)$ such that $\gamma(L)\le C(N)$ for every Lagrangian $L$ contained in the unit co-disc bundle $D^*N$ and Hamiltonian isotopic to $O_N$. At the time of the source paper the conjecture was known only in special cases (Shelukhin: spheres and other compact rank-one symmetric spaces, tori, and a class defined by a string-topology condition); it remains open for arbitrary closed $N$.

Problem Statement

The problem consists of two linked questions posed in the source paper (Remarks 1.4 and 4.4). A complete answer settles both parts; a proof or a counterexample settling one part — including a solution of (B) by an independent construction — is an admissible answer to that part, and the closing paragraph specifies results that do not settle the problem.

(A) Viterbo's conjecture (2007) for arbitrary closed bases. Prove, or refute by an explicit counterexample, the following statement: for every closed smooth manifold $N$ there exists a constant $C(N)<\infty$ such that $\gamma(L)\le C(N)$ for every Lagrangian submanifold $L$ contained in the unit co-disc bundle $D^*N\subset T^*N$ and Hamiltonian isotopic to the zero section $O_N$. Here $\gamma(L)=\ell([N],L)-\ell([\mathrm{pt}],L)$ is the spectral width defined from the Lagrangian spectral invariants $\ell(a,L)$, $a\in H_*(N)\setminus\{0\}$, where $[N]$ and $[\mathrm{pt}]$ are the fundamental class and the point class. Equivalently, by the linear scaling of spectral invariants under the fiber dilation $(q,p)\mapsto(q,\lambda p)$: whenever a sequence $(L_i)$ of Lagrangians of $T^*N$, each Hamiltonian isotopic to $O_N$, converges to $O_N$ in the Hausdorff distance on compact subsets, one has $\gamma(L_i)\to 0$.

(B) The Arnold-type principle for Hausdorff limits. Determine whether Theorem 1.1 of the source paper remains valid when the hypothesis that $L=\phi(O_N)$ for a compactly supported Hamiltonian homeomorphism $\phi$ is weakened to: $L$ is a compact subset of $T^*N$ that is the Hausdorff limit of a sequence $(L_i)$ of Lagrangians each Hamiltonian isotopic to $O_N$. Concretely: define spectral values for such a limit set $L$ — for instance as the limits of the spectral invariants $\ell(a,L_i)$, a construction for which the $\gamma$-control of part (A) is the standard route — and settle whether the following holds: if the number of distinct spectral values of $L$ is smaller than the cup length $\mathrm{cl}(N)$ (one plus the largest number of positive-degree cohomology classes of $N$ with a non-zero cup product), then $L\cap O_N$ is homologically non-trivial, i.e. for every open neighborhood $U$ of $L\cap O_N$ there is some $j>0$ such that the map $H_j(U)\to H_j(T^*N)$ induced by the inclusion $U\hookrightarrow T^*N$ is non-trivial; in particular $L\cap O_N$ is infinite.

A positive solution of (A) yields (B) along the route indicated in Remark 4.4 of the source; a solution of (B) by an independent construction is equally admissible, as is a counterexample to either part. Answers restricted to special classes of base manifolds (rank-one symmetric spaces, homogeneous spaces, string point-invertible manifolds), to Riemannianly-bounded Lagrangians, or to variants obtained by replacing the zero section by a fiber, by a Legendrian in a contactization, or by modifying the co-disc bundle — the settings already treated or refuted in the literature above — do not settle this problem.

References

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