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arXiv · 2012.03134

On the minimal symplectic area of Lagrangians

Zhengyi Zhou
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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 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 $\ge 4$.

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Zhengyi Zhou. 2020-12-05. On the minimal symplectic area of Lagrangians. https://arxiv.org/abs/2012.03134

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