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Conformal CR positive mass theorem Pak Tung Ho Sogang University, Korea 2018 Taipei Conference on Geometric Invariance and Partial Differential Equations, Institute of Mathematics, Academia Sinica, Taipei, Taiwan 17th-20th January, 2018 Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem Suppose ( M , g ) is an n -dimensional Riemannian manifold. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem Suppose ( M , g ) is an n -dimensional Riemannian manifold. ( M , g ) is asymptotically flat if there is a compact subset K ⊂ M such that M − K is diffeomorphic to R n − {| x | ≤ 1 } , and the metric g satisfies Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem Suppose ( M , g ) is an n -dimensional Riemannian manifold. ( M , g ) is asymptotically flat if there is a compact subset K ⊂ M such that M − K is diffeomorphic to R n − {| x | ≤ 1 } , and the metric g satisfies g ij = δ ij + O ( | x | − τ ) , | x || g ij , k | + | x | 2 | g ij , kl | = O ( | x | − τ ) for some τ > ( n − 2) / 2. Here, g ij , k and g ij , kl are the covariant derivatives of g ij . Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem Suppose ( M , g ) is an n -dimensional Riemannian manifold. ( M , g ) is asymptotically flat if there is a compact subset K ⊂ M such that M − K is diffeomorphic to R n − {| x | ≤ 1 } , and the metric g satisfies g ij = δ ij + O ( | x | − τ ) , | x || g ij , k | + | x | 2 | g ij , kl | = O ( | x | − τ ) for some τ > ( n − 2) / 2. Here, g ij , k and g ij , kl are the covariant derivatives of g ij . We also require R g = O ( | x | − q ) for some q > n . Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem The ADM mass of ( M , g ) is defined as n 1 � � m ADM = lim ( g ij , i − g ii , j ) 4( n − 1) ω n − 1 Λ →∞ {| x | =Λ } i , j =1 Here, ω n − 1 is the volume of the ( n − 1)-dimensional unit sphere. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem The ADM mass of ( M , g ) is defined as n 1 � � m ADM = lim ( g ij , i − g ii , j ) 4( n − 1) ω n − 1 Λ →∞ {| x | =Λ } i , j =1 Here, ω n − 1 is the volume of the ( n − 1)-dimensional unit sphere. Example: ( R n , δ ) is asymptotically flat. The ADM mass of ( R n , δ ) is zero. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem Theorem (Positive Mass Theorem) If ( M , g ) is asymptotically flat with R g ≥ 0 , then m ADM ≥ 0 and equality holds if and only if ( M , g ) ≡ ( R n , δ ) . Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem Theorem (Positive Mass Theorem) If ( M , g ) is asymptotically flat with R g ≥ 0 , then m ADM ≥ 0 and equality holds if and only if ( M , g ) ≡ ( R n , δ ) . When 3 ≤ n ≤ 7, Schoen-Yau (1979, 1981) proved the positive mass theorem by using minimal hypersurfaces. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem Theorem (Positive Mass Theorem) If ( M , g ) is asymptotically flat with R g ≥ 0 , then m ADM ≥ 0 and equality holds if and only if ( M , g ) ≡ ( R n , δ ) . When 3 ≤ n ≤ 7, Schoen-Yau (1979, 1981) proved the positive mass theorem by using minimal hypersurfaces. When ( M , g ) is spin, Witten (1981) proved the positive mass theorem by using spinor. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem Theorem (Positive Mass Theorem) If ( M , g ) is asymptotically flat with R g ≥ 0 , then m ADM ≥ 0 and equality holds if and only if ( M , g ) ≡ ( R n , δ ) . When 3 ≤ n ≤ 7, Schoen-Yau (1979, 1981) proved the positive mass theorem by using minimal hypersurfaces. When ( M , g ) is spin, Witten (1981) proved the positive mass theorem by using spinor. Recently, Schoen-Yau claimed to prove the positive mass theorem in general. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem W. Simon (1999) proved the following: Theorem (Conformal Positive Mass Theorem) If ( M , ˜ g ) and ( M , g ) are 3 -dimensional asymptotically flat g = φ 4 g such that R g − φ 4 R ˜ Riemannian manifolds with ˜ g ≥ 0 , Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem W. Simon (1999) proved the following: Theorem (Conformal Positive Mass Theorem) If ( M , ˜ g ) and ( M , g ) are 3 -dimensional asymptotically flat g = φ 4 g such that R g − φ 4 R ˜ Riemannian manifolds with ˜ g ≥ 0 , m ADM ( g ) − m ADM (˜ g ) ≥ 0 Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem W. Simon (1999) proved the following: Theorem (Conformal Positive Mass Theorem) If ( M , ˜ g ) and ( M , g ) are 3 -dimensional asymptotically flat g = φ 4 g such that R g − φ 4 R ˜ Riemannian manifolds with ˜ g ≥ 0 , m ADM ( g ) − m ADM (˜ g ) ≥ 0 and equality holds if and only if ( M , ˜ g ) and ( M , g ) are isometric. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem W. Simon (1999) proved the following: Theorem (Conformal Positive Mass Theorem) If ( M , ˜ g ) and ( M , g ) are 3 -dimensional asymptotically flat g = φ 4 g such that R g − φ 4 R ˜ Riemannian manifolds with ˜ g ≥ 0 , m ADM ( g ) − m ADM (˜ g ) ≥ 0 and equality holds if and only if ( M , ˜ g ) and ( M , g ) are isometric. Taking M = R 3 and ˜ g = δ . Then we have: Theorem If ( R 3 , g = φ − 4 δ ) is 3 -dimensional asymptotically flat manifold such that R g ≥ 0 , then m ADM ( g ) ≥ 0 Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

