FORMATION OF TRAPPED SURFACES II. 1. Introduction
|
|
- Edgar Caldwell
- 6 years ago
- Views:
Transcription
1 FORMATION OF TRAPPED SURFACES II SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI 1. Introduction. Geometry of a null hypersurface As in [?] we consider a region D = D(u, u ) of a vacuum spacetime (M, g) spanned by a double null foliation generated by the optical functions (u, u) increasing towards the future, 0 u u and 0 u u. We denote by H u the outgoing null hypersurfaces generated by the level surfaces of u and by H u the incoming null hypersurfaces generated level hypersurfaces of u. We write S u,u = H u H u and denote by H (u 1,u u ), and H (u 1,u ) u the regions of these null hypersurfaces defined by u 1 u u and respectively u 1 u u. Let L = g αβ α u β, L = g αβ α u β, L be the geodesic vectorfields associated to the two foliations and define, g(l, L) := Ω = g αβ α u β u (1) Observe that the flat value 1 of Ω is 1. As well known, our space-time slab D(u, u ) is completely determined (for small values of u, u ) by data along the null, characteristic, hypersurfaces H 0, H 0 corresponding to u = 0, respectively u = 0. Following [?] we assume that our data is trivial along H 0, i.e. assume that H 0 extends for u < 0 and the spacetime (M, g) is Minkowskian for u < 0 and all values of u 0. Moreover we can construct our double null foliation such that Ω = 1 along H 0, i.e., Ω(0, u) = 1, 0 u u. () We denote by r = r(u, u) the radius of the -surfaces S = S(u, u), i.e. S(u, u) = 4πr. We denote by r 0 the value of r for S(0, 0), i.e. r 0 = r(0, 0). For simplicity we assume r 0 = 1. Throughout this paper we work with the normalized null pair (e 3, e 4 ), e 3 = ΩL, e 4 = ΩL, g(e 3, e 4 ) = Mathematics Subject Classification. 35J10. 1 Note that our normalization for Ω differ from that of [?] 1
2 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI Given a -surfaces S(u, u) and (e a ) a=1, an arbitrary frame tangent to it we define the Ricci coefficients, Γ (λ)(µ)(ν) = g(e (λ), D e(ν) e (µ) ), λ, µ, ν = 1,, 3, 4 (3) These coefficients are completely determined by the following components, χ ab = g(d a e 4, e b ), χ ab = g(d a e 3, e b ), η a = 1 g(d 3e a, e 4 ), η a = 1 g(d 4e a, e 3 ) ω = 1 4 g(d 4e 3, e 4 ), ω = 1 4 g(d 3e 4, e 3 ), (4) ζ a = 1 g(d ae 4, e 3 ) where D a = D e(a). We also introduce the null curvature components, α ab = R(e a, e 4, e b, e 4 ), α ab = R(e a, e 3, e b, e 3 ), β a = 1 R(e a, e 4, e 3, e 4 ), β a = 1 R(e a, e 3, e 3, e 4 ), (5) ρ = 1 4 R(Le 4, e 3, e 4, e 3 ), σ = 1 R(e 4, e 3, e 4, e 3 ) 4 Here R denotes the Hodge dual of R. We denote by the induced covariant derivative operator on S(u, u) and by 3, 4 the projections to S(u, u) of the covariant derivatives D 3, D 4. Observe that, ω = 1 4(log Ω), ω = 1 3(log Ω), η a = ζ a + a (log Ω), η a = ζ a + a (log Ω) (6) We recall the integral formulas for a scalar function f in D, d ( df f = du S(u,u) S(u,u) du + Ωtrχf) = d ( df f = du du + Ωtrχf) = In particular, S(u,u) S(u,u) dr du = 1 Ωtrχ, 8π S(u,u) S(u,u) S(u,u) Ω ( e 4 (f) + trχf ) Ω ( e 3 (f) + trχf ) (7) dr du = 1 Ωtrχ (8) 8π S(u,u) We also recall the following commutation formulas: We record below commutation formulae between and 4, 3 : see for example Lemma in [?]
3 TRAPPED SURFACES 3 Lemma.1. For a scalar function f: [ 4, ]f = 1 (η + η)d 4f χ f (9) [ 3, ]f = 1 (η + η)d 3f χ f, (10) For a 1-form tangent to S: [ 4, a ]U b = χ ac c U b + ac β b U c + 1 (η a + η a )D 4 U b χ ac η b U c + χ ab η U [ 3, a ]U b = χ ac c U b + ac β b U c + 1 (η a + η a )D 3 U b χ ac η b U c + χ ab η U In particular, [ 4, div ]U = 1 trχ div U ˆχ U β U + 1 (η + η) 4U η ˆχ U [ 3, div ]U = 1 trχ div U ˆχ U + β U + 1 (η + η) 3U η ˆχ U.. Christodoulou s heuristic argument. We recall here the assumptions needed in Christodoulou s heuristic argument for the formation of a trapped surface as described in [?]. As mentioned above we assume that our data is trivial along H 0, i.e. assume that H 0 extends for u < 0 and the spacetime (M, g) is Minkowskian for u < 0 and all values of u 0. We introduce a small parameter δ > 0 and restrict the values of u to 0 u δ, i.e. u = δ. Main Assumptions. We assume that throughout D = D(u, u ) we have the following estimates: MA1. For small δ, Ω is comparable with its standard value in flat space, i.e. MA. The Ricci coefficients χ, ω, η, χ, ω verify Ω = 1 + O(δ 1/ ). MA3. Also for some c > 0, ˆχ, ω = O(δ 1/ ), trχ, η = O(1), ˆχ, trχ + r, ω = O(δ1/ ). η = O(δ 1/+c ).
