Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space

Using the sub- and super-solution technique, there is a unique solution φ[γ,kTT,τ] of the Lichnerowicz equation.

“Using the sub and super solution technique [38, 40], it is established that there is a unique solution φ [ γ, k TT , τ ] of the Lichnerowicz equation.”

• cites [38] 2002, Asymptotic behavior in polarized and half-polarized U(1) symmetric vacuum spacetimes, Classical Quantum Gravity sourcePartly supported

  The Asymptotic Constraint Equations section (p. 28) says that the Einstein superHamiltonian constraint is a nonlinear elliptic equation for the conformal factor ψ and that it has a unique solution for nonzero constant mean curvature and generic conformal data. It does not identify this as a Lichnerowicz equation for φ[γ,kTT,τ] or say that the result is established using sub- and super-solutions; its stated conditions also limit the uniqueness claim. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites [40] 2020, Future attractors in 2 + 1 dimensional Λ gravity, Phys. Rev. Lett. sourceNot checked — no full text of the source

• AssumptionRests on an assumption: For genus(Σp) > 1, there exists a unique conformal metric γ with R(γ) = -1.

The phase space of the reduced 2+1 gravity dynamics is the cotangent bundle T*TΣ_p (≈ R^{12p-12}).

“where the phase space of the dynamics was identified with the co-tangent bundle of Teichmüller space ( ≈ R 12 p -12 ).”

• Also stated in: § 2.1. Background on Teichmüller space · p. 7 · plain

  “T ∗ T Σ p is indeed the phase space of the reduced dynamics.”

• cites [1] 1986, Reduction of Einstein’s equations for vacuum space-times with spacelike U(1) isometry groups, Ann. Phys. sourceNot checked — no full text of the source
• cites [13] 2011, Einstein spaces as attractors for the Einstein flow, J. Differ. Geom. sourcePartly supported

  Section 2.5.2 (page unavailable) identifies Teichmüller space as the deformation space for a genus-p surface and gives its dimension as 6p−6, consistent with the stated dimension of a cotangent bundle. But our copy does not identify the reduced 2+1 gravity phase space as that cotangent bundle or state the claimed global equivalence to R^(12p−12). (LLM, conf=0.83)

  Experimental — accuracy not yet measured

The space of measured foliations MF is homeomorphic to the space of measured laminations ML.

“Given a measured foliation, one may obtain a measured lamination via a suitable straightening map [29, 33] (or collapsing a lamination yields a foliation). Therefore, there is a homeomorphism between MF and ML .”

• cites [29] 1997, Hamiltonian reduction of Einstein’s equations of general relativity, Nucl. Phys. B sourceCould not verify

  The Introduction and Section 3 discuss Teichmüller space and conformal structures, but do not address measured foliations or measured laminations. Searches for those terms, straightening, and a homeomorphism between MF and ML found no relevant passage in our copy. Page numbers were not available. (LLM, conf=0.77)

  Experimental — accuracy not yet measured
• cites [33] 1997Not checked yet

  Not checked in the free run.

[37] proved a local well-posedness theorem for the Cauchy problem for a family of elliptic-hyperbolic systems that included the n+1 dimensional vacuum Einstein equations in CMCSH gauge, n >= 2.

“[37] proved a local well-posedness theorem for the Cauchy problem for a family of elliptichyperbolic systems that included the ' n + 1 ' dimensional vacuum Einstein equations in CMCSH gauge, n ≥ 2 .”

• cites [37] 2013, Asymptotics of the Teichmüller harmonic map flow, Adv. Math. sourceCould not verify

  The Introduction (page unavailable) describes the Teichmüller harmonic map flow, and Section 3 studies its asymptotic convergence. Searches for CMCSH gauge, the vacuum Einstein equations, and local well-posedness found no passage addressing the claimed theorem in our copy. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

The space of holomorphic quadratic differentials QD on Σ_p is homeomorphic to the space of measured foliations MF on Σ_p.

