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

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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

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Asymptotically flat (Riemannian) manifolds

A manifold (M3, g) is said to be asymptotically flat if ∃ K ⊆⊆ M s.t. M \ K is diffeomorphic to R3 \ B1(0) and s.t., for some m ∈ R gij = δij + hij, ∂lhij = O(|x|−1−l), l = 0, 1, 2.

M

M \ K

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

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Some examples

Example 1: Schwartzschild metric. It describes a static black hole

f total mass m. The expression is
1 + m

2r 4 dr2 + r2dξ2 . 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 M, ˆ g) and p ∈ ˆ M, one can consider a conformal metric

n ˜

g on ˆ M \ {p} of the following form ˜ g = f(x) ˆ g; f(x) ≃ 1 d(x, p)4 . Then, in normal coordinates x at p, setting y =

x |x|2 (Kelvin inversion)

ne has an asymptotically flat manifold in y-coordinates

˜ g(x) ≃ dx2 |x|4 ≃ dy2, (y large).

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 3 / 29

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Einstein’s equation

It governs the structure of space-time according to general relativity Eab := Rab − 1 2Rg gab = Tab. Here Rab is the Ricci tensor, Rg the scalar curvature, and Tab the stress- energy tensor, generated by matter. In vacuum (Tab ≡ 0), this equation has variational structure, with Euler- Lagrange functional given by A(g) :=

M

Rg dVg Einstein-Hilbert functional. In fact, one has d dg (Rg dVg) [h] = −

hijEij + ∇∗ζ
dVg;

ζ = − (∇∗h + ∇(trgh)) =

h

,k jk − hk k,j

dxj.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 4 / 29

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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 dg(A(g) + m(g))[h] =

M

hijEij dVg. The quantity m(g), called ADM mass ([ADM, ’60]), is defined as m(g) := lim

r→∞

Sr

(∂k gjk − ∂j gkk) νjdσ. Example 1: Schwartzschild. mADM = black-hole mass. Example 2: Conformal blow-ups. If Gp is the Green’s function of an elliptic operator on ˆ M with pole at p, then Gp(x) ≃ d(x, p)−1. If f(x) = G4

p ≃ d(x, p)−4, then

mADM = lim

x→p

Gp(x) −

1 d(x, p)

.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 5 / 29

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The Positive Mass Theorem

Theorem ([Schoen-Yau, ’79]) If Rg ≥ 0 then m(g) ≥ 0. In case m(g) = 0, then (M, g) is isometric to flat Euclidean space (R3, dx2). Physically, this means that a positive local energy density implies a positive global energy for the system. (But you cannot just integrate!) A simplified model In Newtonian gravity, the gravitational potential is described by the Poisson equation. If −∆f = ρ ∈ Cc(R3) = ⇒ f(y) ≃∞ A |y|; A =

R3 ρ.

The issue is that Rg and m(g) are nonlinear in the metric.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 6 / 29

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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 CR of (asymptotic) radius R.

M CR

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 Rg ≥ 0 implies instability by the second variation

formula for the area.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 7 / 29

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Witten’s approach on spin manifolds

Witten (’81) used Dirac’s equation in a different proof. He solved for (∗) Dψ :=

i

ei · ∇eiψ = 0; ψ → ψ0 as |x| → ∞. The square of the D operator satisfies Lichnerowitz’s formula D2 = ∇∗∇ + 1 4Rg. Integrating (∗) by parts, the mass appears in the boundary terms m(g) = cn

M
|∇ψ|2 + 1

4Rg|ψ|2

dVg.

The last formula allows also to characterize R3 as the unique asympto- tically flat space with (Rg ≥ 0 and) zero mass.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 8 / 29

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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

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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. J2 = −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 Z1 such that JZ1 = iZ1; JZ1 = −iZ1 where Z1 = (Z1). We also define (θ, θ1, θ1) as the dual triple to (T, Z1, Z1), so that dθ = iθ1 ∧ θ1.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 10 / 29

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The Tanaka-Webster connection 1-form ω11 and the torsion A11 are uniquely determined by the structure equations

dθ1 = θ1 ∧ ω11 + A11 θ ∧ θ1;

ω11 + ω11 = 0. The Webster curvature is then defined by the formula dω11 = W θ1 ∧ θ1 (mod θ). Basic (flat) example: the Heisenberg group H1 = {(z, t) ∈ C × R}         

θ= dt + izdz − izdz;
θ1=

√ 2dz;

θ1=

√ 2dz,       

T= ∂

∂t;

Z1=

1 √ 2

∂

∂z + iz ∂ ∂t

;
Z1=

1 √ 2

∂

∂z − iz ∂ ∂t

.