4 4 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI Note that in view of (8) we also have, dr du = 1 + O(rδ1/ ), dr du = O(r) (11) Thus, for δ sufficiently small, we infer that r is decreasing along the incoming null hypersurfaces and remains bounded, 0 r r =, in D. Christodoulou s argument for the formation of trapped surfaces in [?] rests on the equations, 4 trχ + 1 (trχ) = ˆχ 1 (trχ) ωtrχ 3 ˆχ + 1 trχˆχ = η + ω ˆχ 1 trχˆχ + η η In view of our Ricci coefficients assumptions we can rewrite, Multiplying the second equation by ˆχ, 4 trχ = ˆχ + O(δ 1/ ) 3 ˆχ + 1 trχˆχ = O(δ 1/+c ) Using also our assumptions for u, u, Ω we deduce, Integrating (1) we obtain, 4 ˆχ + trχ ˆχ = O(δ 1+c ) d du trχ = ˆχ + O(δ 1/ ) (1) d du ˆχ + trχ ˆχ = O(δ 1+c ) (13) trχ(u, u) = In view of our assumptions for trχ and dr du Therefore, r(u, 0) u 0 ˆχ (u, u ) du + O(δ 1/ ) (14) d du (r ˆχ ) = r d du ˆχ + r dr du ˆχ = r [ trχ ˆχ + O(δ 1+c ) ] + r [ 1 + O(rδ c ) ] ˆχ = r O(δ 1+c ). r ˆχ (u, u) = r (0, u) ˆχ (0, u) + r O(δ 1+c ) As in [] we freely prescribe ˆχ along the initial hypersurface H (0,δ) 0, i.e. ˆχ(0, u) = ˆχ 0 (u) = O(δ 1/ ) (15)
5 for some traceless tensor ˆχ 0. We deduce, (need 0 < c 1 ), or, since u δ and r(u, u) = r 0 + u u + O(δ c ), Thus, returning to (14), TRAPPED SURFACES 5 ˆχ (u, u) = r (0, u) r (u, u) ˆχ 0 (u) + O(δ 1+c ) ˆχ (u, u) = r 0 r (u, 0) ˆχ 0 (u) + O(δ 1+c ) trχ(u, δ) = = We have thus proved the following. r(u, 0) r 0 r (u, 0) r(u, 0) r 0 r (u, 0) u 0 u 0 ˆχ 0 (u )du + O(δ c ) ˆχ 0 (u )du + O(δ c ) Proposition.3. Under the assumptions MA1- MA3 we have, for sufficiently small δ > 0 and fixed c > 0, trχ(u, δ) = δ r(u, 0) r 0 ˆχ r 0 (u )du + O(δ c ) (16) (u, 0) 0 Since r(u, u) = r 0 u + u + O(δ c ) formula (16) can also be written in the form, trχ(u, δ) = r(u, δ) r 0 r (u, δ) Corollary.4. The necessary condition to have trχ(u, u = δ) < 0 δ δ 0 ˆχ 0 (u )du + O(δ c ) (17) r(u, 0) < ˆχ r0 0 + O(δ c ) (18) 0 for sufficiently small δ > 0. Since r(u, 0) = r 0 u + O(δ c ), condition (18) can also be written in the form, (r 0 u) r 0 < δ 0 ˆχ 0 + O(δ c ) (19) 3. Change of foliation 3.1. Main transformation formula. To improve on (18) we plan to change the u foliation along u = δ and compute the corresponding incoming expansion trχ. More precisely, given the foliation induced by u, we look for a new foliation v = v(u, ω) defined by the equations u v = e f, v S0 = u S0 = 0 u f = 0, f S = f 0 (0)
6 6 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI with f 0 a function on S 0 = S(0, δ) to be carefully chosen later. NOTE CHANGE: BEFORE WE HAD 3 v = e f WHICH LEADS TO THE UNDESIRED TERM e f log Ω IN THE EQUATION FOR G. We introduce the new null frame adapted to the v-foliation, e 3 = e 3, e a = e a e f Ωe a (v)e 3, e 4 = e 4 e f Ωe a (v)e a + e f Ω v e 3 (1) Indeed since u = Ω 3 we have, e a(v) = e a (v) e f Ωe a (v)e 3 (v) = e a (v) e f e a (v) u (v) = 0. Also, ince e 3 is orthogonal to any vector tangent to H we easily check that We prove the following. g(e a, e b) = g(e a, e b ) = δ ab, g(e 4, e a) = g(e 4, e 4) = 0, g(e 3, e 4) =. Lemma 3.. The new incoming expansion trχ verifies the transformation formula, trχ = trχ e f div (e f F ) trχ F 4ˆχ bc F b F c (η + ζ) F () where F a = e f Ω a v and trχ, ζ, trχ, ˆχ, ω are connection coefficients for the given double null foliation (u, u). Proof. We have, χ (e a, e b) := g(d ae 4, e b) = g(d a e 4, e b) e f Ω 1 e a (v)g(d 3 e 4, e b) Now, writing e 4 = e 4 F + F e 3 with F = F c e c and e b = e b F b e 3, g(d a e 4, e b) = g ( D a (e 4 F + F e 3 ), e b F b e 3 ) = χ(e a, e b ) F b ζ a a F b + F b g(d a F, e 