“By the famous theorem of Hubbard and Masur [31], there is a homeomorphism between the space of holomorphic quadratic differential QD and the space of measured foliations MF on Σ p .”

• cites [31] 2007, The Schwarzian derivative and measured laminations on Riemann surfaces, Duke Math. J. sourceSupported

[13] explicitly showed that every non-trivial solution curve of the reduced dynamics in the configuration space TΣp approaches a point ρ lying in the interior of Teichmüller space as τ -> 0-.

“Analyzing the associated Monge-Ampere equation, [13] explicitly showed that every non-trivial solution curve of the reduced dynamics in the configuration space ( T Σ p ) approaches a point ( ρ ) lying in the interior of the Teichmüller space, that is,”

• cites [13] 2011, Einstein spaces as attractors for the Einstein flow, J. Differ. Geom. sourcePartly supported

  Corollary 8.3 states that solution curves covered by Theorem 8.1 converge to a limiting metric in the deformation space. But Theorem 8.1 is restricted to sufficiently small data near a suitable Einstein metric; the paper does not establish this for every non-trivial curve in TΣp, nor does it derive the result by analyzing a Monge–Ampère equation. Its discussion of convergence to a point in Teichmüller space (Additional information; page unavailable) describes analogous results from other works. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

[13] showed through studying the Hamilton-Jacobi equation that (g,K,N,X) solves the Einstein equations for all τ given initial data satisfying the constraints.

“This is exactly shown in [13] through studying the associated Hamilton Jacobi equation for the reduced dynamics.”

• cites [13] 2011, Einstein spaces as attractors for the Einstein flow, J. Differ. Geom. sourcePartly supported

  Section 5 (Local Existence) establishes a solution for small initial data and says that constraints satisfied initially remain satisfied during evolution. Theorem 8.1 (Section 8, Future Complete Spacetimes) extends this to the future only under additional small-data and geometric assumptions. The paper uses energy estimates and a continuation argument, not an analysis of the Hamilton–Jacobi equation; it also does not establish existence for all τ without those qualifications. (LLM, conf=0.91)

  Experimental — accuracy not yet measured

[12] proved a global existence theorem for expanding solutions in the same gauge by controlling the Dirichlet energy of an associated harmonic map for any τ in (-∞,0).

“In addition to the local well-posedness, [12] proved a global existence theorem for the expanding solutions in the same gauge by controlling the Dirichlet energy of an associated harmonic map for any τ ∈ ( -∞ , 0) .”

• cites [12] 1999, The Einstein flow, the sigma-constant and the geometrization of 3-manifolds, Classical Quantum Gravity sourceCould not verify

  The Letter to the Editor describes τ ∈ (−∞, 0) as the expected maximal range, not a range established by a global-existence theorem. Section 5.1 discusses long-time existence as a potential future application. Searches for Dirichlet energy, harmonic maps, and a global-existence theorem found no passage establishing the attributed result. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

There is a natural homeomorphism between the space of transverse traceless tensors on (Σp,γ) and the Teichmüller space from the Einstein flow.

“In addition, we have a natural homeomorphism between the space of transverse traceless tensors on (Σ p , γ ) and the Teichmüller space from the Einstein flow (for a detailed analysis see [13]).”

• cites [13] 2011, Einstein spaces as attractors for the Einstein flow, J. Differ. Geom. sourcePartly supported

  Section 1.4 (page unavailable) says that, for higher-genus surfaces in two dimensions, TT tensors describe the tangent space to the Teichmüller space of self-similar Einstein-flow solutions. Section 2.5.2 identifies Teichmüller space as the deformation space in that setting. These passages support a tangent-space correspondence, but do not establish the claimed natural homeomorphism between the full space of TT tensors and Teichmüller space. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

In(τ) and S(τ) are continuous functions of τ by existence-uniqueness-continuity (well-posedness) of Einstein's equations.

