On certain quantifications of Gromov’s nonsqueezing theorem

Abstract

Let R > 1 and let B be the Euclidean 4-ball of radius R with a closed subset E removed. Suppose that B embeds symplectically into the unit cylinder D 2 R 2 . By Gromov's nonsqueezing theorem, E must be nonempty. We prove that the Minkowski dimension of E is at least 2, and we exhibit an explicit example showing that this result is optimal at least for R Ä p 2. In the appendix by Joé Brendel, it is shown that the lower bound is optimal for R < p 3. We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball. 53D05, 53D35We use the usual asymptotic notation f 2 O.g/ to mean jf j Ä Cg for some constant C , and f 2 o.g/ to mean that lim t !0 f .t/=g.t/ D 0. We write f 2 ‚.g/ if f 2 O.g/ and g 2 O.f /.Let S R n be any bounded subset. Let N t .S/ denote the open t-neighborhood of S with respect to the standard metric. If † is a compact submanifold (possibly with boundary), let V t . †/ be the exponential t-tube of †, ie the image by the normal exponential map of the open t-neighborhood of the zero-section in the normal bundle of † (which is endowed with the natural metric).We denote by Vol n the Euclidean n-volume of a set and setNote that when n is a natural number, ˛n D Vol n .B n / is precisely the Euclidean volume of the unit n-dimensional ball B n R n . For s 0, the s-dimensional lower Minkowski content of S is defined asNote that the normalization is chosen so that if † k R n is a closed k-dimensional submanifold, then ᏹ k . †/ D Vol k . †/ coincides with the Euclidean k-volume of †.