Symplectic inflation techniques and applications to embeddings, topological stability, and cones

<p>This article studies the geometry of symplectic and almost complex manifolds through the lens of inflation techniques. Originating in the 1990’s, inflation has played a central role in understanding the existence and connectedness of symplectic forms, as well as their applications to symplectic embeddings, symplectomorphism groups, and symplectic capacities. We provide a systematic account of different versions of inflation, including recent singular tame-to-tame refinements, and describe how each leads to distinct applications.</p> <p>As main results, we establish a strengthened singular inflation theorem and apply it to prove stability of symplectic embedding spaces in ruled surfaces with big symplectic forms. We further apply our methods to the study of almost Kähler cones, extending Zhang’s cone theorem [Proc. Lond. Math. Soc. 115 (2017), 1227–1275] from minimal ruled surfaces with a negative section to arbitrary ruled surfaces with a negative section. These results highlight new connections between the topology of embedding spaces, the structure of symplectic cones, and the geometry of almost Kähler manifolds.</p>