“Both I n ( τ ) and S ( τ ) are continuous functions of τ by existence-uniqueness-continuity (or well-posedness) of the Einstein's equations [12].”

• cites [12] 1999, The Einstein flow, the sigma-constant and the geometrization of 3-manifolds, Classical Quantum Gravity sourceCould not verify

  The Letter to the Editor defines a time-dependent Hamiltonian in terms of τ, and section 4 discusses its infimum, but neither passage identifies I_n(τ) or S(τ) or establishes their continuity. Searches for those functions and for well-posedness, existence, uniqueness, and continuous dependence found no passage making the attributed argument. (LLM, conf=0.74)

  Experimental — accuracy not yet measured

|KTT|^2_g satisfies an estimate for all τ in (-∞,0).

“Lemma 3.2. | K TT | 2 g satisfies the estimate”

For φ solving the Lichnerowicz equation, e^{2φ} verifies a point-wise estimate.

“Lemma 3.1. Let φ : Σ p → R solves the Lichnerowicz equation (3.18). Then e 2 φ := e 2 φ ( τ,k TT ,γ ) verifies the following point-wise estimate”

Compactness yields the extraction of a convergent subsequence from any minimizing sequence on the solution ray in the forward-in-time direction.

“Compactness yields the extraction of a convergent subsequence from any minimizing sequence on the solution ray in the forwardin-time direction (minimizing the Dirichlet energy).”

Every non-trivial solution curve of the reduced Einsteinian dynamics runs off the edge of Teichmüller space at the limit of the big-bang singularity τ→-∞.

“every such non-trivial solution curve runs off the edge of the configuration space (Teichmüller space) in the limit of the big-bang singularity τ →-∞ (past).”

• Also stated in: § 1. Introduction · p. 5 · plain

  “Every non-trivial solution curve of the reduced Einsteinian dynamics runs off the edge of the Teichmüller space at the limit of big-bang singularity”

Every non-trivial solution asymptotically approaches the fixed point solution (R(γ)=-1, k^TT=0, N=2, X^i=0) of the dynamical equations in the future limit τ:=tr_g K→0.

“Every such non-trivial solution asymptotically approaches the fixed point solution ( R ( γ ) = -1 , k TT = 0 , N = 2 , X i = 0 ) of the dynamical equations in the limit τ := tr g K → 0 (future)”

• AssumptionRests on an assumption: Every Riemannian metric on a closed genus p>1 surface Σ_p is uniquely globally conformal to a metric γ satisfying R(γ)=-1.

The ratio of the transverse measure of any non-trivial element C of π_1(Σ_p) with respect to the vertical measured foliation of the holomorphic quadratic differential φ := (k^TT_11 - √-1 k^TT_12) dz^2 to its hyperbolic length with respect to γ approaches a finite constant independent of the homotopy class in the limit τ→-∞ along every sequence on the solution curve.

“The ratio of the transverse measure of any non-trivial element C of π 1 (Σ p ) with respect to the vertical measured foliation of the natural holomorphic quadratic differential ϕ := ( k TT 11 -√ -1 k TT 12 ) dz 2 , 4 and its hyperbolic length that is the length with respect to the metric γ approaches to a finite constant independent of any homotopy class in the limit of the big-bang singularity i.”

• Also stated in: § 1. Introduction · p. 4 · plain

  “More precisely, the ratio of hyperbolic length and transverse measure against the vertical foliation approaches a constant independent of the homotopy class of curves as the big bang is approached.”

• AssumptionRests on an assumption: Every Riemannian metric on a closed genus p>1 surface Σ_p is uniquely globally conformal to a metric γ satisfying R(γ)=-1.

The reduced Hamiltonian can be related to the Dirichlet energy of the Gauss map.

