A positive mass theorem in three dimensional Cauchy–Riemann geometry

Abstract

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.

Convergence of the CR Yamabe flow

“…Moreover, the Green's function satisfies the estimates (see Proposition 5.2 and Proposition 5.3 in [11])…”

“…Give any x ∈ M , we can find a contact form θ x = ϕ 2 n x θ 0 conformal to θ 0 , where θ x is a contact form defined in (z, t) which is the CR normal coordinates centered at x. On the other hand, θ x satisfies the following properties: (see Theorem 3.7 in P.172 of [15] and Proposition 6.5 in [11])…”

“…The remaining case, namely, when n = 1 or M is spherical, was solved by Gamara and Yacoub in [18,19] by using critical point at infinity. See also the recent work by Cheng-Chiu-Yang [10] and Cheng-Malchiodi-Yang [11] for these cases.…”

“…We refer the readers to (3.6) in [10] for the precise definition of s(M ). When n = 1, the definition of the CR Panetiz operator was included in section 2, but we also refer the readers to [11] for more properties of the CR Panetiz operator. Similar to the Yamabe flow, we need to use the CR positive mass theorem to prove the convergence of the CR Yamabe flow.…”

“…The CR positive mass theorem when M is spherical was obtained by Cheng-Chiu-Yang in [10] for n ≥ 2, with further assumption that s(M ) < 1 for n = 2. On the other hand, the CR positive mass theorem for the case when n = 1 was obtained by Cheng-Malchiodi-Yang in [11], under the assumption that the CR Panetiz operator of M is nonnegative. Therefore, we have included the assumptions in Theorem 1.1 so that we can apply the CR positive mass theorem.…”