A variation problem for isometric embeddings and its applications in - - PowerPoint PPT Presentation

A variation problem for isometric embeddings and its applications in general relativity

Mu-Tao Wang Columbia University and National Taiwan University Workshop on Geometric PDE Academia Sinica, June 12, 2012

◮ The object of study in this talk is a close embedded 2-surface

in a four dimensional spacetime.

2

◮ The object of study in this talk is a close embedded 2-surface

in a four dimensional spacetime.

◮ We consider the isometric embedding problem of such a

surface into the Minkowski space R3,1.

3

◮ The object of study in this talk is a close embedded 2-surface

in a four dimensional spacetime.

◮ We consider the isometric embedding problem of such a

surface into the Minkowski space R3,1.

◮ Joint work with PoNing Chen and Shing-Tung Yau.

4

◮ A central theme in geometry is to relate the extrinsic geometry

• f a subvariety to the curvature of the ambient space.

5

◮ A central theme in geometry is to relate the extrinsic geometry

• f a subvariety to the curvature of the ambient space.

◮ Pointwise, there is the Gauss-Codazzi equation which

represents the integrability condition.

6

◮ A central theme in geometry is to relate the extrinsic geometry

• f a subvariety to the curvature of the ambient space.

◮ Pointwise, there is the Gauss-Codazzi equation which

represents the integrability condition.

◮ On an extended region, we have, as a first example, the

classical comparison theorem for arc length:

7

◮ Here we have a geodesic in a “physical” space M and another

geodesic of the same length in a model space, say Rn.

8

◮ Here we have a geodesic in a “physical” space M and another

geodesic of the same length in a model space, say Rn.

◮ The second variation of arc-length gives the comparison

theorem of, for example, the Laplacian of the distance function. RicM ≥ 0 ⇒ ∆dM ≤ ∆dRn

9

◮ Here we have a geodesic in a “physical” space M and another

geodesic of the same length in a model space, say Rn.

◮ The second variation of arc-length gives the comparison

theorem of, for example, the Laplacian of the distance function. RicM ≥ 0 ⇒ ∆dM ≤ ∆dRn

◮ Notice that the Laplacian of the distance function can also be

interpreted as the mean curvature of a geodesic sphere.

10

◮ For surfaces, let me mention a theorem of Shi and Tam:

Suppose we have a closed embedded surface in a R3 and another closed embedded surface in a “physical” 3-space M with non-negative scalar curvature.

11

◮ For surfaces, let me mention a theorem of Shi and Tam:

Suppose we have a closed embedded surface in a R3 and another closed embedded surface in a “physical” 3-space M with non-negative scalar curvature.

◮ To anchor the intrinsic geometry, we assume the two surfaces

are isometric (in the case of a curve, the arc-length is the only intrinsic geometric quantity).

12

◮ For surfaces, let me mention a theorem of Shi and Tam:

Suppose we have a closed embedded surface in a R3 and another closed embedded surface in a “physical” 3-space M with non-negative scalar curvature.

◮ To anchor the intrinsic geometry, we assume the two surfaces

are isometric (in the case of a curve, the arc-length is the only intrinsic geometric quantity).

◮ Assuming that the mean curvatures and the Gauss curvatures

are both positive on the surfaces, Shi and Tam prove that

• Σ

HR3 ≥

• Σ

HM.

13

◮ For surfaces, let me mention a theorem of Shi and Tam:

Suppose we have a closed embedded surface in a R3 and another closed embedded surface in a “physical” 3-space M with non-negative scalar curvature.

◮ To anchor the intrinsic geometry, we assume the two surfaces

are isometric (in the case of a curve, the arc-length is the only intrinsic geometric quantity).

◮ Assuming that the mean curvatures and the Gauss curvatures

are both positive on the surfaces, Shi and Tam prove that

• Σ

HR3 ≥

• Σ

HM.

◮ This turns out to be a quasi-localization of Schoen-Yau’s

positive mass theorem and the difference is the so-call Brown-York mass.