3 ) + F g(d a e 3, e b F b e 3 ) = χ ab ζ a F b a F b F b χ(f, e a ) + F χ ab Also, = χ ab ζ a F b a F b F b F c χ ac + F χ ab g(d 3 e 4, e b) = g ( D 3 (e 4 F + F e 3 ), e b F b e 3 ) = g(d 3 e 4, e b ) F b g(d 3 e 4, e 3 ) 3 F b Hence, = η b + 4F b ω 3 F b χ ab = χ ab ζ b F a a F b F b χ(f, e b ) + F χ ab F a ( ηb + 4F b ω 3 F b ) = χ ab a F b + F a 3 F b ζ b F a F a η b + ( ) F χ ab F b F c χ ac 4ωFa F b By symmetry in a, b we deduce the formula, χ ab = χ ab ( a F b + b F a ) + 3 (F a F b ) (ζ b + η b )F a + (ζ a + η a )F b (3) + ( ) F χ ab F b F c χ ac F a F c χ bc 4ωFa F b
7 TRAPPED SURFACES 7 and, taking the trace, trχ = trχ div F + 3 F (η + ζ) F + ( F trχ χ bc F b F c ) 4ω F = trχ div F + 3 F (η + ζ) F ˆχ bc F b F c 4ω F We next calculate 3 F using (0) and the commutation formula [ 3, ]h = ( log Ω) 3 h χ h or, [ u, ]h = Ωχ h Since u f = 0 and F = Ω 1 e f v we deduce, or, i.e., from which, Therefore, as desired. u F a = u (Ωe f v) = Ωe f u v + u Ωe f v = Ωe f u v Ωe f Ωχ v + u Ωe f v = Ω f Ω e f χ v + u Ωe f v = Ω f Ωχ F Ω 1 u ΩF 3 F = F χ F Ω 1 3 ΩF = F χ F + ωf 3 F + 1 trχf = f ˆχ F + ωf (4) 3 F = trχ F + F f ˆχ bc F b F c + 4ω F trχ = trχ div F (η + ζ) F ˆχ bc F b F c 4ω F trχ F + F f ˆχ bc F b F c + 4ω F = trχ div F + F f trχ F 4ˆχ bc F b F c (η + ζ) F = trχ e f div (e f F ) + F f trχ F 4ˆχ bc F b F c (η + ζ) F Remark. Note that we can eliminate ζ from the formula () by writing the term (η + ζ) F = 4η F Ω 1 Ω F Thus, trχ = trχ e f Ωdiv (Ω 1 e f F ) trχ F 4ˆχ bc F b F c 4η F (5)
8 8 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI To understand how trχ differs from trχ it only remains to derive a transport equation for div G with G = e f F Transport equation for div G. In view of (4) and e 3 (f) = 0 we have for G := e f F. 3 G + 1 trχg = e f f ˆχ G + ωg (6) To derive a transport equation for div G we make us of the following Lemma 3.4. Assume that the S-tangent vectorfield V verifies an equation of the form, Then, 3 V + 1 trχv = ˆχ V + W 3 (div V ) + 1 trχdiv V = div W + W (log Ω) ˆχ V trχ V + ( trχζ ˆχ ζ ˆχ (log Ω) ) V Proof. 3 (div V ) + 1 trχdiv V = div ( ˆχ V + W ) 1 trχ V + [ 3, div ]V We make use of the commutation formula, see lemma.1, Therefore, [ 3, div ]V = 1 trχdiv V ˆχ V + ( β η ˆχ ) V + (log Ω) 3 V 3 (div V ) + trχdiv V = div ( ˆχ V + W ) ˆχ V + ( β 1 trχ η ˆχ ) V + (log Ω) 3 V = div W ˆχ V + ( div ˆχ + β 1 trχ η ˆχ ) V + (log Ω) ( 1 ) trχv ˆχ V + W = div W + W (log Ω)) ˆχ V + ( div ˆχ 1 trχ + β η ˆχ 1 trχ (log Ω) ˆχ (log Ω)) V
9 TRAPPED SURFACES 9 Using the Codazzi equation, div ˆχ = 1 trχ + β + ζ (ˆχ 1 trχ) as well as η = ζ + (log Ω) we derive, div ˆχ 1 trχ + β η ˆχ 1 trχ (log Ω) ˆχ (log Ω)) = trχ ζ (ˆχ 1 trχ η ˆχ 1 trχ (log Ω) ˆχ (log Ω) Hence, as desired. = trχ ˆχ (ζ + η + (log Ω)) + 1 trχ(ζ + (log Ω)) = trχ ˆχ (ζ + (log Ω)) + trχη 3 (div V ) + trχdiv V = div W + W (log Ω) ˆχ V trχ V + ( trχη ˆχ ζ ˆχ (log Ω) ) V Applying the lemma to equation (6) we derive, 3 (div G) + trχdiv G = div W + W (log Ω) ˆχ G trχ G with W = e f f + ωg. Thus, + ( trχη ˆχ ζ ˆχ (log Ω) ) G div W + W (log Ω) = div (e f f) + e f (log Ω) f + div (ωg) + (log Ω)ωG We deduce the following transport equation for div G, with error term, 3 (div G) + trχdiv G = div (e f f) + ωdiv G + Err 1 (7) Err 1 = e f (log Ω) f ˆχ G trχ G + ( trχη ˆχ ζ ˆχ (log Ω) + ω + ω log Ω ) G In the same manner we deduce a transport equation for the principal term div (e f f) on the right hand side of (7). Indeed, since 3 f = 0 we derive, Therefore, using lemma 3.4, 3 ( f) + 1 trχ f = ˆχ f 3 div (e f f) + trχdiv (e f f) = ˆχ (e f f) trχ e f f + ( trχη ˆχ ζ ˆχ (log Ω) ) e f f.