ξ0 on H1 is spanned by real and imaginary parts of

Z1. The standard

CR structure J0 : ξ0 → ξ0 verifies J0

Z1= i
Z1.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 11 / 29

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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 = u2θ with Wˆ

θ > 0.

The conformal sublaplacian rules trasformation of Webster’s curvature u → Lu := −4∆bu + Wu = −4(u,11 + u,11) + W u. In fact, Lu = Wˆ

θ u3 if ˆ

θ = u2θ. If M(ξ, J) has positive Webster class, then the Green’s function G(x, y)

f L is positive, and for p ∈ M we can consider the form ˆ

θ = G(p, ·)2θ. This means that we are solving for −4∆bG(p, ·) + WG(p, ·) = δp, namely we get zero Webster curvature on M \ {p}.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 12 / 29

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In CR normal coordinates ([Jerison-Lee, ’89]) (z(x), t(x)) at p one has the following asymptotics for G G(p, x) = 1 32πρ(x)−2 + A + ox(1); ρ4(x) = |z|4 + t2, for some A ∈ R, where ox(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

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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 v; t∗ = − t |v|2 ;

n 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 θ =

1 + 4πAρ−2

∗

+ O(ρ−3

∗ )

(
θ)∗ + O(ρ−3

∗ )dz∗ + O(ρ−3 ∗ )dz∗;

ˆ θ1 =

O(ρ−3

∗ ) + O(ρ−5 ∗ )

(
θ)∗ + O(ρ−4

∗ )dz∗

+

1 + 2πAρ−2

∗

+ O(ρ−3

∗ )

√ 2dz∗, converging to the standard forms

θ and
θ1 of H1 as ρ∗ → +∞.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 14 / 29

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Asymptotically flat CR manifolds

Motivated by the above computations we introduce the Definition (asymptotically flat CR manifolds) A three dimensional CR manifold (N, J, θ) is said to be asymptotically flat CR if N = N0 ∪ N∞, with N0 compact and N∞ diffeomorphic to H1 \ Bρ0 on which (J, θ) ≃ (J0,

θ) in the sense that

θ =

1 + 4πAρ−2 + O(ρ−3)

θ +O(ρ−3)dz + O(ρ−3)dz; θ1 = O(ρ−3)

θ +O(ρ−4)dz +
1 + 2πAρ−2 + O(ρ−3)

√ 2dz, for some A ∈ R and a unitary coframe θ1 in some system of coordinates (asymptotic coordinates). We also require W ∈ L1(N).

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 15 / 29

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A notion of CR-mass

Given a one-parameter family of CR structures J(s), we have ˙ J = 2E = 2E11θ1 ⊗ Z1 + 2E11θ1 ⊗ Z1. If W(s) is the corresponding Webster curvature, then d ds |s=0

N

W(s) θ ∧ dθ =

N

˙ W θ ∧ dθ = −

N

d

E11,1 θ ∧ θ1

+ conj. −

N

(A11E11 + conj.) θ ∧ dθ =

∞

i ˙ ω1¯

1 ∧ θ −

N

(A11E11 + conj.) θ ∧ dθ. Definition (CR mass) Let N be an asympt. flat CR manifold. Define m(J, θ) := i

∞

ω1¯

1 ∧ θ :=

lim

Λ→+∞ i

SΛ

ω1¯

1 ∧ θ,

where SΛ = {ρ = Λ} (a large sphere).

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 16 / 29

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The Paneitz operator

The CR Paneitz operator P is a fourth-order operator defined by Pϕ := 4(ϕ,¯

111 + iA11ϕ,¯ 1)¯ 1 + conj.

