A positive mass theorem in CR geometry Joint work with J.H.Cheng and - PowerPoint PPT Presentation

A positive mass theorem in CR geometry Joint work with J.H.Cheng and P.Yang Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 1 / 29

Asymptotically flat (Riemannian) manifolds A manifold ( M 3 , g ) is said to be asymptotically flat if ∃ K ⊆⊆ M s.t. M \ K is diffeomorphic to R 3 \ B 1 (0) and s.t., for some m ∈ R ∂ l h ij = O ( | x | − 1 − l ) , g ij = δ ij + h ij , l = 0 , 1 , 2 . M \ K M In general relativity these manifolds represent time-slices of static space- times where gravity is present. Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 2 / 29

Some examples Example 1: Schwartzschild metric . It describes a static black hole of total mass m . The expression is 1 + m � 4 � � dr 2 + r 2 dξ 2 � . 2 r At r = m 2 there is a minimal surface, representing an event horizon . Example 2: Conformal blow-ups . Given a compact Riemannian three-manifold ( ˆ g ) and p ∈ ˆ M, ˆ M , one can consider a conformal metric g on ˆ on ˜ M \ { p } of the following form 1 ˜ g = f ( x ) ˆ g ; f ( x ) ≃ d ( x, p ) 4 . x Then, in normal coordinates x at p , setting y = | x | 2 (Kelvin inversion) one has an asymptotically flat manifold in y -coordinates g ( x ) ≃ dx 2 | x | 4 ≃ dy 2 , ˜ ( y large ) . Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 3 / 29

Einstein’s equation It governs the structure of space-time according to general relativity E ab := R ab − 1 2 R g g ab = T ab . Here R ab is the Ricci tensor, R g the scalar curvature, and T ab the stress- energy tensor, generated by matter. In vacuum ( T ab ≡ 0 ), this equation has variational structure, with Euler- Lagrange functional given by � A ( g ) := R g dV g Einstein-Hilbert functional . M In fact, one has d h ij E ij + ∇ ∗ ζ � � dg ( R g dV g ) [ h ] = − dV g ; � � ζ = − ( ∇ ∗ h + ∇ ( tr g h )) = jk − h k ,k dx j . h k,j Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 4 / 29

The mass of an asymptotically flat manifold If we consider variations which preserve asymptotic flatness, then the divergence term has a role (flux at infinity), and d � h ij E ij dV g . dg ( A ( g ) + m ( g ))[ h ] = M The quantity m ( g ) , called ADM mass ([ADM, ’60]), is defined as � ( ∂ k g jk − ∂ j g kk ) ν j dσ. m ( g ) := lim r →∞ S r Example 1: Schwartzschild . m ADM = black-hole mass. Example 2: Conformal blow-ups . If G p is the Green’s function of an elliptic operator on ˆ M with pole at p , then G p ( x ) ≃ d ( x, p ) − 1 . If f ( x ) = G 4 p ≃ d ( x, p ) − 4 , then � 1 � m ADM = lim G p ( x ) − . d ( x, p ) x → p Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 5 / 29

The Positive Mass Theorem Theorem ([Schoen-Yau, ’79]) If R g ≥ 0 then m ( g ) ≥ 0 . In case m ( g ) = 0 , then ( M, g ) is isometric to flat Euclidean space ( R 3 , dx 2 ) . Physically, this means that a positive local energy density implies a positive global energy for the system. (But you cannot just integrate!) In Newtonian gravity, the gravitational potential is A simplified model described by the Poisson equation. If A � − ∆ f = ρ ∈ C c ( R 3 ) = ⇒ f ( y ) ≃ ∞ | y | ; A = R 3 ρ. The issue is that R g and m ( g ) are nonlinear in the metric. Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 6 / 29

Idea of the proof The main idea relies on constructing an asymptotically planar minimal surface Σ in M . This is done by solving Plateau problems on larger and larger circles C R of (asymptotic) radius R . C R M - If m ( g ) < 0 there would be a uniform control on the height, and one can find ( R → + ∞ ) a stable asymptotically planar minimal surface. Needs n ≤ 7 for regularity reasons. Still open in general dimension. - On the other hand R g ≥ 0 implies instability by the second variation formula for the area. Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 7 / 29

Witten’s approach on spin manifolds Witten (’81) used Dirac’s equation in a different proof. He solved for � ( ∗ ) � Dψ := e i · ∇ e i ψ = 0; ψ → ψ 0 as | x | → ∞ . i The square of the � D operator satisfies Lichnerowitz’s formula � D 2 = ∇ ∗ ∇ + 1 4 R g . Integrating ( ∗ ) by parts, the mass appears in the boundary terms � � |∇ ψ | 2 + 1 � 4 R g | ψ | 2 m ( g ) = c n dV g . M The last formula allows also to characterize R 3 as the unique asympto- tically flat space with ( R g ≥ 0 and) zero mass. Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 8 / 29

