Positive mass theorem and the Yamabe equation on CR manifolds

Abstract: Our goal is to survey the development of positive mass theorem and the Yamabe equation on CR manifolds in recent years. We introduce the notion of the mass in several complex variables or CR geometry. We then consider the Yamabe problem on CR manifolds to find a minimizer for the CR-Sobolev quotient. The positive mass theorem plays a key role in finding a solution to the Yamabe equation with minimum energy for the positive curvature case. We mainly focus on the team works in the following three papers [CMY17], [CMY23] and [CC22], on a positive mass theorem in 3-dimensional CR geometry, the CR-Sobolev quotient of Rossi spheres, and the 5-dimensional situation, respectively.

• The paper demonstrates that the CR positive mass theorem under scalar curvature and torsion constraints ensures rigidity and embeddability in three-dimensional cases.
• It employs variational techniques on the CR Yamabe equation, leveraging invariants like the CR Paneitz operator to compute the pseudohermitian mass.
• The study extends the analysis to higher dimensions, including spin manifolds, and identifies negative mass phenomena in non-embeddable CR structures such as Rossi spheres.

Positive Mass Theorem and the Yamabe Equation on CR Manifolds

Introduction and Context

The manuscript "Positive mass theorem and the Yamabe equation on CR manifolds" (2606.19801) delineates substantial advances in the analysis of geometric partial differential equations within CR (Cauchy-Riemann) geometry. The survey synthesizes a sequence of key results regarding the positive mass theorem, the variational Yamabe problem, and their interplay in the setting of strongly pseudoconvex, compact CR manifolds. The paper introduces the notion of $p$-mass (pseudohermitian mass) in several complex variables, formalizes its connection to the CR Yamabe minimization, and explores its computation and implications in both three- and higher-dimensional cases, including scenarios involving non-embeddable CR structures and contact spin manifolds.

Foundational Constructs in CR Geometry

A CR manifold $(M^{2n+1}, J)$ is an odd-dimensional contact manifold equipped with a complex structure $J$ on its contact distribution, reflecting the boundary geometry of domains in complex Euclidean spaces. The pseudohermitian structure is defined by a choice of contact form $\theta$ and the Levi metric, with the Tanaka-Webster connection providing the curvature and torsion tensors relevant for geometric analysis. The sublaplacian $\Delta_b$, Tanaka-Webster scalar curvature $W$, and CR invariant sublaplacian $L_b$ are central in variational problems and conformal deformations.

The Yamabe equation in the CR context aims to find a positive smooth function $u$ solving

$$-\left(2 + \frac{2}{n} \right) \Delta_b u + W u = u^{1 + \frac{2}{n}}$$

on compact CR manifolds, with energy measured by the CR-Sobolev quotient. The minimization of this quotient, $\mathcal{Y}(M,J)$, is intricately related to the geometry and topology of $(M^{2n+1}, J)$0.

CR Positive Mass Theorem

The $(M^{2n+1}, J)$1-mass is defined for asymptotically Heisenberg manifolds as an ADM-type geometric invariant:

$(M^{2n+1}, J)$2

where $(M^{2n+1}, J)$3 involves connection forms associated with the Tanaka-Webster connection. The positivity of $(M^{2n+1}, J)$4-mass under scalar curvature and torsion constraints forms the backbone of the CR positive mass theorem. This result is pivotal for establishing strict inequalities needed in the solvability of the CR Yamabe minimization problem.

Dimension 3: The survey references [CMY17], which proves non-negativity (and rigidity) of $(M^{2n+1}, J)$5-mass for embeddable three-dimensional CR manifolds with positive Yamabe constant, utilizing the CR Paneitz operator and embeddability equivalence. The mass is shown to be the zeroth-order term in the Green's function expansion for the conformal sublaplacian.

Dimension 5 (Contact Spin Case): By extending the analysis to five-dimensional contact spin manifolds, the manuscript presents a Weitzenböck-type formula and leverages Clifford algebra cancellations, thereby resolving technical obstructions in the Dirac equation. The CR positive mass theorem is proved for spherical, spin contact manifolds using subelliptic theory and decay estimates for spinor solutions.

Higher Dimensions: For dimensions $(M^{2n+1}, J)$6, positive mass is obtained for spherical CR structures, subject to nontrivial integrability criteria for Green's functions (cf. $(M^{2n+1}, J)$7) and the injectivity of the developing map, as established in [CCY14].

CR Paneitz Operator and Embeddability

The CR Paneitz operator, a fourth-order conformally covariant operator, plays a central role in the mass formula and embeddability theory. For three-dimensional CR manifolds, non-negativity of $(M^{2n+1}, J)$8 is proven equivalent to embeddability when the Yamabe constant is positive (cf. [Tak20]). This equivalence is critical in ensuring solvability for CR Yamabe minimizer problems and is established through the connection between operator positivity and the closed range property for $(M^{2n+1}, J)$9.

Rossi Spheres and Counterexamples

The paper discusses Rossi spheres as canonical examples of non-embeddable CR structures on $J$0. For these, it is shown that the $J$1-mass is negative for small deformation parameters, and the infimum of the CR-Sobolev quotient coincides with that of the standard sphere but is not attained. This signals a strong divergence from the Riemannian case and provides explicit non-attainment phenomena for the CR Yamabe minimization (cf. [CMY23]). These results underscore the nuanced behavior of CR geometry in the presence of non-embeddability and exhibit cases where minimal variational solutions cannot exist.

Strong Numerical and Structural Claims

• Three-Dimensional Rigidity: $J$2 implies CR equivalence with the standard $J$3.
• Negative Mass in Non-Embeddable Cases: Rossi spheres demonstrate $J$4, confirming negative mass for $J$5.
• Sobolev Quotient Non-Attainment: For Rossi spheres, the infimum matches the standard sphere, $J$6, but is not achieved by any function.
• Spin Case Extension: For dimension 5, the positive mass theorem holds and the CR Yamabe minimizer problem is solvable under spin conditions.

Practical and Theoretical Implications

The synthesis of the positive mass theorem and Yamabe minimization deepens the analogy between CR and conformal Riemannian geometries, affirming the role of geometric invariants in scalar curvature prescription problems. The embeddability conditions, characterized via analytic properties of the CR Paneitz operator, are vital for understanding geometric obstructions to mapping domains in higher-dimensional complex spaces. These results pave the way for further exploration of $J$7 mass phenomena, mass inequality extensions, and variational problems on contact spin manifolds.

The explicit negative mass phenomena for non-embeddable CR structures indicate new challenges and directions in geometric analysis, especially in the classification of critical PDEs and their relation to global geometric invariants.

Future Directions

Potential developments include:

• Extension of the positive mass theorem to broader classes of CR manifolds and their moduli.
• Systematic classification of mass behavior in critical PDE contexts, especially for non-embeddable or exotic CR structures.
• Deeper investigation into the geometric and analytic interplay underlying the invariance properties and the spectral theory of the Paneitz operator.
• Progress in the analysis of the Yamabe equation in higher dimensions, particularly regarding compactness, blow-up phenomena, and variational landscape in the presence of torsion and topological constraints.

Conclusion

The survey provides an authoritative account of major advances in CR geometric analysis, integrating mass invariants, scalar curvature prescription, and embeddability theory. The positive mass theorem for CR manifolds, its sharp rigidity, and its profound implications for the Yamabe minimization problem constitute key contributions to the landscape of geometric partial differential equations, with extensive repercussions in both theoretical and applied contexts in several complex variables and contact geometry.