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# Journal of Symplectic Geometry

## Volume 20 (2022)

### Number 6

### On the minimal symplectic area of Lagrangians

Pages: 1385 – 1413

DOI: https://dx.doi.org/10.4310/JSG.2022.v20.n6.a5

#### Author

#### Abstract

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 the Arnol’d chord conjecture holds for the following four cases: **(1)** $Y$ admits an exact filling with $SH^\ast (W)=0$ (for some nonzero ring coefficient); **(2)** $Y$ admits a symplectically aspherical filling with $SH^\ast (W)=0$ 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 $S^\ast Q$ 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 $\geq 4$.

Received 6 April 2021

Accepted 20 May 2022

Published 26 April 2023