On the minimal symplectic area of Lagrangians

We show that the minimal symplectic area of Lagrangian submanifolds 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 $K(\pi,1)$-Lagrangians. As a corollary, we show that Arnol'd chord conjecture holds for the following four cases: (1) $Y$ admits an exact filling with $SH^*(W)=0$ (for some 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 $K(\pi,1)$ space; (4) $Y$ is the cosphere bundle $S^*Q$ with $\pi_2(Q)\to H_2(Q)$ nontrivial and the Legendrian has trivial $\pi_2$. 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 $\ge 4$.