Volume functionals on pseudoconvex hypersurfaces

The focus of this paper is on a volume form defined on a pseudoconvex hypersurface $M$ in a complex Calabi–Yau manifold (that is, a complex $n$-manifold with a nowhere-vanishing holomorphic $n$-form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in ${\mathrm{\text{R}}}^n$. We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces $M$ bounding compact domains $\mathrm{\Omega }\subset Z$. That is, we study critical points of the volume functional $A(M)$ where the ordinary volume $V(\mathrm{\Omega })$ is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere $S^{2n-1}\subset {\mathrm{\text{C}}}^n$ does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in “most” directions but non-negative in directions corresponding to deformations of $S^{2n-1}$ by holomorphic diffeomorphisms. We are led to conjecture a “minimax” characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case $n=3$ and the embedding problem studied in our previous paper [S. Donaldson and F. Lehmann, Closed 3-forms in five dimensions and embedding problems, preprint (2022), arXiv:2210.16208]. There are some special features in this case. The volume functional can be defined without reference to the embedding in $Z$ using only a closed “pseudoconvex” real $3$-form on $M$. In Sec. 4, we review this and develop some of the theory from the point of the symplectic structure on exact $3$-forms on $M$ and the moment map for the action of the diffeomorphisms of $M$.