10 10 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI We summarize the results of this subsection in the following proposition. We summarize the results of this subsection in the following proposition. Proposition 3.5. Let v, f be defined according to (0), F = Ω 1 e f v and G = e f F. The trace of the null second fundamental form χ, relative to the new frame (1), is given by the formula (), i.e., F verifies the transport equation and div G verifies, where, Also, with error term, trχ = trχ e f div G trχ F 4ˆχ bc F b F c (η + ζ) F (8) 3 F + 1 trχf = f ˆχ F + ωf (9) 3 (div G) + trχdiv G = div (e f f) + Err 1 (30) Err 1 = e f (log Ω) f ˆχ G trχ G + ( trχζ ˆχ ζ ˆχ (log Ω) + ω + ω log Ω ) G 3 f = 0 (31) 3 ( f) + 1 trχ f = ˆχ f (3) 3 [e f div (e f f)] + trχ[e f div (e f f)] = Err (33) Err = ˆχ ( f f f ) trχ f + ( trχη ˆχ ζ ˆχ (log Ω) ) f 3.6. Additional assumptions. To proceed we need to make stronger assumptions than those of section.. More precisely, we need, in addition MA1 -MA3 the following, MA-S. The Ricci coefficients η, η, log Ω verify the stronger assumptions, η, η = O(δ c ) MA3-S For a fixed c > 0, η, η = O(δ 1/+c ), χ, β = O(δ c )
11 TRAPPED SURFACES 11 As a corollary of proposition 3.5 and these assumptions we deduce first, trχ = trχ e f div (G) + r F + F O(δ c ) (34) From the equation (3) 3 ( f) + 1 trχ f = ˆχ f we deduce, Therefore, u (r f ) = O(δ 1/ ) r f r f = r 0 f 0 (1 + O(δ 1/ ) ) (35) We can also deduce in the same manner an estimate for r f. Indeed, differentiating (3) and commuting with 3, according to lemma.1 we deduce, 3 ( f) + trχ f = ˆχ f + O(δ c )(1 + 1 r ) f Note that the 1 term is due to the contribution of the term trχη f which appear in the commutation r lemma. Hence, since r r 0 r we deduce using (35), and we infer that, u (r f) = O(δ 1/ )r f + O(δ c )(1 + 1 r )r f = O(δ 1/ )r f + O(rδ c )r 0 f 0 r f C ( r 0 f 0 + O(δ c )r 0 f 0 ) (36) Proceeding in the same manner with (33) we derive, for H = e f div (e f f) We deduce, Hence, or, Now, from equation (38), 3 H + trχh = O(δ 1/ ) ( f + f ) + O(δ c )(1 + 1 r ) f u (r H) = O(δ 1/ )r f + O(δ c )r f = O(δ c ) [ r 0 f 0 + r 0 f 0 ] r H = r 0H 0 + O(δ c ) [ r 0 f 0 + r 0 f 0 ] r div (e f ) f = r 0 (e f 0 ) + O(δ c ) [ r 0 f 0 + r 0 f 0 ] e f 0 (37) 3 F + 1 trχ F = f + O(δ1/ )F we deduce, u (r F ) = O(δ 1/ )r F + r f = O(δ 1/ )r F + r 0 f 0 ( 1 + O(δ c ) )
12 1 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI and therefore, since F 0 = e f 0 v 0 = 0, r F = ur 0 f 0 ( 1 + O(δ c ) ) (38) with C > 0 independent of δ or f 0. We next calculate F. Using the commutation lemma.1 we deduce, 3 F + trχ F f + O(δ 1/ ) F + O(δ c ) F Thus, according to (36) u (r F ) r f + O(δ 1/ )r F + O(δ c )r F O(δ 1/ )r F + C ( r0 f 0 + O(δ c )r 0 f 0 ) + O(δ c )rr 0 f 0 We deduce, r F C ( r0 f 0 + O(δ c )r 0 f 0 ) (39) Since G = e f F we also deduce, r G C ( r0 f 0 + O(δ c )r 0 f 0 ) e f 0 (40) Next we calculate div G from (30) which we write in the form, 3 (div G) + trχdiv G = div (e f f) + O(δ c )I 0 e f 0 I 0 := ( r0 f 0 + O(δ c )r 0 f 0 ). Hence, making use of (37) u (r div G) = r div (e f f) + O(δ c )r I 0 e f 0 = r0 (e f 0 ) + O(δ c )I 0 + O(δ c )r I 0 e f 0 Therefore, r div G = ur0 (e f 0 ) + O(δ c )I 0 (41) Finally, going back to (34), and formula (38) for F, trχ = trχ e f div (G) + r F + F O(δ c ) = trχ + ur r0e f 0 (e f 0 ) + O(r δ c )I 0 + r 3 u r0 f 0 ( 1 + O(δ c ) ) = trχ + ur r0( f0 + f 0 ) + r 3 u r0 f 0 ( 1 + O(δ c ) ) + O(r δ c )I 0 ( = trχ + ur 0 f r 0 + [ 1 + u/r ( 1 + O(δ c ) ] f 0 ]) + O(r δ c )I 0 We summarize the result in the following proposition.
