[Paper Review] Remarks on variational problems for Fefferman's measure

David E. Barrett, Christopher Hammond|arXiv (Cornell University)|Sep 27, 2011

Geometric Analysis and Curvature Flows13 references3 citations

TL;DR

This paper investigates variational problems—specifically the Plateau and isoperimetric problems—associated with Fefferman's measure on strongly pseudoconvex hypersurfaces in ℂ². It establishes that the isoperimetric quotient shares features of both Euclidean and Blaschke's equiaffine geometry, and introduces a new biholomorphically invariant isoperimetric quantity, 𝒬*, which is strictly positive and extremal for the sphere, offering a refined invariant for CR geometry.

ABSTRACT

We investigate the Plateau and isoperimetric problems associated to Fefferman's measure for strongly pseudoconvex real hypersurfaces in $\mathbb C^n$ (focusing on the case $n=2$), showing in particular that the isoperimetric problem shares features of both the euclidean isoperimetric problem and the corresponding problem in Blaschke's equiaffine geometry in which the key inequalities are reversed. The problems are invariant under constant-Jacobian biholomorphism, but we also introduce a non-trivial modified isoperimetric quantity invariant under general biholomorphism.

Motivation & Objective

• To analyze the Plateau problem for Fefferman's measure on strongly pseudoconvex hypersurfaces in ℂⁿ, focusing on n=2.
• To investigate the isoperimetric problem associated with Fefferman's measure, particularly the behavior of the isoperimetric quotient 𝒬(Z).
• To construct a new biholomorphically invariant isoperimetric quantity 𝒬*(Z) that remains unchanged under general biholomorphic maps, not just constant-Jacobian ones.
• To establish the positivity and extremality of 𝒬*(Z), especially in relation to the unit sphere.
• To resolve normalization and curvature issues in Fefferman's measure and related geometric invariants.

Proposed method

• Uses Fefferman's measure defined via a Monge-Ampère-type determinant and a volume form on the hypersurface, ensuring invariance under biholomorphic maps with constant Jacobian.
• Applies the first variation formula to derive criticality conditions for hypersurfaces minimizing Fefferman's measure, leading to a PDE involving the Levi form and mean curvature.
• Introduces the modified isoperimetric quotient 𝒬*(Z) as an infimum over all biholomorphic images of the domain, ensuring invariance under general biholomorphisms.
• Employs Sobolev embedding estimates and the Jerison-Lee inequality to prove the positivity of 𝒬*(Z), relying on L² and L⁴/³ norms on the boundary.
• Uses normalization techniques and curvature conventions from Webster and Li-Luk to reconcile differing definitions of the contact form and curvature in the literature.
• Applies complex geometric analysis, including CR geometry and the theory of the Kohn Laplacian, to derive sharp inequalities and extremality conditions.

Experimental results

Research questions

• RQ1 Does the isoperimetric quotient 𝒬(Z) exhibit properties intermediate between Euclidean and Blaschke's equiaffine geometry?
• RQ2 Can a biholomorphically invariant isoperimetric quantity be constructed that remains finite and positive under general biholomorphic maps, not just volume-preserving ones?
• RQ3 Is the unit sphere the maximizer of the new invariant 𝒬*(Z) among all strongly pseudoconvex boundaries in ℂ²?
• RQ4 Are extremal hypersurfaces guaranteed to exist in the definition of 𝒬*(Z), i.e., does the infimum achieve a minimum?
• RQ5 How do different normalization choices in Fefferman's measure and Webster curvature affect the resulting geometric inequalities?

Key findings

• The isoperimetric quotient 𝒬(Z) is invariant under constant-Jacobian biholomorphisms, and for the unit sphere in ℂ², 𝒬(S⁵) = 8π.
• The modified isoperimetric quantity 𝒬*(Z) is strictly positive and invariant under all biholomorphic maps, providing a true geometric invariant.
• The sphere maximizes 𝒬*(Z) among all compact strongly pseudoconvex boundaries in ℂ², assuming the conjecture in Question 26 holds.
• The inequality ‖h‖_{L²(Ω)} ≤ C_Ω ‖h‖_{L⁴/³(Z)} holds for holomorphic functions on the domain Ω bounded by Z, which implies the positivity of 𝒬*(Z).
• The Jerison-Lee Sobolev inequality is recovered and refined via a complex geometric approach, with equality achieved when the function is constant.
• The paper resolves normalization ambiguities in Fefferman's measure and relates curvature conventions to known results in CR geometry, particularly those of Li and Luk.