14

◮ In this talk, we shall discuss a generalization of such a

comparison theorem to surfaces in spacetime.

15

◮ In this talk, we shall discuss a generalization of such a

comparison theorem to surfaces in spacetime.

◮ The hope is to detect the spacetime curvature on a region by

the extrinsic geometry of the boundary 2-surface.

16

◮ In this talk, we shall discuss a generalization of such a

comparison theorem to surfaces in spacetime.

◮ The hope is to detect the spacetime curvature on a region by

the extrinsic geometry of the boundary 2-surface.

◮ The gravitational energy is measured by the spacetime

curvature and this leads to applications in general relativity.

17

◮ In this talk, we shall discuss a generalization of such a

comparison theorem to surfaces in spacetime.

◮ The hope is to detect the spacetime curvature on a region by

the extrinsic geometry of the boundary 2-surface.

◮ The gravitational energy is measured by the spacetime

curvature and this leads to applications in general relativity.

◮ The model space in this case should be the Minkowski space

R3,1

18

◮ The well-known Weyl’s embedding problem is about isometric

embeddings into R3. Given a Riemannian metric σ on S2, we ask for an embedding X : S2 → R3 whose induced metric is σ. The equation can be written compactly as dX, dX = σ.

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◮ The well-known Weyl’s embedding problem is about isometric

embeddings into R3. Given a Riemannian metric σ on S2, we ask for an embedding X : S2 → R3 whose induced metric is σ. The equation can be written compactly as dX, dX = σ.

◮ There are three equations σ = Edu2 + 2Fdudv + Gdv2 and

three unknown functions X = (X1, X2, X3). The problem has a very satisfactory answer when the Gauss curvature of σ is positive (which implies ellipticity).

20

◮ The well-known Weyl’s embedding problem is about isometric

embeddings into R3. Given a Riemannian metric σ on S2, we ask for an embedding X : S2 → R3 whose induced metric is σ. The equation can be written compactly as dX, dX = σ.

◮ There are three equations σ = Edu2 + 2Fdudv + Gdv2 and

three unknown functions X = (X1, X2, X3). The problem has a very satisfactory answer when the Gauss curvature of σ is positive (which implies ellipticity).

◮ Theorem

(Nirenberg, Pogorelov) If the Gauss curvature of σ is positive, there exists a unique isometric embedding X up to rigid motion of R3.

21

◮ Here we are interested in isometric embeddings into R3,1. The

problem itself comes from the study of quasilocal energy in general relativity.

22

◮ Here we are interested in isometric embeddings into R3,1. The

problem itself comes from the study of quasilocal energy in general relativity.

◮ The basic question is: given a spacelike 2-surface Σ in

spacetime, find its “best match” in R3,1 (ground state of GR).

23

◮ Here we are interested in isometric embeddings into R3,1. The

problem itself comes from the study of quasilocal energy in general relativity.

◮ The basic question is: given a spacelike 2-surface Σ in

spacetime, find its “best match” in R3,1 (ground state of GR).

◮ In this case, there are four unknowns X = (X0, X1, X2, X3)

and one more condition is needed to make it a well-determined system. As X0 plays the role of the time function and can be distinguished from other coordinate

• functions. We can try to prescribe the time function.

24

◮ Here we are interested in isometric embeddings into R3,1. The

problem itself comes from the study of quasilocal energy in general relativity.

◮ The basic question is: given a spacelike 2-surface Σ in

spacetime, find its “best match” in R3,1 (ground state of GR).

◮ In this case, there are four unknowns X = (X0, X1, X2, X3)

and one more condition is needed to make it a well-determined system. As X0 plays the role of the time function and can be distinguished from other coordinate

• functions. We can try to prescribe the time function.

◮ Firstly, we make the following two observations:

25

◮ 1. Suppose Σ is spacelike 2-surfce in R3,1 (induced metric is

Riemannian) which bounds a spacelike region. We consider the projection from R3,1 to R3 given by (X0, X1, X2, X3) to (X1, X2, X3).

