[Paper Review] A positive mass theorem in three dimensional Cauchy-Riemann geometry

Jih-Hsin Cheng, Andrea Malchiodi+1 more|arXiv (Cornell University)|Dec 30, 2013

Geometry and complex manifolds4 citations

TL;DR

This paper introduces a p-mass invariant for asymptotically flat pseudohermitian 3-manifolds, analogous to the ADM mass in Riemannian geometry, and proves its positivity under conditions of positive Tanaka-Webster curvature and non-negative CR Paneitz operator. The key result establishes that the p-mass is non-negative, with zero mass characterizing the standard CR 3-sphere, and applies this to solve the CR Yamabe problem with minimal energy.

ABSTRACT

We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the expansion of the Green function for the CR Laplacian. We deduce an integral formula for the p-mass, and we reduce its positivity to a solution of Kohn's equation. We prove that the p-mass is non-negative for (blow-ups of) compact 3-manifolds of positive Tanaka-Webster class and with non-negative CR Paneitz operator. Under these assumptions, we also characterize the zero mass case as the standard three dimensional CR sphere. We then show the existence of (non-embeddable) CR 3-manifolds having nonpositive Paneitz operator or negative p-mass through a second variation formula. Finally, we apply our main result to find solutions of the CR Yamabe problem with minimal energy.

Motivation & Objective

• To define a p-mass invariant analogous to the ADM mass in asymptotically flat pseudohermitian 3-manifolds.
• To establish the non-negativity of the p-mass under the assumption of positive Tanaka-Webster class and non-negative CR Paneitz operator.
• To characterize the zero p-mass case as the standard three-dimensional CR sphere.
• To demonstrate the existence of non-embeddable CR 3-manifolds with nonpositive Paneitz operator or negative p-mass via a second variation formula.
• To apply the positive p-mass theorem to find solutions of the CR Yamabe problem with minimal energy.

Proposed method

• Define the p-mass as the first nontrivial coefficient in the Green's function expansion for the CR Laplacian on the blow-up of a compact pseudohermitian 3-manifold.
• Derive an integral formula for the p-mass using the divergence structure of the CR Einstein tensor and the Kohn-Knapp-Weiss formula.
• Reduce the positivity of the p-mass to solving Kohn’s equation for a certain (1,1)-form on the manifold.
• Use the conformal covariance of the conformal sublaplacian to relate the p-mass to the Tanaka-Webster curvature and CR Paneitz operator.
• Apply a second variation formula to construct examples of non-embeddable CR 3-manifolds with nonpositive Paneitz operator or negative p-mass.
• Utilize CR normal coordinates and asymptotic expansions to analyze the behavior of contact forms and connection forms near infinity.

Experimental results

Research questions

• RQ1 Can a notion of mass analogous to the ADM mass be defined in three-dimensional Cauchy-Riemann geometry?
• RQ2 Under what geometric conditions is the p-mass non-negative in asymptotically flat pseudohermitian 3-manifolds?
• RQ3 What characterizes the zero p-mass case in terms of CR geometry?
• RQ4 Can non-embeddable CR 3-manifolds with negative p-mass or nonpositive CR Paneitz operator be constructed?
• RQ5 How can the positive p-mass theorem be applied to solve the CR Yamabe problem with minimal energy?

Key findings

• The p-mass is non-negative for the blow-up of a compact 3-dimensional pseudohermitian manifold with positive Tanaka-Webster class and non-negative CR Paneitz operator.
• The zero p-mass case occurs if and only if the manifold is CR equivalent to the standard three-dimensional CR sphere.
• Non-embeddable CR 3-manifolds with nonpositive Paneitz operator or negative p-mass exist, as shown via a second variation formula.
• The p-mass is identified as the first nontrivial coefficient in the Green function expansion of the CR Laplacian on the blow-up of a compact pseudohermitian manifold.
• The integral formula for the p-mass is derived from the divergence structure of the CR Einstein tensor and the Kohn-Knapp-Weiss formula.
• The positive p-mass theorem enables the construction of solutions to the CR Yamabe problem with minimal energy by ensuring the Yamabe quotient is bounded below by the standard sphere.