“Therefore, we have the following relation between the Dirichlet energy of the Gauss map and the reduced Hamiltonian of the dynamics”

The Dirichlet energy decays monotonically in the time-forward direction.

“First we observe the following monotonic decay of the Dirichlet energy in the time-forward direction”

The lapse function N verifies the estimate 1 <= N <= 2.

“Corollary 3.1. The Lapse function N verifies the estimate 1 ≤ N ≤ 2”

The transverse measure with respect to the horizontal foliation associated with the holomorphic quadratic differential φ collapses to zero in the limit τ→-∞.

“The transverse measure with respect to the horizontal foliation associated with the holomorphic quadratic differential ϕ collapses to zero in the same limit.”

• Also stated in: § 1. Introduction · p. 4 · plain

  “The corresponding horizontal measure approaches zero at the same limit.”

• AssumptionRests on an assumption: Every Riemannian metric on a closed genus p>1 surface Σ_p is uniquely globally conformal to a metric γ satisfying R(γ)=-1.

If K := KTT + 1/2 tr_g K g solves the momentum constraint in CMC gauge, then |KTT|^2_g satisfies a quasi-linear elliptic equation on a constant time hypersurface.

“Proposition 3.1. Let K := K TT + 1 2 tr g Kg be a solution of the momentum constraint equation (3.8) in CMC gauge ∂ i τ := ∂ i tr g K = 0 , then | K TT | 2 g satisfies the following quasi-linear elliptic equation on a constant time hypersurface”

Along any solution ray with initial data (γ0,kTT0,τ0,φ0), E[γ,ρ] lies in the interval [-4πχ,E0] due to the monotonicity property.

“Along any solution ray with initial data ( γ 0 , k TT 0 , τ 0 , φ 0 ) , E [ id ; γ, ρ ] lies in the interval [ -4 πχ, E 0 ] due to the monotonicity property 3.56.”

Since the real dimension of the space of holomorphic quadratic differentials is 6p-6 (finite), the L1 norm is equivalent to Ber's supremum norm.

“Since, the dimension (real) of this space is 6 p -6 (therefore, finite), the L 1 norm is equivalent to Ber's supremum norm.”

The constants Cf and C∞ are uniform in that they do not depend on the chosen homotopy class [C].

“An important point to notice is that the constants C f and C ∞ are uniform in the sense that they do not depend on the chosen homotopy class [ C ] .”

At the big-bang limit τ → -∞, an important expression is obtained with 0 < Cf^2 < C∞^2 < ∞.

“with 0 &lt; C 2 f &lt; C 2 ∞ &lt; ∞ . This is an important expression obtained at the limit of the big-bang ( τ → -∞ ).”

• AssumptionThe authors state this as an assumption: The infimum of √|kTT|^2γ is Cf, and it is strictly positive provided one stays away from the finite zeros of the quadratic differential ϕ.
• AssumptionThe authors state this as an assumption: The quadratic differential has a finite number of zeros (z1, z2, ..., zn), n < ∞.

The transverse-traceless property of kTT precisely implies ∂φ/∂z̄ = 0, i.e., φ is holomorphic.

“The transverse-traceless property of k TT ij precisely implies ∂ϕ ∂ ¯ z = 0 i.e., ϕ is holomorphic.”

Under the assumption |kTTj(u,u)| > δ for all j, K1 → √a/√2 and K2 → √a/√2 as j → ∞.

“Under the assumption | k TT j ( u, u ) | &gt; δ ∀ j , we observe that K 1 → √ a √ 2 and K 2 → √ a √ 2 as j →∞ .”

• AssumptionThe authors state this as an assumption: In the proof of Lemma 4.3, |kTTj(u,u)| > δ for all j for a fixed δ > 0.

In the big-bang limit τ → -∞, a crucial relation holds for a suitable constant C with |Cf|^{1/2}2^{1/4} ≤ C ≤ |C∞|^{1/2}2^{1/4}.

