We state and prove a Cheeger-like inequality for coexact 1-forms on closed orientable Riemannian manifolds.

This paper concerns classical nonlinear scalar field models on the real line. If the potential is a symmetric double well, then such a model admits static solutions called kinks and antikinks, which are among the simplest examples of topological solitons. We study pure kink-antikink pairs, which are solutions that converge in one infinite time direction to a superposition of one kink and one antikink, without radiation. Our main result is a complete classification of all kink-antikink pairs in the strongly interacting regime, which means the speeds of the kinks tend asymptotically to zero. We show that up to translation there is exactly one such solution, and we give a precise description of the dynamics of the kink separation.

Using topological cyclic homology, we give a refinement of Beilinson’s p-adic Goodwillie isomorphism between relative continuous K-theory and cyclic homology. As a result, we generalize results of Bloch–Esnault–Kerz and Beilinson on the p-adic deformations of K-theory classes. Furthermore, we prove structural results for the Bhatt–Morrow–Scholze filtration on $\mathrm{TC}$ and identify the graded pieces with the syntomic cohomology of Fontaine–Messing.