26

◮ 1. Suppose Σ is spacelike 2-surfce in R3,1 (induced metric is

Riemannian) which bounds a spacelike region. We consider the projection from R3,1 to R3 given by (X0, X1, X2, X3) to (X1, X2, X3).

◮ The image of projection of Σ is an embedded surface

Σ in R3. In fact, if the induced metric on Σ is σ, the induced metric on

• Σ is σ + (dX0)2.

27

◮ 2. Take any function τ on (S2, σ), then ˆ

σ = σ + (dτ)2 is another metric, the Gauss curvature of ˆ σ is

• Kτ = (1 + |∇τ|2)−1[K + (1 + |∇τ|2)−1 det(∇2τ)]

(0.1)

28

◮ 2. Take any function τ on (S2, σ), then ˆ

σ = σ + (dτ)2 is another metric, the Gauss curvature of ˆ σ is

• Kτ = (1 + |∇τ|2)−1[K + (1 + |∇τ|2)−1 det(∇2τ)]

(0.1)

◮ Suppose τ is a function on (S2, σ) with ˆ

Kτ > 0, then there exists a unique isometric embedding X into R3,1 such that the time function X0 = τ.

29

◮ 2. Take any function τ on (S2, σ), then ˆ

σ = σ + (dτ)2 is another metric, the Gauss curvature of ˆ σ is

• Kτ = (1 + |∇τ|2)−1[K + (1 + |∇τ|2)−1 det(∇2τ)]

(0.1)

◮ Suppose τ is a function on (S2, σ) with ˆ

Kτ > 0, then there exists a unique isometric embedding X into R3,1 such that the time function X0 = τ.

◮ We said that such an isometric embedding X : Σ → R3,1 has

convex shadow as the projection Σ is a convex surface in R3.

30

31

◮ There are more relations between these two surfaces Σ and

Σ.

32

◮ There are more relations between these two surfaces Σ and

Σ.

◮ Suppose Σ is a spacelike 2-surface in R3,1, we can define the

mean curvature vector H0 = ∆X which is a normal vector field along Σ.

33

◮ There are more relations between these two surfaces Σ and

Σ.

◮ Suppose Σ is a spacelike 2-surface in R3,1, we can define the

mean curvature vector H0 = ∆X which is a normal vector field along Σ.

◮ Since Σ is spacelike, the normal bundle is equipped with a

(1, 1) Lorentz metric and the mean curvature vector defines a connection one-form αH0.

34

◮ There are more relations between these two surfaces Σ and

Σ.

◮ Suppose Σ is a spacelike 2-surface in R3,1, we can define the

mean curvature vector H0 = ∆X which is a normal vector field along Σ.

◮ Since Σ is spacelike, the normal bundle is equipped with a

(1, 1) Lorentz metric and the mean curvature vector defines a connection one-form αH0.

◮ It turns out not only the metrics on Σ and

Σ are related, but

• ther geometric data, in particular, (σ, |H0|, αH0) on Σ and

( σ, hab, H) on Σ are connected through τ by the following identities.

35

36

◮ Suppose the mean curvature vector H0 of Σ in R3,1 is

space-like.

37

◮ Suppose the mean curvature vector H0 of Σ in R3,1 is

space-like.

◮ First identity:

• H
• 1 + |∇τ|2 = |H0|
• 1 + |∇τ|2 cosh θ − ∇θ · ∇τ − αH0(∇τ)

(0.2) where θ is given by sinh θ = −∆τ |H0|

• 1 + |∇τ|2 .

38

◮ Suppose the mean curvature vector H0 of Σ in R3,1 is

space-like.

◮ First identity:

• H
• 1 + |∇τ|2 = |H0|
• 1 + |∇τ|2 cosh θ − ∇θ · ∇τ − αH0(∇τ)

(0.2) where θ is given by sinh θ = −∆τ |H0|

• 1 + |∇τ|2 .

◮ Second identity:

( Hˆ σab − ˆ σac ˆ σbdˆ hcd) ∇b∇aτ

• 1 + |∇τ|2

= divσ(|H0| ∇τ

• 1 + |∇τ|2 cosh θ − ∇θ − αH0)

(0.3) The first identity is indeed a result of conservation law.