Applications and extensions of the PMT - Most notably, it was fundamental to solve Yamabe’s conjecture in low dimensions: find conformal metrics with constant scalar curvature ([Schoen, ’84]). Done previously in [Aubin, ’76] in high dimensions. - In turn, Yamabe’s invariant (vaguely: the largest scalar curvature one can put on a given manifold) leads to compactness and finite-topology theorems ([Bray-Neves, ’04], [Chang-Qing-Yang, ’07]). - CMC foliations at infinity ([Huisken-Yau, ’96], [Qing-Tian, ’07]) and and isoperimetric sets of large volume ([Eichmair-Metzger, ’13]). - Rigitity of model geometries (e.g. [P.Miao, ’05] for the hyperbolic one). - Penrose’s conjecture ([Huisken-Ilmanen, ’91], [Bray, ’01]): a positive lower bound on the mass depending on event horizon areas. Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 9 / 29

CR and pseudo-hermitian manifolds We deal with three-dimensional manifolds with a non-integrable two- dimensional distribution (contact structure) ξ . We also have a CR structure (complex rotation) J : ξ → ξ s.t. J 2 = − 1 . A contact form θ is a 1-form annihilating ξ : we assume that θ ∧ dθ � = 0 everywhere on M (pseudoconvexity). The Reeb vector field is the unique vector field T for which θ ( T ) ≡ 1; T � dθ = 0 . Given J as above, we have locally a vector field Z 1 such that JZ 1 = iZ 1 ; JZ 1 = − iZ 1 where Z 1 = ( Z 1 ) . We also define ( θ, θ 1 , θ 1 ) as the dual triple to ( T, Z 1 , Z 1 ) , so that dθ = iθ 1 ∧ θ 1 . Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 10 / 29

The Tanaka-Webster connection 1-form ω 11 and the torsion A 11 are uniquely determined by the structure equations dθ 1 = θ 1 ∧ ω 11 + A 11 θ ∧ θ 1 ; � ω 11 + ω 11 = 0 . The Webster curvature is then defined by the formula dω 11 = W θ 1 ∧ θ 1 ( mod θ ) . Basic (flat) example: the Heisenberg group H 1 = { ( z, t ) ∈ C × R } ◦  ◦  θ = dt + izdz − izdz ; T = ∂ ∂t ;     ◦ √  ◦ � ∂   1 ∂z + iz ∂ � θ 1 = Z 1 = ; 2 dz ; √ ∂t 2 ◦ ◦ � ∂ √     1 ∂z − iz ∂ � Z 1 = √ .  θ 1 =  2 dz,  ∂t 2 ◦ ξ 0 on H 1 is spanned by real and imaginary parts of Z 1 . The standard ◦ ◦ CR structure J 0 : ξ 0 → ξ 0 verifies J 0 Z 1 = i Z 1 . Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 11 / 29

Blow-up of a compact pseudo-hermitian manifold M We will consider manifolds of positive Webster class , namely for which there exists a conformal change ˆ θ = u 2 θ with W ˆ θ > 0 . The conformal sublaplacian rules trasformation of Webster’s curvature u �→ L u := − 4∆ b u + Wu = − 4( u , 11 + u , 11 ) + W u. θ u 3 if ˆ θ = u 2 θ . In fact, L u = W ˆ If M ( ξ, J ) has positive Webster class, then the Green’s function G ( x, y ) of L is positive, and for p ∈ M we can consider the form ˆ θ = G ( p, · ) 2 θ . This means that we are solving for − 4∆ b G ( p, · ) + W G ( p, · ) = δ p , namely we get zero Webster curvature on M \ { p } . Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 12 / 29

In CR normal coordinates ([Jerison-Lee, ’89]) ( z ( x ) , t ( x )) at p one has the following asymptotics for G 1 32 πρ ( x ) − 2 + A + o x (1); ρ 4 ( x ) = | z | 4 + t 2 , G ( p, x ) = for some A ∈ R , where o x (1) → 0 as ( z, t ) → 0 . • Again, the main term ρ − 2 will give the right behavior to obtain an asymptotically flat structure. • We will also need to keep track of the second order term A . Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 13 / 29

CR inversion If ( z, t ) are CR normal coordinates in a neighborhood U of p , we define inverted CR normal coordinates ( z ∗ , t ∗ ) as z ∗ = z t ∗ = − t v ; | v | 2 ; on U \ { p } , where v = t + i | z | 2 . Notice that ρ ∗ ( z ∗ , t ∗ ) = ρ ( z, t ) − 1 . In these coordinates the new forms become ◦ ˆ 1 + 4 πAρ − 2 + O ( ρ − 3 θ ) ∗ + O ( ρ − 3 ∗ ) dz ∗ + O ( ρ − 3 � � θ = ∗ ) ( ∗ ) dz ∗ ; ∗ ◦ ˆ O ( ρ − 3 ∗ ) + O ( ρ − 5 θ ) ∗ + O ( ρ − 4 θ 1 � � = ∗ ) ( ∗ ) dz ∗ � √ 1 + 2 πAρ − 2 + O ( ρ − 3 � + ∗ ) 2 dz ∗ , ∗ ◦ ◦ θ 1 of H 1 as ρ ∗ → + ∞ . converging to the standard forms θ and Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 14 / 29