On the minimal symplectic area of Lagrangians

We show that the minimal symplectic area of Lagrangian submani-folds are universally bounded in symplectically aspherical domains with vanishing symplectic cohomology. If an exact domain admits a k-semi-dilation, then the minimal symplectic area is universally bounded for K (p, 1)-Lagrangians. As a corollary, we show that the Arnol'd chord conjecture holds for the following four cases: (1) Y admits an exact filling with SH*(W) = 0 (for some nonzero ring coefficient); (2) Y admits a symplectically aspherical filling with SH*(W) = 0 and simply connected Legendrians; (3) Y admits an exact filling with a k-semi-dilation and the Legendrian is a K(p, 1) space; (4) Y is the cosphere bundle S*Q with p(2)(Q) -> H-2(Q) non-trivial and the Legendrian has trivial p(2). In addition, we obtain the existence of homoclinic orbits in case (1). We also provide many more examples with k-semi-dilations in all dimensions = 4.