“Therefore, we obtain the following crucial relation in the big-bang limit ( τ →-∞ )”

• Also stated in: § 4. Asymptotic behavior of the solution curve at big-bang and Thurston boundary · p. 27 · plain

  “for a suitable constant | C f | 1 / 2 2 1 / 4 ≤ C ≤ | C ∞ | 1 / 2 2 1 / 4 .”

• AssumptionThe authors state this as an assumption: The infimum of √|kTT|^2γ is Cf, and it is strictly positive provided one stays away from the finite zeros of the quadratic differential ϕ.
• AssumptionThe authors state this as an assumption: The quadratic differential has a finite number of zeros (z1, z2, ..., zn), n < ∞.

The limit of the ratio μvert_j(C)/lγ_j(C) is √a/√2, a universal constant independent of the chosen sequence.

“Remark: Note that the limit of the ratio µ vert j ( C ) /l γ j ( C ) is √ a √ 2 which is a universal constant and therefore does not depend on the chosen sequence.”

The evolution equation for |kTT|^2γ is solved by |kTT|^2γ = a^2 - O(1/τ^2) almost everywhere on Σp as τ → -∞.

“Therefore we observe that the evolution equation for | k TT | 2 γ is solved by | k TT | 2 γ = a 2 -O ( 1 τ 2 ) almost everywhere on Σ p as τ → -∞ .”

The true reduced dynamics occur on the quotient space M_-1/D0.

“Therefore, the true reduced dynamics occur on the quotient space M -1 / D 0 .”

The limit of the convergent subsequence is characterized by kTT ≡ 0 since it realizes the infimum of the Dirichlet energy.

“The limit of the convergent subsequence is characterized by k TT ≡ 0 since it realizes the infimum of the Dirichlet energy.”

The corresponding Einstein solution curve leaves every compact set in Teichmüller space.

“An immediate interpretation of such limiting behavior would be that the corresponding Einstein solution curve leaves every compact set in the Teichmüller space (configuration space).”

Any sequence on the solution ray cannot leave compact sets of Teichmüller space in the forward time direction (τ -> 0) due to monotonicity and the properness of the Dirichlet energy.

“Therefore, any sequence on the solution ray can not leave compact sets of the Teichm ¨ ller space in the forward time direction (i.e., as τ → 0 ) due to 3.56 and 3.57 or more explicitly pre-image of the compact set [ -4 πχ, E 0 ] is a compact subset of the Teichmüller space due to the aforementioned 'properness' property of the Dirichlet energy E [ id ; γ, ρ ] .”

Every non-trivial solution curve leaves the Teichmüller space at the limit of the big bang.

“Therefore, every non-trivial solution curve leaves the Teichmüller space at the limit of the big bang.”

The Dirichlet energy of the identity harmonic map is a proper function on Teichmüller space.

“Note that the conformal and diffeomorphism invariance of E [ id ; g, ρ ] allow it to be a function on the Teichmüller space of Σ p and in particular a proper function (i.e., the inverse images of the compact sets are compact)[19, 35].”

• cites [19] Chruściel, 2004, The constraint equations sourceNot checked — no full text of the source
• cites [35] 1995, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Gravity sourceNot checked — no full text of the source

M_-1/D0 is precisely the Teichmüller space of Σp.

“Now, M -1 /D 0 is precisely the Teichmüller space of Σ p and following [19], the transverse-traceless tensor k TT models the tangent space at γ .”

• cites [19] Chruściel, 2004, The constraint equations sourceNot checked — no full text of the source

Even though Σp in question does not have punctures, it forms δ-thin regions in the limit τ -> -∞ as a consequence of the blow-up of the Dirichlet energy.

“Now, even though Σ p in question does not have punctures, it forms δ -thin regions in the limit of τ → -∞ . This is precisely the consequence of the blow-up of the Dirichlet energy.”