13 TRAPPED SURFACES 13 Proposition 3.7. Assume that MA1-MA3 and MA-S, MA3-S are verified in the space-time region D(u, δ) and f, v defined according to (0) The the expansion trχ of the v foliation verifies, for all 0 u u and 0 u δ, with I 0 = ( r0 f 0 + O(δ c )r 0 f 0 ) verifies, ( trχ = trχ + ur 0 f r 0 + [ 1 + u/r ( 1 + O(δ c ) ] f 0 ]) + O(r δ c )I 0 (4) In particular, if δ > 0 is sufficiently small, trχ trχ + ur [ 0 f0 + ( 1 + u )] f0 + O(r δ c )I r 0 (43) r 3.8. Main equation. We now combine the results of propositions.3 and 3.7. For simplicity we shall also assume that r 0 = 1. According to proposition.3 we have, Thus, inserting in (4) trχ (u, δ) r + u r where r = r(u, δ) and trχ(u, δ) = r(u, δ) 1 r (u, δ) δ 0 ˆχ 0 (u )du + O(δ c ) ( f 0 + [ 1 + u/r ( 1 + O(δ c ) ] f 0 ]) 1 r M 0 + O(r δ c )I 0 M 0 = δ 0 ˆχ 0 (u )du. Now, along a level surface 3 S 1 := {v = 1} H u we can express both u and r as functions along S 0 which we denote by U = U(f 0 ) and R = R(f 0 ). In fact, since v = ue f 0 we deduce U = e f 0. Moreover since according to (11) dr = 1 + du O(δ1/ r) = 1 + O(δ 1/ ) we can write, R = 1 U + O(δ 1/ ) U To have trχ non-positive along S v0 we need, R + U ( f R 0 + [ 1 + U/R ( 1 + O(δ c ) ] f 0 ]) 1 ( ) M0 O(δ c )I r 0 We deduce the following. Corollary 3.9. A necessary condition for S 1 to be a trapped surface, is that, f 0 + [ 1 + U/R ( 1 + O(δ c ) ] f 0 ] + R U 1 ( ) M0 O(δ c )I 0 U where, 3 with v the deformation function defined by (0) U = e f 0, R = 1 e f 0 + O(δ 1/ ) e f 0 (44)
14 14 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI Note that the inequality is only meaningful in the domain D, i.e. for U = e f 0 u, or for δ sufficiently small, R > 1 u (45) We now re-express (44) with respect to R = R(f 0 ). We have, On the other hand, In view of formula (7) and the equation, we deduce, Hence, from which we deduce, u (r u r) = 1 8π u = 1 16π R = R df (f 0) f 0 R = R df (f 0) f 0 + d R d f (f 0) f 0 dr df (f 0) = dr du du df = ur e f 0 d R d f (f 0) = ur e f 0 + u re f 0 3 trχ + 1 (trχ) = ωtrχ ˆχ S(u,u) S(u,u) Ωtrχ = 1 8π Ω trχ 1 8π S(u,u) S(u,u) r ur + ( u r) = Ω + ro(δ 1/ ) = 1 + O(δ 1/ ) Ω ( e 3 (Ωtrχ) + Ωtrχtrχ ) Ω ˆχ = 1 + ro(δ 1/ ) r ur = O(δ 1/ ). (46) Hence, R = f 0 e f 0 u r = f 0 e ( f O(δ 1/ ) ) R = f 0 e f 0 u r + ( ur e f 0 + u re ) f 0 f 0 = e f 0 ( f ) O(δ 1/ + ( 1 + O(δ 1/ )e f 0 f 0 + R 1 e f 0 f 0 O(δ 1/ ) = e ( f 0 f 0 f 0 ) + O(δ 1/ ) ( f 0 + (1 + R 1 ) f 0 )
15 TRAPPED SURFACES 15 Thus, Note that, f 0 = e f 0 R + O(δ 1/ ) f 0 f 0 = e f 0 R + f 0 + O(δ 1/ ) ( f 0 + (1 + R 1 ) f 0 ) = e f 0 R + e f 0 R + O(δ 1/ ) ( f 0 + (1 + R 1 ) f 0 ) The left hand side of (44) becomes, L = f 0 + [ 1 + U/R ( 1 + O(δ c ) )] f 0 ] + R U Hence, L = e f 0 = e f 0 R e f 0 R + (1 + e f 0 R 1 )e f 0 R + Re f 0 + O(δ c )R 1 e f 0 R + O(δ 1/ ) ( e f 0 R + e f 0 R ) [ R + R 1 R ( 1 + O(δ c ) ) + R + O(δ 1/ ) ( ] R + e f 0 R ) The inequality (44) becomes, or, R + R 1 R ( 1 + O(δ c ) ) + R + O(δ 1/ ) ( R + e f 0 R 1 (M 0 O(δ c )I 0 ) R + R 1 R ( 1 + O(δ c ) ) + R 1 M 0 + O(δ c )J where J is a fixed smooth function depending only on R and R. We deduce the following. Proposition A necessary condition such that S 1 is a trapped surface is that, for δ > sufficiently small there exists a smooth function R on 0 verifying R > 1 u and the differential inequality, R + ( 1 + O(δ c ) ) R 1 R + R 1 M 0 + O(δ c )J (47) To proceed we need to make the assumption R δ c/. Hence, it suffices to prove the inequality: or, for δ sufficiently small, R + R 1 R + R 1 M 0 + O(δ c/ )J R + R 1 R + R < 1 M 0. (48) In order that the initial surface, corresponding to R = 1, is not trapped we need 1 M 0 < 1. We also note, by the maximum principle that, max R < 1 max M 0