It characterizes CR pluriharmonic functions ([Lee, ’88]) and to the Sze- go kernel expansion ([Hirachi, ’93]). Moreover, under the contact form change ˆ θ = e2fθ one has Pˆ

θϕ = e−4fPθϕ.

The Paneitz operator enters in the assumptions of the following embed- dability theorem (in our notation bu := −2u,11). Theorem ([Chanillo-Chiu-Yang, ’12]) Let M be a compact 3D CR ma-

nifold. If P ≥ 0 and W > 0, then range(b) is closed. Moreover, M can

be embedded into CN for some N ∈ N.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 17 / 29

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An integral formula for the mass

Proposition Let (N, J, θ) be an asymptotically flat CR manifold. For ε ∈ (0, 1) let β : N → C be such that β = z + β−1 + O(ρ−2+ε) and bβ = O(ρ−4) near ∞, where β−1 is a term with homogeneity ρ−1. Then one has 2 3m(J, θ) = −

N

|bβ|2θ ∧ dθ + 2

N

|β,11|2θ ∧ dθ + 2

N

W|β,1|2θ ∧ dθ + 1 2

N

βPβ θ ∧ dθ. The proof uses integration by parts. The correction term on β−1 arises from trying to annihilate bβ, Taylor expanding bz.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 18 / 29

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Solving for bβ = 0

Near infinity one has the expansion bz = 4πA z ρ6 (|z|2 + it) + O(ρ−4) near infinity.

It is possible to add a correction term β−1 s.t.

β−1 = O(ρ−1); b(z + β−1) = O(ρ−4). The construction of β−1 is quite delicate and done using the Szego kernel. One then exploits crucially the following result Theorem ([Hsiao-Yung, ’13]) Suppose N is the blow-up of a compact embeddable CR manifold. Then for ε ∈ (0, 1) and any h = O(ρ−4) in the range of b there exists ˜ β s.t. ˜ β = O(ρ−2+ε); b ˜ β = h.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 19 / 29

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A positive mass theorem

Theorem ([Cheng-M.-Yang, ’13]) Let M be a smooth 3D compact CR manifold. Suppose the Webster class is positive, and that the CR Paneitz operator is non-negative. Let p ∈ M and let θ be a blow-up of contact form at p. Then (a) m(J, θ) ≥ 0; (b) if m(J, θ) = 0 M is CR equivalent to the standard S3.

It turns out that m(J, θ) = 48π2A, where A is the constant term

appearing in the expansion of the Green’s function.

If m(J, θ) = 0, from the integral formula we get

β,11 ≡ 0; β,11 ≡ 0; Pβ ≡ 0. These can be integrated and give congruence of the blow-up to H1.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 20 / 29

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Application: the CR Yamabe problem

In the CR case one looks for constant Webster curvature under a confor- mal change of contact form. If ˆ θ = u2θ, then −4∆bu + Wu = ˆ Wu3. If one wants to solve for constant ˆ W, it is possible to do it looking for solutions of the following extremization problem Y(M, J) := inf

ˆ θ

M WJ,ˆ

θ ˆ

θ ∧ dˆ θ

M ˆ

θ ∧ dˆ θ 1

2

= inf

u≡0

M
2|∇bu|2 + 1

2WJ,θ u2

θ ∧ dθ

M u4 θ ∧ dθ

1

2

. The cases Y(M, J) < 0 and Y(M, J) = 0 are easy, while the positive case is difficult since the embedding S1,2 ֒ → L4 is critical.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 21 / 29

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Positive Webster class

In the positive case, one has always Y(M, J) ≤ Y(S3, J0). To see this, one can exploit the conformality of the map ̟ : S3 \p → H1, p = (0, 1) ∈ C2 ̟(z1, z2) =

z1

1 + z2 , Re

i1 − z2

1 + z2

,

where (z1, z2) are standard coordinates in C2. Composing with a dilation, the conformal factor of the inverse map is given by ωλ(z, t) = 1 λ

t2 + |z|4 + 2

λ2 |z|2 + 1 λ4 − 1

2

; λ > 0, (z, t) ∈ H1. In [Jerison-Lee, ’88] these functions were classified as extremals for the Sobolev-type ratio in H1, equal to Y(S3, J0). Localizing these functions on any manifold with λ large, in the quotient

ne can get arbitrarily close to Y(S3, J0).