The configuration space of the gauge-fixed dynamics of 2+1 gravity on spacetimes of topological type Σ_p × R is the Teichmüller space TΣ_p (≈ R^{6p-6}) of Σ_p.

“The configuration space of the gauge fixed dynamics is identified with the Teichmüller space ( T Σ p ≈ R 6 p -6 ) of Σ p .”

• Also stated in: § 1. Introduction · p. 1 · plain

  “Moncrief [1] reduced the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmüller space”

• cites [1] 1986, Reduction of Einstein’s equations for vacuum space-times with spacelike U(1) isometry groups, Ann. Phys. sourceNot checked — no full text of the source

Every non-trivial solution curve attaches to the Thurston boundary of Teichmüller space, i.e. the space of projective measured laminations/foliations (PML/PMF), at the big-bang limit.

“and attaches to the Thurston boundary of the Teichmüller space, that is, the space of projective measured laminations or foliations ( PML , PMF )”

• Also stated in: § Abstract · p. 1 · boosted

  “we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations ( PML PMF ), the Thurston boundary of the Teichmüller space.”

Since kTT ≡ 0 implies convergence to a point lying in the interior of TΣp, the left inequality in (4.7) is strict by the blow-up of Dirichlet energy in theorem 1.1.

“Since, k TT ≡ 0 implies convergence to a point lying in the interior of T Σ p , the left inequality in (4.7) is strict (by the blow-up of Dirichlet energy in theorem 1.1).”

The space of big-bang singularities is a subset of the Thurston boundary of Teichmüller space.

“Therefore the space of big bang singularities is a subset of the Thurston boundary of the Teichmüller space.”

If Teichmüller space is parameterized in the space of projective currents by the lengths of 9g-9 nontrivial elements of π1(Σp), then the properness of the Dirichlet energy yields geodesics with large hyperbolic length.

“If we parameterize the Teichmüller space in the space of projective currents by the lengths of 9 g -9 nontrivial elements of π 1 (Σ p ) ( 9 g -9 theorem), then the properness of the Dirichlet energy yields geodesics with large hyperbolic length.”

The transverse-traceless tensor kTT models the tangent space at γ.

“Now, M -1 /D 0 is precisely the Teichmüller space of Σ p and following [19], the transverse-traceless tensor k TT models the tangent space at γ .”

• cites [19] Chruściel, 2004, The constraint equations sourceNot checked — no full text of the source

Proposition 4.3 [41]: For a hyperbolic Riemann surface with finite area, for any holomorphic quadratic differential φ, the following are equivalent: ||φ||L1 ≤ C, ||φ||L∞ ≤ C, and at each puncture φ has at worst a simple pole.

“Proposition 4.3. Let (Σ p , γ ) be a hyperbolic Riemann surface with finite area. Then for any holomorphic quadratic differential ϕ of Σ p the following are equivalent”

• cites [41] 2012Not checked yet

  Not checked in the free run.

The Einstein flow drives the solution curve so that the measured foliation behaves in this way at the big-bang limit.

“In a sense, the Einstein flow drives the solution curve in such a way that the measured foliation behaves in this way at the limit of the big-bang singularity.”

The infimum of |kTT|^2γ may be zero because the holomorphic quadratic differential φ has a finite number of zeros.

“Notice that the infimum of | k TT | 2 γ may be zero since the holomorphic quadratic differential ϕ := ( k TT 11 -ik TT 12 ) dz 2 has finite number of zeros.”

The limit of π∘l(γj) lies on the Thurston boundary.

“Therefore the limit of π ◦ l ( γ j ) lies on the Thurston boundary by lemma 6.1. This establishes theorem 3.”

From the Deligne-Mumford compactness theorem, punctured surfaces are achieved as a limit of a Riemann surface degenerating via collapsing non-trivial simply closed geodesics.

“Now consider the case when Σ p completely degenerates and forms punctures (From the Deligne-Mumford compactness theorem, punctured surfaces are achieved as a limit of a Riemann surface degenerating via collapsing non-trivial simply closed geodesics).”

