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 R 3 , 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 R 3 , 1 . ◮ Joint work with PoNing Chen and Shing-Tung Yau. 4
◮ A central theme in geometry is to relate the extrinsic geometry of a subvariety to the curvature of the ambient space. 5
◮ A central theme in geometry is to relate the extrinsic geometry of 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 of 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 R n . 8
◮ Here we have a geodesic in a “physical” space M and another geodesic of the same length in a model space, say R n . ◮ The second variation of arc-length gives the comparison theorem of, for example, the Laplacian of the distance function. Ric M ≥ 0 ⇒ ∆ d M ≤ ∆ d R n 9
◮ Here we have a geodesic in a “physical” space M and another geodesic of the same length in a model space, say R n . ◮ The second variation of arc-length gives the comparison theorem of, for example, the Laplacian of the distance function. Ric M ≥ 0 ⇒ ∆ d M ≤ ∆ d R n ◮ 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 R 3 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 R 3 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 R 3 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 � � H R 3 ≥ H M . Σ Σ 13
◮ For surfaces, let me mention a theorem of Shi and Tam: Suppose we have a closed embedded surface in a R 3 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 � � H R 3 ≥ H M . Σ Σ ◮ 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 R 3 , 1 18
◮ The well-known Weyl’s embedding problem is about isometric embeddings into R 3 . Given a Riemannian metric σ on S 2 , we ask for an embedding X : S 2 → R 3 whose induced metric is σ . The equation can be written compactly as � dX , dX � = σ. 19
◮ The well-known Weyl’s embedding problem is about isometric embeddings into R 3 . Given a Riemannian metric σ on S 2 , we ask for an embedding X : S 2 → R 3 whose induced metric is σ . The equation can be written compactly as � dX , dX � = σ. ◮ There are three equations σ = Edu 2 + 2 Fdudv + Gdv 2 and three unknown functions X = ( X 1 , X 2 , X 3 ). 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 R 3 . Given a Riemannian metric σ on S 2 , we ask for an embedding X : S 2 → R 3 whose induced metric is σ . The equation can be written compactly as � dX , dX � = σ. ◮ There are three equations σ = Edu 2 + 2 Fdudv + Gdv 2 and three unknown functions X = ( X 1 , X 2 , X 3 ). 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 R 3 . 21
◮ Here we are interested in isometric embeddings into R 3 , 1 . The problem itself comes from the study of quasilocal energy in general relativity. 22
◮ Here we are interested in isometric embeddings into R 3 , 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 R 3 , 1 (ground state of GR). 23
◮ Here we are interested in isometric embeddings into R 3 , 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 R 3 , 1 (ground state of GR). ◮ In this case, there are four unknowns X = ( X 0 , X 1 , X 2 , X 3 ) and one more condition is needed to make it a well-determined system. As X 0 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 R 3 , 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 R 3 , 1 (ground state of GR). ◮ In this case, there are four unknowns X = ( X 0 , X 1 , X 2 , X 3 ) and one more condition is needed to make it a well-determined system. As X 0 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 R 3 , 1 (induced metric is Riemannian) which bounds a spacelike region. We consider the projection from R 3 , 1 to R 3 given by ( X 0 , X 1 , X 2 , X 3 ) to ( X 1 , X 2 , X 3 ). 26
◮ 1. Suppose Σ is spacelike 2-surfce in R 3 , 1 (induced metric is Riemannian) which bounds a spacelike region. We consider the projection from R 3 , 1 to R 3 given by ( X 0 , X 1 , X 2 , X 3 ) to ( X 1 , X 2 , X 3 ). ◮ The image of projection of Σ is an embedded surface � Σ in R 3 . In fact, if the induced metric on Σ is σ , the induced metric on � Σ is σ + ( dX 0 ) 2 . 27
σ = σ + ( d τ ) 2 is ◮ 2. Take any function τ on ( S 2 , σ ), then ˆ another metric, the Gauss curvature of ˆ σ is K τ = (1 + |∇ τ | 2 ) − 1 [ K + (1 + |∇ τ | 2 ) − 1 det( ∇ 2 τ )] � (0.1) 28
σ = σ + ( d τ ) 2 is ◮ 2. Take any function τ on ( S 2 , σ ), then ˆ another metric, the Gauss curvature of ˆ σ is K τ = (1 + |∇ τ | 2 ) − 1 [ K + (1 + |∇ τ | 2 ) − 1 det( ∇ 2 τ )] � (0.1) ◮ Suppose τ is a function on ( S 2 , σ ) with ˆ K τ > 0, then there exists a unique isometric embedding X into R 3 , 1 such that the time function X 0 = τ . 29
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