Nonlinear weakly sequentially continuous embeddings between Banach spaces

In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n{)}_n$ of a normalized weakly null sequence in $X$ satisfies $$\mathrm{\parallel }{e}_1+\dots +e_k{\mathrm{\parallel }}_{{\bar{\delta }}_Y}\lesssim \mathrm{\parallel }{e}_1+\dots +e_k{\mathrm{\parallel }}_S,$$ where ${\bar{\delta }}_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding.