surface geometry and general relativity...surface geometry and general relativity mu-tao wang...

Surface geometry and general relativity

Mu-Tao Wang

Columbia University

December 8, 2013

Annual Meeting TMS 2013, Kaohsiung

I I shall discuss two classical theorems for surfaces in R3 andtheir generalizations for surfaces in 4D spacetime.

I These generalizations are not only of mathematical interest,but also of physically relevant importance.

I Closely connected to fundamental problems in GR such asgravitational energy and cosmic censorship.

I First part joint work with Po-Ning Chen and Shing-Tung Yau.

I Second part joint work with Simon Brendle and Pei-Ken Hung.

I All surfaces are of the topological type of an S2.

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Review of surface geometry in R3

I Consider an embedded surface in R3 given by X : Σ → R3,X = (X 1,X 2,X 3).

I Local coordinates uaa=1,2 on Σ and each X i a function ofua.

I The induced metric or the first fundamental form is given by

3∑i=1

∂X i

∂ua∂X i

∂ub.

This is a positive definite symmetric 2-tensor.

I More familiar form is

[E FF G

], EG − F 2 > 0.

I The metric determines all intrinsic geometry of the surface.

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3∑i=1

∂X i

∂ua∂X i

∂ub.



[E FF G

], EG − F 2 > 0.


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3∑i=1

∂X i

∂ua∂X i

∂ub.



[E FF G

], EG − F 2 > 0.


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3∑i=1

∂X i

∂ua∂X i

∂ub.



[E FF G

], EG − F 2 > 0.


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3∑i=1

∂X i

∂ua∂X i

∂ub.



[E FF G

], EG − F 2 > 0.


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3∑i=1

∂X i

∂ua∂X i

∂ub.



[E FF G

], EG − F 2 > 0.


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Surfaces in R3.

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The mean curvature function

I An important extrinsic geometric quantity is the meancurvature H, which is related to the variation of area.

I If we deform the surface Σ in the normal direction at thespeed of s, the area change is∫

Σs H dµ.

I H = 0 corresponds to minimal surfaces.

I H = c corresponds to CMC (soap film).

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Σs H dµ.



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Σs H dµ.



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Σs H dµ.



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Σs H dµ.



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Surfaces in spacetime

I Consider an embedding X : Σ → R3,1 withX = (X 0,X 1,X 2,X 3)

I The induced metric is

−∂X0

∂ua∂X 0

∂ub+

3∑i=1

∂X i

∂ua∂X i

∂ub.

I Σ is spacelike if this is positive definite.

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−∂X0

∂ua∂X 0

∂ub+

3∑i=1

∂X i

∂ua∂X i

∂ub.


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−∂X0

∂ua∂X 0

∂ub+

3∑i=1

∂X i

∂ua∂X i

∂ub.


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−∂X0

∂ua∂X 0

∂ub+

3∑i=1

∂X i

∂ua∂X i

∂ub.


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I There is also the mean curvature vector ~H which is a normalvector field that again measures how area change whensurface is deformed.

I

δ ~V |Σ| = −∫

Σ〈 ~H, ~V 〉dµ.

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I

δ ~V |Σ| = −∫

Σ〈 ~H, ~V 〉dµ.

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I

δ ~V |Σ| = −∫

Σ〈 ~H, ~V 〉dµ.

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Significance of ~H for spacetime surfaces

I The mean curvature vector can be defined on surfaces in anyspacetime and is closely related to gravitational energy.

I When light rays are emanating from a surface in R3,1, theycould be diverging (outward) or converging (inward).

I There are surfaces (trapped surfaces) in a general spacetimesuch that all light rays are converging.

I This is an indication of strong gravitational field.

I Penrose singularity theorem:The existence of a trapped surface implies the formation ofspacetime singularity in the future.

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I Weyl’s isometric embedding problem into R3.

I Given a positive definite symmetric 2-tensor σab, does thereexist an embedding X : Σ→ R3 such that the induced metric

3∑i=1

∂X i

∂ua∂X i

∂ub= σab?

I There are three unknowns X 1,X 2,X 3, all functions of u1, u2.There are also three equations (E, F, G).

I When the Gauss curvature of σab is positive, this is anonlinear elliptic system of PDE’s which was solved byNirenberg and Pogorelov.

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3∑i=1

∂X i

∂ua∂X i

∂ub= σab?



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3∑i=1

∂X i

∂ua∂X i

∂ub= σab?



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3∑i=1

∂X i

∂ua∂X i

∂ub= σab?



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3∑i=1

∂X i

∂ua∂X i

∂ub= σab?



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I Generalization to R3,1 and why?

I An isometric embedding of a 2-surface into spacetime isunder-determined: four unknowns and only three equations(E, F, G).

I One needs to impose one more condition in order to expectany uniqueness.

I How and why? Quasi-local energy in general relativity.

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I A fundamental difficulty in GR is, unlike any other physicaltheory, there is NO mass or energy density for gravitation.

I The naive formula that mass is the bulk integral of massdensity is ultimately false.

I Newtonian gravity ∆Φ = 4πρ, ρ is the mass density. Bydivergence theorem, the total mass

∫Ω ρ is a flux integral on

the boundary surface ∂Ω.

I Perhaps one can define mass or energy on ∂Ω which is a2-dimensional surface?

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I In 1982, Penrose proposed a list of major unsolved problemsin GR, and the first one is:

I Find a suitable definition of quasi-local energy-momentum(mass) for a surface Σ = ∂Ω in general spacetime.

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I Hamilton-Jacobi analysis of the Einstein-Hilbert actionsuggests the following approach (Brown-York):

I Given a surface Σ in a general spacetime N. Find its “groundstate”: an isometric embedding into R3,1 that “best matches”the geometry of Σ in N.

I Hope: anchor the intrinsic geometry by isometric embeddingand read off the “gravitation energy” from the difference ofextrinsic geometries.

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I (W-Yau 2009) Given a 2-surface Σ in a general spacetime Nwith(1) the induced metric σab, and(2) the mean curvature vector ~H,a quasi-local energy can be defined for each isometricembeddings of σab into R3,1.

I The definition satisfies the important positivity and rigidityproperties and agrees with other well-accepted notions.

I Minimizing among all isometric embeddings gives an “optimalembedding equation” which is a fourth order PDE.

I Together with the isometric embedding equations, we have asystem of PDE’s with four unknowns and four equations.

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I Most recent applications: conserved quantities in generalrelativity.

I The second problem on Penrose’s list is:Find a suitable definition of quasi-local angular momentum.

I In special relativity, conserved quantities are derived fromKilling fields which correspond to continuous symmetry(isometry) of R3,1.

I For example, the rotation Killing field K = X 1 ∂∂X 2 − X 2 ∂

∂X 1

gives angular momentum.

I However, a general spacetime dose not have any continuoussymmetry or Killing field.

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∂X 1



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∂X 1



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∂X 1



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∂X 1



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∂X 1



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I The optimal isometric embedding of Σ into R3,1 is applied totransplant Killing fields in R3,1 back to the surface of interestin a physical spacetime.

I Not only can we define energy, linear momentum, angularmomentum, and center of mass, but also study the dynamicsof these conserved quantities along the Einstein equation.

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I Recall the Einstein equation can be formulated as an initialvalue problem of a system of hyperbolic PDE’s.

I Given an initial data (M, g(0), k(0)) with k(0) = ∂∂t g(0).

I To the solution of the Einstein equation (M, g(t), k(t)) weattach conserved quantities e(t), pi (t), Ji (t) and C i (t).

I Theorem (Chen-W.-Yau) In the nonlinear context of Einsteinevolution equation, we have

e(∂tCi (t)) = pi

and∂tJi (t) = 0.

I This is the first time when the classical formula mx = p isshown to be consistent with the Einstein equation.

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e(∂tCi (t)) = pi

and∂tJi (t) = 0.


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e(∂tCi (t)) = pi

and∂tJi (t) = 0.


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e(∂tCi (t)) = pi

and∂tJi (t) = 0.


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e(∂tCi (t)) = pi

and∂tJi (t) = 0.


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e(∂tCi (t)) = pi

and∂tJi (t) = 0.


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I In the second half of the talk, I shall discuss an inequality inclassical differential geometry (Minkowski) and an inequalityin general relativity (Penrose).

I This part is based on joint work with Brendle and Hung.

I Let Σ be a closed embedded surface in R3. We recall that∫Σ H dµ corresponds to the area change by unit speed (s = 1).

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I Minkowski inequality for surfaces in R3.

I For a closed convex surface Σ in R3,∫ΣH dµ ≥

√16π |Σ|,

where |Σ| is the area of Σ.

I True in higher dimensions as well. Generalization to meanconvex, star shaped hypersurfaces by Huisken, Guan-Li.

I For a round 2-sphere of radius R in R3,∫H dµ =

d

dR(4πR2) = 8πR,

and the right hand side is√

16π · 4πR2 = 8πR.

I This inequality is sharp in the sense that equality holds if andonly if Σ is a round sphere, regardless of the radius.

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√16π |Σ|,




d

dR(4πR2) = 8πR,


16π · 4πR2 = 8πR.


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√16π |Σ|,




d

dR(4πR2) = 8πR,


16π · 4πR2 = 8πR.


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√16π |Σ|,




d

dR(4πR2) = 8πR,


16π · 4πR2 = 8πR.


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√16π |Σ|,




d

dR(4πR2) = 8πR,


16π · 4πR2 = 8πR.


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√16π |Σ|,




d

dR(4πR2) = 8πR,


16π · 4πR2 = 8πR.


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I Recall the mean curvature vector field ~H defined by

δ ~V |Σ| = −∫

Σ〈 ~H, ~V 〉dµ.

I In GR, one is interested in measuring the divergence of lightrays emanating from Σ.

I We can take two null normal vector fields L and L of aspacelike 2-surface such that 〈L, L〉 = 0, 〈L, L〉 = 0,〈L, L〉 = −2.

I For example, for a surface Σ ⊂ R3 ⊂ R3,1 with outward unitnormal ν. One can take L = ∂

∂t + ν (outward) and L = ∂∂t − ν

(inward).

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δ ~V |Σ| = −∫

Σ〈 ~H, ~V 〉dµ.





(inward).

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δ ~V |Σ| = −∫

Σ〈 ~H, ~V 〉dµ.





(inward).

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δ ~V |Σ| = −∫

Σ〈 ~H, ~V 〉dµ.





(inward).

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δ ~V |Σ| = −∫

Σ〈 ~H, ~V 〉dµ.





(inward).

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I −∫

Σ〈 ~H, L〉dµ and −∫

Σ〈 ~H, L〉dµ are called the nullexpansions of a 2-surface in a general spacetime.

I For a surface Σ ⊂ R3 ⊂ R3,1, one expansion (outward) ispositive, the other (inward) is negative.

I For a trapped surface, both expansions are negative

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I −∫





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I −∫





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I −∫





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I In his original paper on cosmic censorship (every spacetimesingularity is shielded behind a black hole and thus invisible),Penrose proposed the following:

I Conjecture (Penrose [1973, Naked singularity] )For a closed embedded spacelike 2-surface in R3,1 which ispast null convex. Suppose we normalize such that〈 ∂∂t , L〉 = −1, then

−∫

Σ〈 ~H, L〉dµ ≥

√16π|Σ| (∗).

I “Past null convexity” means the past null cone along thedirection of −L extends to infinity regularly.

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−∫

Σ〈 ~H, L〉dµ ≥

√16π|Σ| (∗).


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−∫

Σ〈 ~H, L〉dµ ≥

√16π|Σ| (∗).


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−∫

Σ〈 ~H, L〉dµ ≥

√16π|Σ| (∗).


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I Gibbons’ observation: If Σ ⊂ R3 ⊂ R3,1 with the above choiceof L and L. (∗) is exactly the classical Minkowski inequality.

I (∗) is a null counterpart of the Riemannian Penrose inequality(Huisken-Ilmanen, Bray)

16π(ADM mass) ≥√

16π|Σ|

for spatial infinity (asymptotically flat).

I Tod showed (∗) holds for surfaces lie in a light cone (of apoint) of R3,1. Generalizations by Mars and Mars-Soria.

I For general surfaces in the Minkowski spacetime, theconjecture (∗) remains open.

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16π|Σ|




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16π|Σ|




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16π|Σ|




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16π|Σ|




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I Returning to the Minkowski inequality, there has been greatinterest among differential geometers in generalizing theinequality to surfaces in other space forms.

I In particular, the case of hyperbolic space was studied byGallego and Solanes.

I They showed that, for a convex surface Σ in H3∫ΣH dµ ≥ 2|Σ|.

I A geodesic sphere of radius r has area 4π sinh2 r and thus∫Σ H dµ = d

dr (4π sinh2 r) = 8π sinh r cosh r , while on the righthand side, 2|Σ| = 8π sinh2 r .

I This is a beautiful inequality but equality is never achieved.

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I On the other hand, we can embed the hyperbolic space H3

isometrically into R3,1 as(t, x , y , z) | t > 0,−t2 + x2 + y2 + z2 = −1.

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I On the other hand, we can embed the hyperbolic space H3

isometrically into R3,1 as(t, x , y , z) | t > 0,−t2 + x2 + y2 + z2 = −1.

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I It is not even clear why the expression −∫

Σ〈 ~H, L〉dµ shouldbe positive, but the Penrose inequality would predict aMinkowski type inequality on H3 ⊂ R3,1 which is sharp.

I The inequality (and more general and higher dimensionalversions) was proved by Brendle-Hung-W.

I The proof involves inverse mean curvature flows, a newmonotonicity formula on static vacuum spacetime, aHeitze-Karcher type inequality, and Beckner’s sharp Sobolevinequality on spheres.

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I The proof allows us to generalize the inequality to spacetimesof more physical interest and make the following conjecture:

I Conjecture (Brendle-W.): The following inequality holds forany spacelike past null convex 2-surface Σ in theSchwarzschild spacetime,

− 1

16π

∫Σ〈 ~H, L〉dµ+ m ≥

√|Σ|16π

. (0.1)

where m is the total mass of the Schwarzschild spacetime. Lis chosen so that the dual null normal L ( 〈L, L〉 = −2)satisfies 〈L, ∂

∂t 〉 = −1 with respect to the Killing field ∂∂t .

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− 1

16π

∫Σ〈 ~H, L〉dµ+ m ≥

√|Σ|16π

. (0.1)



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− 1

16π

∫Σ〈 ~H, L〉dµ+ m ≥

√|Σ|16π

. (0.1)



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I Conclusion.

I 1) There are close connections between surface geometry inR3 and surfaces in spacetime (codimension 2).

I 2) Physical prediction inspires new results in mathematics. Onthe other hand, natural consequences of mathematics givefeedback to the physical picture. Cross-fertilization ofmathematics and physics.

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I Conclusion.



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I Conclusion.



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I Conclusion.



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Thank you!

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surface geometry and general relativity...surface geometry and general relativity mu-tao wang...

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