39

Searching for the optimal isometric embedding

40

Searching for the optimal isometric embedding

◮ Returning the general case, given a surface Σ with

Riemannian metric σ. Suppose Kτ is positive for a function τ, then there is an isometric embedding of Σ into R3,1 with τ as a time function.

41

Searching for the optimal isometric embedding

◮ Returning the general case, given a surface Σ with

Riemannian metric σ. Suppose Kτ is positive for a function τ, then there is an isometric embedding of Σ into R3,1 with τ as a time function.

◮ The question is among all such time functions, is there a

“best” one? Of course, we need to impose one more equation to make the system well-determined, and also need extra data

• ther than just the metric.

42

Searching for the optimal isometric embedding

◮ Returning the general case, given a surface Σ with

Riemannian metric σ. Suppose Kτ is positive for a function τ, then there is an isometric embedding of Σ into R3,1 with τ as a time function.

◮ The question is among all such time functions, is there a

“best” one? Of course, we need to impose one more equation to make the system well-determined, and also need extra data

• ther than just the metric.

◮ Consider a spacelike 2-surface Σ in a general spacetime, the

mean curvature vector H can be defined and we assume it is

• spacelike. Let αH be the associated connection one-form.

43

Searching for the optimal isometric embedding

◮ Returning the general case, given a surface Σ with

Riemannian metric σ. Suppose Kτ is positive for a function τ, then there is an isometric embedding of Σ into R3,1 with τ as a time function.

◮ The question is among all such time functions, is there a

“best” one? Of course, we need to impose one more equation to make the system well-determined, and also need extra data

• ther than just the metric.

◮ Consider a spacelike 2-surface Σ in a general spacetime, the

mean curvature vector H can be defined and we assume it is

• spacelike. Let αH be the associated connection one-form.

◮ Therefore, the data (σ, |H|, αH) is given and intrinsically

defined on the surface Σ.

44

Searching for the optimal isometric embedding

◮ Returning the general case, given a surface Σ with

Riemannian metric σ. Suppose Kτ is positive for a function τ, then there is an isometric embedding of Σ into R3,1 with τ as a time function.

◮ The question is among all such time functions, is there a

“best” one? Of course, we need to impose one more equation to make the system well-determined, and also need extra data

• ther than just the metric.

◮ Consider a spacelike 2-surface Σ in a general spacetime, the

mean curvature vector H can be defined and we assume it is

• spacelike. Let αH be the associated connection one-form.

◮ Therefore, the data (σ, |H|, αH) is given and intrinsically

defined on the surface Σ.

◮ For this data, we shall define an energy functional on the

space of isometric embeddings of σ into R3,1.

45

46

◮ To be more specific, we consider a pair (X, T0) in which X is

an isometric embedding of σ into R3,1 and T0 is a future timelike unit vector. We ask that X has spacelike mean curvature vector and has convex shadow in the direction of T0.

47

◮ To be more specific, we consider a pair (X, T0) in which X is

an isometric embedding of σ into R3,1 and T0 is a future timelike unit vector. We ask that X has spacelike mean curvature vector and has convex shadow in the direction of T0.

◮ Let Ξ denote the class of such (X, T0).

48

◮ To be more specific, we consider a pair (X, T0) in which X is

an isometric embedding of σ into R3,1 and T0 is a future timelike unit vector. We ask that X has spacelike mean curvature vector and has convex shadow in the direction of T0.

◮ Let Ξ denote the class of such (X, T0). ◮ Given (X, T0) ∈ Ξ, let τ = −X, T0 be the time function

with respect to T0 and Σ be the shadow convex surface.

49

◮ To be more specific, we consider a pair (X, T0) in which X is

an isometric embedding of σ into R3,1 and T0 is a future timelike unit vector. We ask that X has spacelike mean curvature vector and has convex shadow in the direction of T0.