16 16 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI Thus, 1 u < 1 max S 0 M 0. (49) is a necessary condition for the formation of a trapped surface. Recall that u is the maximum of u for which our assumptions are satisfied in D(u, δ). Is it sufficient? Performing the transformation R = e φ we derive, φ + 1 < 1 M 0 e φ 4. Solutions to the deformation equation on a fixed sphere (S, γ). In this section we provide examples of solutions to our main deformation equation, φ + 1 < Me φ (50) on a smooth, compact, dimensional Riemnannian manifold S, diffeomorphic to the standard sphere, with strictly positive Gaussian curvature K. We define r = r(s) such that S = 4πr and define, k m = min r K, k M = max r K S S We consider geodesic balls B(p, ɛ) = B(p, ɛ) for sufficiently small ɛ > 0. We start by proving the following lemma. Lemma 4.1. Given a ball B(p, ɛ) S, there exists a function w ɛ, smooth outside the point p, such that w ɛ + K = 4πδ p (51) where δ p is the Dirac measure at p. Moreover, if λ denotes the distance function from p, w ɛ = χ ɛ log λ + v (5) with v C 1.1 (S), v(p) = 0, smooth in S \ {p} and χ ɛ a smooth cutoff function, { χ ɛ = 1 on B(p, ɛ) χ ɛ = 0 on B(p, ɛ) Assuming the lemma true we consider the cut-off function { ϕ ɛ = 0 on B(p, ɛ/) ϕ ɛ = 1 on B ɛ (p)
17 TRAPPED SURFACES 17 and define w ɛ = ϕ ɛ w ɛ. Note that w ɛ verifies the following properties: w ɛ = 0, on B(p, ɛ/) w ɛ = log ɛ + O(1), on S \ B(p, ɛ/) w ɛ = O(ɛ log ɛ) on S \ B(p, ɛ/) w ɛ + K = 0 on S \ B(p, ɛ) Consider now the function w ɛ = Λw ɛ, for a fixed constant Λ and observe that, on S \ B(p, ɛ), we must have, w ɛ + 1 = ΛK + 1 < 0 provided that Λ > km 1. Thus, with a fixed choice of Λ > km 1 we have, w ɛ = 0, on B(p, ɛ/) w ɛ = Λ log ɛ + O(1), on S \ B(p, ɛ/) (54) w ɛ = O(ɛ log ɛ) on S \ B(p, ɛ/) w ɛ + 1 < 0 on S \ B(p, ɛ) It remains to check under what conditions for M, the function w ɛ verifies (50). Clearly, on S \B(p, ɛ), (50) is trivially verified in view of the fact that M 0. Now let M ɛ = inf B(p,ɛ) M. Thus, for some constant C, Me wɛ M ɛ e wɛ Cɛ Λ M ɛ, w ɛ + 1 = O(ɛ log ɛ) Hence, to have (50) verified in B(p, ɛ) we need, This proves the following. O(ɛ log ɛ) < M ɛ ɛ Λ Proposition 4.. Let M ɛ = min B(p,ɛ) M and let Λ > (min S K) 1. Assume that, for some universal constant C > 0, (53) M ɛ > Cɛ Λ log ɛ (55) Then, for sufficiently small ɛ > 0, there exists a function φ ɛ verifying the inequality (50) and such that min φ ɛ > log ɛ + O(1) (56) φ ɛ = O(ɛ 1 log ɛ), φ ɛ = O(ɛ log ɛ). (57) In remains to prove lemma 4.1. This is a standard argument, see for example chapter in [?], which we sketch below. Let λ be the geodesic distance function from p. In a neighborhood of p we can write, ds = dλ + a (λ, θ)dθ, a(0) = 0, da (0) = 1. dλ
18 18 SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI Let h be the geodesic curvature of the level curves of λ, i.e. denoting, e = γ( θ, θ ) 1/ θ the unit tangent vector to these curves, h verifies the second variation formula, or, h = γ( e λ, e) = a 1 λ a λ h = h K λa + Ka = 0. (58) Since a(0) = 0, da(0) = 1 we deduce from (58) that dλ λa(0) = 0. Consequently, Now, a(λ) = λ + O(λ 3 ), λ a(λ) = 1 + O(λ ), h(λ) = λ 1 + O(λ) Thus, for δ < ɛ converging to 0, Hence, passing to the limit, Note also that, S\B(p,δ) log λ = λ 1 h λ = O(1) S (χ ɛ log λ)dv γ = π + O(δ) S (χ ɛ log λ)dv γ = π for any smooth test function χ supported in B(p, ɛ). We now solve the equation, where, f is the bounded function { f = K + (χ ɛ log λ) f = 0 χ (χ ɛ log λ)dv γ = πχ(p) (59) S v = f. (60) on S \ {p} at p Note that (60) admits a C 1,1 solution in view of the fact that f L (S) and, fdv γ = Kdv γ + (χ ɛ log λ)dv γ = 4π 4π = 0. S We can also normalized v such v(p) = 0. We now define, S S w ɛ = χ ɛ log λ v