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 22 / 29

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Compactness recovery

Compactness of minimizing sequences indeed holds provided one has the strict inequality Y(M, J) < Y(S3, J0) ([Jerison-Lee, ’87]). There would not be enough energy for blow-up. This condition was proved in [Jerison-Lee, ’89] in (real) dimension greater

r equal to 5 and non locally spherical manifolds using local expansions

near the functions ωλ. In low dimension the decay of extremals is slower, so a global argument is needed. Here the positive mass enters. Following the argument in [Schoen, ’84], one can glue a highly peaked ωλ at p ∈ M to a scaled Green’s function.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 23 / 29

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More precisely, we set u(z, t) ≃    ωλ(z, t) in {ρ ≤ ρ0}; ε0 ˜ Gp(z, t) in M \ {ρ ≤ ρ0}, where ε0 = 1 λ(1 + 2πAρ2

0).

Then one finds

M
2|∇bu|2 + 1

2WJ,θ u2

θ ∧ dθ

M u4θ ∧ dθ

1

2

≤ Y(S3, J0) − c0A λ2ρ2 , which implies strict inequality and compactness of minimizing sequences. In [Gamara, ’01], [Gamara-Yacoub, ’01] constant curvature structures were found using topological methods (no information on minimality).

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 24 / 29

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An example with positive W and negative mass

We consider a deformation J(s) of the CR structure of H1 such that at s = 0 E11,1 = 0. We can take for example E11(z, z, t) =

t + i(|z|2 + 1)

−k , with k large to have fast decay at infinity and compactify to S3. Then ˙ m(J(0), θ) = 0; ¨ m(J(0), θ) = −3 2

H1 |E11,1|2 ◦

θ ∧d

θ < 0.

We can transport the latter example to S3 using the CR stereographic

projection. Likely, this structure is non embeddable for s small.

Therefore, the positivity of the Paneitz operator is necessary for the positivity of the mass, peculiar to the CR case.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 25 / 29

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Some open problems

Is there any role of minimal surfaces for the positivity of the CR mass? In CR geometry a natural notion of area is given by A(Ω) =

Ω

θ ∧ e1, where e1 is dual to e1 ∈ TΩ ∩ ξ, unit vector. In H1, it coincides with the relative perimeter (in the sense of GMT, see [Garofalo-Nhieu, ’96]). It can also be identified with the 3-d Hausdorff measure, see [Franchi- Serapioni-Serra Cassano,’01] and [Balogh,’03]. The second variation of the area at a minimal surface has the following expression ([Cheng-Hwang-M.-Yang, ’05]): e2 = Je1 A′′

fe2 =

Σ

{(e1(f))2 + f2[−2W + 2 ImA11 − 4e1(α) − 4α2]} θ ∧ e1. Here α is such that αe2 + T ∈ TΣ, e2 = Je1.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 26 / 29

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There are also quantitative lower bounds on the ADM mass. One is Penrose’s inequality, concerning outermost minimal surfaces

M

uter minimizing

non outer minimizing

If A is the total area of the outermost minimal surfaces, Penrose’s ine- quality asserts that m ≥

A

16π.

The inequality was proved in [Huisken-Ilmanen, ’97], [Bray, ’01] using geometric flows. Are there any analogues in the CR case?

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 27 / 29

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Another problem recently settled is the compactness of solutions to Ya- mabe’s equation ([Brendle-Marques, ’09], [Khuri-Marques-Schoen, ’09]). Compactness holds if and only if n ≤ 24. The compactness issue for the Webster-Yamabe problem is entirely open. One reason is a lack of the positive mass theorem in higher dimensions. At an even more basic level, the profile of blow-ups has not been classified. This concerns entire positive solutions to −∆bu = u

Q+2 Q−2

in Hn; Q = 2n + 2. Assuming finite volume, these are known to be CR bubbles, by [Jerison- Lee, ’88]. However blow-up profiles do not satisfy this assumption in

general. Moving plane methods do not work either.

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 28 / 29

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Thanks for your attention

Andrea Malchiodi (SISSA, Trieste) IHP, October 2nd, 2014 29 / 29