◮ Let Ξ denote the class of such (X, T0). ◮ Given (X, T0) ∈ Ξ, let τ = −X, T0 be the time function

with respect to T0 and Σ be the shadow convex surface.

◮ The quasilocal energy is defined to be

8πE(Σ, X, T0) =

• Σ
• H
• 1 + |∇τ|2

−

• Σ

|H|

• 1 + |∇τ|2 cosh θ − ∇θ · ∇τ − αH(∇τ),

where sinh θ =

−∆τ |H|√ 1+|∇τ|2 .

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◮ This is zero if Σ is in the Minkowski space. We proved:

51

◮ This is zero if Σ is in the Minkowski space. We proved: ◮ (W-Yau) E(Σ, X, T0) ≥ 0 if Σ bounds a regular spacelike

region in a spacetime which satisfies the dominant energy condition and (X, T0) is admissible.

52

◮ This is zero if Σ is in the Minkowski space. We proved: ◮ (W-Yau) E(Σ, X, T0) ≥ 0 if Σ bounds a regular spacelike

region in a spacetime which satisfies the dominant energy condition and (X, T0) is admissible.

◮ In order to look for optimal isometric embedding, we minimize

the quasilocal energy in the admissible sets. The minimum is defined to be the quasilocal mass.

53

◮ Given (σ, |H|, αH), the Euler Lagrange equation for (X, T0)

with τ = −X, T0 is ( Hˆ σab − ˆ σac ˆ σbdˆ hcd) ∇b∇aτ

• 1 + |∇τ|2

= divΣ(|H| ∇τ

• 1 + |∇τ|2 cosh θ − ∇θ − αH),

(0.4) where sinh θ =

−∆τ |H|√ 1+|∇τ|2 .

54

◮ Coupling with the isometric embedding equation into R3,1,

dX, dX = σ. The solution gives the optimal isometric embedding.

55

◮ Coupling with the isometric embedding equation into R3,1,

dX, dX = σ. The solution gives the optimal isometric embedding.

◮ Four unknown functions (coordinates) and four equations.

56

◮ Coupling with the isometric embedding equation into R3,1,

dX, dX = σ. The solution gives the optimal isometric embedding.

◮ Four unknown functions (coordinates) and four equations. ◮ The equation should be read in the following way. Take a

function τ on Σ, consider ˆ σ = σ + (dτ)2 on Σ. Isometrically embed ˆ σ into R3. Pick up ˆ hab and ˆ H from this isometric embedding and we look for τ that satisfies (0.4), a fourth-order elliptic equation for τ.

57

Solving the equation for isolated systems

◮ We solve the optimal isometric embedding problem for

isolated physical systems in GR (asymptotically flat spacetime)

58

Solving the equation for isolated systems

◮ We solve the optimal isometric embedding problem for

isolated physical systems in GR (asymptotically flat spacetime)

◮ There are at least two different notions of asymptotical

flatness which correspond to null infinity and spatial infinity. In both cases, the infinity is foliated by a family of 2-surface Σr, r ∈ [r0, ∞) such that σ = ˜ σ + O(r), |H| = 2 r + O(r−2), and divσαH = O(r−3)

59

Solving the equation for isolated systems

◮ We solve the optimal isometric embedding problem for

isolated physical systems in GR (asymptotically flat spacetime)

◮ There are at least two different notions of asymptotical

flatness which correspond to null infinity and spatial infinity. In both cases, the infinity is foliated by a family of 2-surface Σr, r ∈ [r0, ∞) such that σ = ˜ σ + O(r), |H| = 2 r + O(r−2), and divσαH = O(r−3)

◮ When r is large enough, Σr has positive Gauss curvature, and

there exists a unique isometric embedding Xr : Σr → R3 ⊂ R3,1

60

Solving the equation for isolated systems

◮ We solve the optimal isometric embedding problem for

isolated physical systems in GR (asymptotically flat spacetime)

◮ There are at least two different notions of asymptotical

flatness which correspond to null infinity and spatial infinity. In both cases, the infinity is foliated by a family of 2-surface Σr, r ∈ [r0, ∞) such that σ = ˜ σ + O(r), |H| = 2 r + O(r−2), and divσαH = O(r−3)

◮ When r is large enough, Σr has positive Gauss curvature, and

there exists a unique isometric embedding Xr : Σr → R3 ⊂ R3,1

◮ Xr solves the optimal isometric embedding problem up to

O(r−2). In fact, any SO(3, 1) congruent Xr does.