19 and note that, Moreover, in view of (59), as desired. TRAPPED SURFACES 19 w ɛ + K = 0, on S \ {p} w ɛ + K = 4πδ Remark 4.3. Note that the proof of the proposition only requires the existence of a smooth function w on S which verifies w + 1 < 0 in the complement of a closed domain D for which inf D M : M D > 0. Indeed if such a function exists we can produce a solution to our inequality simply by taking φ = log s + w for a sufficiently small constant s > 0. Indeed, with such a choice (50) is automatically satisfied in the complement of D. In D, φ = w, e φ = s 1 e w and therefore we need max D [ e w ( w + 1) ] < s 1 M D. Finally we note that such solutions can easily be constructed for balls B(p, δ) with δ < i p, the radius of injectivity of S at p. Department of Mathematics, Princeton University, Princeton NJ address: seri@@math.princeton.edu Department of Mathematics, Princeton University, Princeton NJ address: irod@@math.princeton.edu
DYNAMICAL FORMATION OF BLACK HOLES DUE TO THE CONDENSATION OF MATTER FIELD
DYNAMICAL FORMATION OF BLACK HOLES DUE TO THE CONDENSATION OF MATTER FIELD PIN YU Abstract. The purpose of the paper is to understand a mechanism of evolutionary formation of trapped surfaces when there
More informationAsymptotic Behavior of Marginally Trapped Tubes
Asymptotic Behavior of Marginally Trapped Tubes Catherine Williams January 29, 2009 Preliminaries general relativity General relativity says that spacetime is described by a Lorentzian 4-manifold (M, g)
More informationRIGIDITY OF STATIONARY BLACK HOLES WITH SMALL ANGULAR MOMENTUM ON THE HORIZON
RIGIDITY OF STATIONARY BLACK HOLES WITH SMALL ANGULAR MOMENTUM ON THE HORIZON S. ALEXAKIS, A. D. IONESCU, AND S. KLAINERMAN Abstract. We prove a black hole rigidity result for slowly rotating stationary
More informationNull Cones to Infinity, Curvature Flux, and Bondi Mass
Null Cones to Infinity, Curvature Flux, and Bondi Mass Arick Shao (joint work with Spyros Alexakis) University of Toronto May 22, 2013 Arick Shao (University of Toronto) Null Cones to Infinity May 22,
More informationarxiv: v2 [gr-qc] 24 Apr 2017
TRAPPED SURFACES IN VACUUM ARISING DYNAMICALLY FROM MILD INCOMING RADIATION XINLIANG AN AND JONATHAN LUK arxiv:409.670v [gr-qc] 4 Apr 07 Abstract. In this paper, we study the minimal requirement on the
More informationUmbilic cylinders in General Relativity or the very weird path of trapped photons
Umbilic cylinders in General Relativity or the very weird path of trapped photons Carla Cederbaum Universität Tübingen European Women in Mathematics @ Schloss Rauischholzhausen 2015 Carla Cederbaum (Tübingen)
More informationThe Bounded L 2 curvature conjecture in general relativity
The Bounded L 2 curvature conjecture in general relativity Jérémie Szeftel Département de Mathématiques et Applications, Ecole Normale Supérieure (Joint work with Sergiu Klainerman and Igor Rodnianski)
More informationStationarity of non-radiating spacetimes
University of Warwick April 4th, 2016 Motivation Theorem Motivation Newtonian gravity: Periodic solutions for two-body system. Einstein gravity: Periodic solutions? At first Post-Newtonian order, Yes!
