Riemannian distance and symplectic embeddings in cotangent bundle

Abstract

Given an open neighborhood $W$ of the zero section in the cotangent bundle of $N$ we define a distance-like function ${\rho }_W$ on $N$ using certain symplectic embeddings from the standard ball $B^{2n}(r)$ to $W$. We show that when $W$ is the unit disc-cotangent bundle of a Riemannian metric on $N$, ${\rho }_W$ recovers the metric. As an intermediate step, we give a new construction of the ball of capacity 4 to the product of Lagrangian discs $P_L:=B^n(1)\times B^n(1)$, and we give a new proof of the strong Viterbo conjecture about normalized capacities for $P_L$. We also give bounds of the symplectic packing number of two balls in a unit disc-cotangent bundle relative to the zero section $N$.

corrected a mistake on the page 19 below Equation (8), small changes, typos corrected

References in corpus (1)

• Gromov width of the disk cotangent bundle of spheres of revolution

Cited by in corpus (2)

• Symplectic capacities of disc cotangent bundles of flat tori
• A note on the capacities of Lagrangian $p$-sum