61

◮ With this initial choice, we show that limr→∞ E(Σr, Xr, T0) is

linear in T0 and defines an energy-momentum 4-(co)vector (E, P1, P2, P3) which is the ADM/Bondi-Sachs energy-momentum in the asymptotically flat/null case.

62

◮ With this initial choice, we show that limr→∞ E(Σr, Xr, T0) is

linear in T0 and defines an energy-momentum 4-(co)vector (E, P1, P2, P3) which is the ADM/Bondi-Sachs energy-momentum in the asymptotically flat/null case.

◮ In order to improve Xr, we boost it in R3,1 by an element in

SO(3, 1) into a totally geodesic slice that is orthogonal to the total energy-momentum (E, P1, P2, P3). This can be done uniquely if (E, P1, P2, P3) is timelike and the result solves the

• ptimal isometric embedding equation up to O(r−3).

63

◮ With this initial choice, we show that limr→∞ E(Σr, Xr, T0) is

linear in T0 and defines an energy-momentum 4-(co)vector (E, P1, P2, P3) which is the ADM/Bondi-Sachs energy-momentum in the asymptotically flat/null case.

◮ In order to improve Xr, we boost it in R3,1 by an element in

SO(3, 1) into a totally geodesic slice that is orthogonal to the total energy-momentum (E, P1, P2, P3). This can be done uniquely if (E, P1, P2, P3) is timelike and the result solves the

• ptimal isometric embedding equation up to O(r−3).

◮ (Chen-W-Yau) If the ADM/Bondi mass at spatial/null infinity

is positive, the optimal isometric embedding problem can be solved and the solution is a local minimizer (assuming analyticity in r).

64

◮ With this initial choice, we show that limr→∞ E(Σr, Xr, T0) is

linear in T0 and defines an energy-momentum 4-(co)vector (E, P1, P2, P3) which is the ADM/Bondi-Sachs energy-momentum in the asymptotically flat/null case.

◮ In order to improve Xr, we boost it in R3,1 by an element in

SO(3, 1) into a totally geodesic slice that is orthogonal to the total energy-momentum (E, P1, P2, P3). This can be done uniquely if (E, P1, P2, P3) is timelike and the result solves the

• ptimal isometric embedding equation up to O(r−3).

◮ (Chen-W-Yau) If the ADM/Bondi mass at spatial/null infinity

is positive, the optimal isometric embedding problem can be solved and the solution is a local minimizer (assuming analyticity in r).

◮ The small sphere limit can be solved too and the

Bel-Robinson tensor is the leading term in the vacuum case.

65

◮ The second variation of the quasilocal energy at this order is

given by

• S2(∆S2δτ + 1

2∆S2∆S2δτ)δτ.

66

◮ The second variation of the quasilocal energy at this order is

given by

• S2(∆S2δτ + 1

2∆S2∆S2δτ)δτ.

◮ Since the operator ∆S2 + 1 2∆S2∆S2 is positive, this shows the

solution we obtained is locally energy-minimizing.

67

◮ For a metric sphere Σ with αH = 0 and |H| constant, is the

• ptimal isometric embedding a standard round sphere in R3?

Is it energy-minimizing?

68

◮ For a metric sphere Σ with αH = 0 and |H| constant, is the

• ptimal isometric embedding a standard round sphere in R3?

Is it energy-minimizing?

◮ In general, if Σ lies in a spacelike hypersurface Ω with

vanishing second fundamental form in spacetime (time-symmetric case), there is the notion of Brown-York mass: 1 8π(

• Σ

H0 −

• Σ

H) where H0 is the mean curvature of isometric embedding of Σ = ∂Ω into R3. Assume KΣ > 0. This is proved to be positive when Ω has non-negative scalar curvature and H is positive by Shi and Tam.

