Shadow–flux lower-bound conjecture for Lagrangian cobordisms

Let $L_0$ and $L_1$ be Lagrangian isotopic submanifolds, and let $K:L_1\rightsquigarrow L_0$ be a Lagrangian cobordism. Let $d(L_0,L_1)$ denote the minimal flux distance, defined by

$$d(L_0,L_1):=\inf_{\substack{i_t\in\mathcal I(L_0,L_1)\\ B\in\mathcal B}}\sum_{c\in B}|\operatorname{Flux}_{i_t}(c)|,$$

where $\mathcal I(L_0,L_1)$ is the set of Lagrangian homotopies between $L_0$ and $L_1$, $\mathcal B$ is the set of bases of $H_1(L,\mathbb Z)$, and $\underline b_1(K)$ is the first Betti number of the cobordism. Shadow–flux lower-bound conjecture. There exists a constant $A$ such that, for every such pair and cobordism,

$$\operatorname{Area}(K)>\frac{A\cdot d(L_0,L_1)}{\underline b_1(K)}.$$

This conjecture proposes a quantitative trade-off between the shadow area and topological complexity of a Lagrangian cobordism. The surrounding discussion explains that cobordisms with smaller shadow can be constructed at the cost of increasing their first Betti number; whether the stated universal lower bound holds remains open.