• cites Deligne-Mumford compactness theoremNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Lemma Appendix A.3: The map L is injective.

“Lemma Appendix A.3.”

• Also stated in: § 6. Proof of theorem 3 · p. 33 · plain

  “is injective.”

Analyzing the asymptotic behavior of the Monge-Ampere equation is in principle the same as analyzing the Gauss map equation while satisfying Einstein's equations through the associated Hamilton-Jacobi equation.

“Therefore analyzing the asymptotic behavior of the Monge-Ampere equation is in principle the same as analyzing the Gauss map equation while satisfying Einstein's equations through the associated Hamilton-Jacobi equation.”

• AssumptionRests on an assumption: The Monge-Ampere equation has a unique solution.

At the big bang, the conformal geometry degenerates via pinching and wringing of (Σp, γ).

“At the big bang, the conformal geometry degenerates via pinching and wringing of (Σ p , γ ) .”

The space of big-bang singularity is characterized by a subset of the Thurston boundary of the Teichmüller space.

“Now we prove the final theorem that essentially indicates that the space of big-bang singularity is characterized by a subset of the Thurston boundary of the Teichmüller space.”

Each solution curve starts at τ → 0 and approaches the Thurston boundary of TΣp in PCurr (or equivalently PΩ) in the big-bang limit.

“Each solution curve starts at τ → 0 and approaches the Thurston boundary of T Σ p in P Curr (or equivalently P Ω ) in the limit of big-bang.”

If each member of a sequence on the solution curve satisfies ||φτj||L1 ≤ C with the limit satisfying lim ||φτj||L1 ≤ C for a uniform C, then from Propositions 4.1-4.3 the L∞ norm of the limit is also bounded.

“If each member of the sequence satisfies || ϕ τ j || L 1 (Σ p ) ≤ C with the limit satisfying lim j →∞ || ϕ τ j || L 1 (Σ p ) ≤ C (for a uniform C ; this is precisely what we have proved in lemma (4.1), then from the propositions (4.1-4.3) we conclude that the L ∞ norm of the limit is also bounded”

Letting any one of the Fenchel-Nielsen lengths l_i tend to infinity implies approaching the boundary ∂TΣ.

“Letting any one of the l i tend to infinity i.e., lim n →∞ l n i = ∞ implies approaching the boundary ∂ T Σ .”

Through the unique solution of the Monge-Ampere equation, the Gauss map equation defines a ray structure of the Einstein equations.

“Through the unique solution of the Monge-Ampere equation, the Gauss map equation defines a ray structure of the Einstein equations.”

• AssumptionRests on an assumption: The Monge-Ampere equation has a unique solution.

PMF and TΣp are disjoint in the space of projective currents PΩ.

“Recall that PMF and T Σ p are disjoint in the space of projective currents i.e., P Ω (see Appendix A.3).”

• cites Appendix A.3Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The image of Teichmüller space under L and PML are disjoint.

“Another important observation is to note that the image of the Teichmüller space under the map ˆ L i.e., ˆ L ( T Σ p ) and PML are disjoint.”

The constant C is independent of the homotopy class of loops at the big-bang limit τ → -∞.

“In fact, we need to show that C does not depend on the homotopy class of loops at the limit τ →-∞ . This follows since we will show that | k TT | 2 γ behaves as a constant modulo factor involving inverse power of the mean extrinsic curvature τ as τ approaches -∞ (i.e., big bang).”

The flow of the vector field -∂/∂θ_i preserves the length l_i of the geodesic about which Σp is twisted.

“Therefore, flow of the vector field -∂ ∂θ i preserves the length l i of the geodesic about which Σ p is twisted.”

• cites [43] 1995Not checked yet

  Not checked in the free run.

• cites [44] 2010Not checked yet

  Not checked in the free run.