Positive Mass Theorem W. Simon (1999) proved the following: Theorem (Conformal Positive Mass Theorem) If ( M , ˜ g ) and ( M , g ) are 3 -dimensional asymptotically flat g = φ 4 g such that R g − φ 4 R ˜ Riemannian manifolds with ˜ g ≥ 0 , m ADM ( g ) − m ADM (˜ g ) ≥ 0 and equality holds if and only if ( M , ˜ g ) and ( M , g ) are isometric. Taking M = R 3 and ˜ g = δ . Then we have: Theorem If ( R 3 , g = φ − 4 δ ) is 3 -dimensional asymptotically flat manifold such that R g ≥ 0 , then m ADM ( g ) ≥ 0 and equality holds if and only if ( M , g ) is flat. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

CR Positive Mass Theorem Suppose ( N , J , θ ) be a 3-dimensional CR manifold with a contact structure ξ and a CR structure J : ξ → ξ such that J 2 = − 1 . Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

CR Positive Mass Theorem Suppose ( N , J , θ ) be a 3-dimensional CR manifold with a contact structure ξ and a CR structure J : ξ → ξ such that J 2 = − 1 . Let T be the unique vector field such that θ ( T ) = 1 and d θ ( T , · ) = 0 . Also, let Z 1 be vector field such that JZ 1 = iZ 1 and JZ 1 = − iZ 1 . Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

CR Positive Mass Theorem Suppose ( N , J , θ ) be a 3-dimensional CR manifold with a contact structure ξ and a CR structure J : ξ → ξ such that J 2 = − 1 . Let T be the unique vector field such that θ ( T ) = 1 and d θ ( T , · ) = 0 . Also, let Z 1 be vector field such that JZ 1 = iZ 1 and JZ 1 = − iZ 1 . Let ( θ, θ 1 , θ 1 ) be dual to ( T , Z 1 , Z 1 ) so that d θ = ih 11 θ 1 ∧ θ 1 with h 11 = 1. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

CR Positive Mass Theorem The connection 1-form ω 1 1 and the torsion are determined by d θ 1 = θ 1 ∧ ω 1 1 + A 1 1 θ ∧ θ 1 , ω 1 1 + ω 1 1 = 0 . Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

CR Positive Mass Theorem The connection 1-form ω 1 1 and the torsion are determined by d θ 1 = θ 1 ∧ ω 1 1 + A 1 1 θ ∧ θ 1 , ω 1 1 + ω 1 1 = 0 . The Tanaka-Webster curvature is given by 1 = R θ 1 ∧ θ 1 (mod θ ) . d ω 1 Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

CR Positive Mass Theorem Example: The Heisenberg group H 1 = { ( z , t ) : z ∈ C , t ∈ R } , J 0 : C → C the standard complex structure, and ◦ θ = dt + izdz − izdz . Then � ∂ � ∂ � � ◦ 1 ∂ z + iz ∂ ◦ 1 ∂ z − iz ∂ Z 1 = √ , Z 1 = √ . ∂ t ∂ t 2 2 ◦ ◦ √ √ θ 1 = θ 1 = 2 dz , 2 dz . The Tanaka-Webster curvature R = 0. Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

CR Positive Mass Theorem ( N , J , θ ) is called asymptotically flat pseudohermitian if there is a compact subset K ⊂ N such that N − K is diffeomorphic to H 1 − { ρ ≤ 1 } , such that Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem

CR Positive Mass Theorem ( N , J , θ ) is called asymptotically flat pseudohermitian if there is a compact subset K ⊂ N such that N − K is diffeomorphic to H 1 − { ρ ≤ 1 } , such that ◦ θ = (1 + 4 π A ρ − 2 + O ( ρ − 3 )) θ + O ( ρ − 3 ) dz + O ( ρ − 3 ) dz , √ ◦ θ 1 = O ( ρ − 3 ) θ + O ( ρ − 4 ) dz + (1 + 2 π A ρ − 2 + O ( ρ − 3 )) 2 dz for some constant A . Here, � | z | 4 + t 2 . 4 ρ = Pak Tung Ho Sogang University, Korea Conformal CR positive mass theorem