التفاصيل البيبلوغرافية
| العنوان: |
Volume functionals on pseudoconvex hypersurfaces. |
| المؤلفون: |
Donaldson, Simon, Lehmann, Fabian |
| المصدر: |
International Journal of Mathematics; Aug2024, Vol. 35 Issue 9, p1-19, 19p |
| مصطلحات موضوعية: |
AFFINE geometry, PSEUDOCONVEX domains, CALABI-Yau manifolds, COMPLEX manifolds, FUNCTIONALS, HYPERSURFACES, SUBMANIFOLDS |
| مستخلص: |
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 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 Ω ⊂ Z. That is, we study critical points of the volume functional A (M) where the ordinary volume V (Ω) 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 2 n − 1 ⊂ 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 2 n − 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. [ABSTRACT FROM AUTHOR] |
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