We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space.

Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies.

We study asymptotics for solutions of Maxwell’s equations, in fact, of the Hodge–de Rham equation $(d+\delta)u = 0$ without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which, in particular, include Schwarzschild—de Sitter spaces of all spacetime dimensions $n \geq 4$. We prove that solutions decay exponentially to $0$ or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and, in particular, prove analogous results on Schwarzschild–de Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwell’s equations, the Hodge–de Rham equation and the wave equation on differential forms on Kerr–de Sitter spacetimes with small angular momentum.

For a nullhomologous Legendrian knot in a closed contact $3$-manifold $Y$ we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of $Y$ is mapped by a surgery cobordism to the contact invariant of the result of contact surgery, and we characterize the $\mathrm{spin}^c$ structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery on a Legendrian knot in the standard $3$-sphere, generalizing previous results of Lisca–Stipsicz and Golla. In fact, our methods allow direct calculation of the contact invariant in terms of the rational surgery mapping cone of Ozsváth and Szabó. The proof involves a construction called reducible open book surgery, which reduces in special cases to the capping-off construction studied by Baldwin.

In “Simple closed geodesics on convex surfaces” [J. Differential Geom., 36(3):517–549, 1992], E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the curvature. More precisely, in a closed $(n+1)$-manifold with $2 \leq n \leq 6$, a least area closed minimal hypersurface exists and any such hypersurface is embedded.

As an application, we give a short proof of the fact that if a closed three-manifold $M$ has scalar curvature at least $6$ and is not isometric to the round three-sphere, then $M$ contains an embedded closed minimal surface of area less than $4 \pi$. This confirms a conjecture of F. C. Marques and A. Neves.