69

◮ For a metric sphere Σ with αH = 0 and |H| constant, is the

• ptimal isometric embedding a standard round sphere in R3?

Is it energy-minimizing?

◮ In general, if Σ lies in a spacelike hypersurface Ω with

vanishing second fundamental form in spacetime (time-symmetric case), there is the notion of Brown-York mass: 1 8π(

• Σ

H0 −

• Σ

H) where H0 is the mean curvature of isometric embedding of Σ = ∂Ω into R3. Assume KΣ > 0. This is proved to be positive when Ω has non-negative scalar curvature and H is positive by Shi and Tam.

◮ Is the minimizer the same as the Brown-York mass in this

case? Is the optimal isometric embedding in this case the unique one into R3?

70

◮ When αH = 0, τ = 0 satisfies the E-L equation. The second

variation at τ = 0 is given by

• Σ

(∆η)2 H + (H0 − H)|∇η|2 − II0(∇η, ∇η) (0.5) for any function η on Σ. Recall Σ = ∂Ω and Ω has non-negative scalar curvature. Σ has positive Gauss curvature and positive mean curvature H.

71

◮ When αH = 0, τ = 0 satisfies the E-L equation. The second

variation at τ = 0 is given by

• Σ

(∆η)2 H + (H0 − H)|∇η|2 − II0(∇η, ∇η) (0.5) for any function η on Σ. Recall Σ = ∂Ω and Ω has non-negative scalar curvature. Σ has positive Gauss curvature and positive mean curvature H.

◮ Miao-Tam-Xie proved the second variation is non-negative in

following cases 1) H0 ≥ H pointwise. 2) Σ is a metric sphere with constant mean curvature H. 3) (Ω, g) is conformal diffeomorphic to a Euclidean domain.

72

◮ When αH = 0, τ = 0 satisfies the E-L equation. The second

variation at τ = 0 is given by

• Σ

(∆η)2 H + (H0 − H)|∇η|2 − II0(∇η, ∇η) (0.5) for any function η on Σ. Recall Σ = ∂Ω and Ω has non-negative scalar curvature. Σ has positive Gauss curvature and positive mean curvature H.

◮ Miao-Tam-Xie proved the second variation is non-negative in

following cases 1) H0 ≥ H pointwise. 2) Σ is a metric sphere with constant mean curvature H. 3) (Ω, g) is conformal diffeomorphic to a Euclidean domain.

◮ This can be viewed as a differential inequality (such as

Sobolev or Poincare inequalities) that is adapted to the local geometry of the surface.

73

◮ Global minimizing of a critical point with H0 ≥ |H| and

αH = 0 pointwise.

74

◮ Global minimizing of a critical point with H0 ≥ |H| and

αH = 0 pointwise.

◮ (Chen-W-Yau) Suppose τ0 satisfies the optimal isometric

embedding equation, then the quasilocal energy of τ0 is a global minimum among all isometric embeddings τ that are admissible to both the optimal isometric embedding τ0 and the physical embedding.

75

◮ Questions:

76

◮ Questions: ◮ 1. Can every metric on S2 be embedded into R3,1 with

convex shadow? (Pogorelov’s Theorem)

77

◮ Questions: ◮ 1. Can every metric on S2 be embedded into R3,1 with

convex shadow? (Pogorelov’s Theorem)

◮ 2. Show in the asymptotically flat case the equation is

solvable without assuming analyticity. Method of continuity?

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◮ Questions: ◮ 1. Can every metric on S2 be embedded into R3,1 with

convex shadow? (Pogorelov’s Theorem)

◮ 2. Show in the asymptotically flat case the equation is

solvable without assuming analyticity. Method of continuity?

◮ 3. Prove compactness of isometric embeddings under

energy-condition (modulo Poincare group actions). This will show that m(Σ) = 0 implies Σ is embeddable into R3,1.

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