More informationThe causal structure of microlocalized rough Einstein metrics
Annals of Mathematics, 6 005, 95 43 The causal structure of microlocalized rough Einstein metrics By Sergiu Klainerman and Igor Rodnianski Abstract This is the second in a series of three papers in which
More informationOverview of the proof of the Bounded L 2 Curvature Conjecture. Sergiu Klainerman Igor Rodnianski Jeremie Szeftel
Overview of the proof of the Bounded L 2 Curvature Conjecture Sergiu Klainerman Igor Rodnianski Jeremie Szeftel Department of Mathematics, Princeton University, Princeton NJ 8544 E-mail address: seri@math.princeton.edu
More informationarxiv: v3 [math.ap] 10 Oct 2014
THE BOUNDED L 2 CURVATURE CONJECTURE arxiv:1204.1767v3 [math.ap] 10 Oct 2014 SERGIU KLAINERMAN, IGOR RODNIANSKI, AND JEREMIE SZEFTEL Abstract. This is the main paper in a sequence in which we give a complete
More information1 First and second variational formulas for area
1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on
More informationNon-existence of time-periodic vacuum spacetimes
Non-existence of time-periodic vacuum spacetimes Volker Schlue (joint work with Spyros Alexakis and Arick Shao) Université Pierre et Marie Curie (Paris 6) Dynamics of self-gravitating matter workshop,
More informationQuasi-local Mass and Momentum in General Relativity
Quasi-local Mass and Momentum in General Relativity Shing-Tung Yau Harvard University Stephen Hawking s 70th Birthday University of Cambridge, Jan. 7, 2012 I met Stephen Hawking first time in 1978 when
More informationRigidity of Black Holes
Rigidity of Black Holes Sergiu Klainerman Princeton University February 24, 2011 Rigidity of Black Holes PREAMBLES I, II PREAMBLE I General setting Assume S B two different connected, open, domains and
More informationMyths, Facts and Dreams in General Relativity
Princeton university November, 2010 MYTHS (Common Misconceptions) MYTHS (Common Misconceptions) 1 Analysts prove superfluous existence results. MYTHS (Common Misconceptions) 1 Analysts prove superfluous
More informationHAWKING S LOCAL RIGIDITY THEOREM WITHOUT ANALYTICITY
HAWKING S LOCAL RIGIDITY THEOREM WITHOUT ANALYTICITY S. ALEXAKIS, A. D. IONESCU, AND S. KLAINERMAN Abstract. We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local,
More informationRiemannian Curvature Functionals: Lecture I
Riemannian Curvature Functionals: Lecture I Jeff Viaclovsky Park City athematics Institute July 16, 2013 Overview of lectures The goal of these lectures is to gain an understanding of critical points of
More informationRigidity of outermost MOTS: the initial data version
Gen Relativ Gravit (2018) 50:32 https://doi.org/10.1007/s10714-018-2353-9 RESEARCH ARTICLE Rigidity of outermost MOTS: the initial data version Gregory J. Galloway 1 Received: 9 December 2017 / Accepted:
More informationON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES
ON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES CLAUS GERHARDT Abstract. We prove that the mean curvature τ of the slices given by a constant mean curvature foliation can be used
More informationFoliations of hyperbolic space by constant mean curvature surfaces sharing ideal boundary
Foliations of hyperbolic space by constant mean curvature surfaces sharing ideal boundary David Chopp and John A. Velling December 1, 2003 Abstract Let γ be a Jordan curve in S 2, considered as the ideal
More informationarxiv: v1 [math.ap] 8 Apr 2012
Parametrix for wave equations on a rough background I: regularity of the phase at initial time Jérémie Szeftel DMA, Ecole Normale Supérieure, 45 rue d Ulm, 755 Paris, jeremie.szeftel@ens.fr arxiv:124.1768v1
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationSingularity formation in black hole interiors
Singularity formation in black hole interiors Grigorios Fournodavlos DPMMS, University of Cambridge Heraklion, Crete, 16 May 2018 Outline The Einstein equations Examples Initial value problem Large time
More informationTHE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS
THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic
More informationÜbungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.
Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση
More informationAnalysis in weighted spaces : preliminary version
Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.
More informationStable shock formation near plane-symmetry for nonlinear wave equations
Stable shock formation near plane-symmetry for nonlinear wave equations Willie WY Wong wongwwy@member.ams.org Michigan State University University of Minnesota 9 March 2016 1 Based on: arxiv:1601.01303,
More informationThe Penrose inequality on perturbations of the Schwarzschild exterior
The Penrose inequality on perturbations of the Schwarzschild exterior Spyros Alexakis Abstract We prove a version the Penrose inequality for black hole space-times which are perturbations of the Schwarzschild
More informationEnergy method for wave equations
Energy method for wave equations Willie Wong Based on commit 5dfb7e5 of 2017-11-06 13:29 Abstract We give an elementary discussion of the energy method (and particularly the vector field method) in the
More informationThe Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant
The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant Jared Speck & Igor Rodnianski jspeck@math.princeton.edu University of Cambridge & Princeton University October
More informationClassification theorem for the static and asymptotically flat Einstein-Maxwell-dilaton spacetimes possessing a photon sphere
Classification theorem for the static and asymptotically flat Einstein-Maxwell-dilaton spacetimes possessing a photon sphere Boian Lazov and Stoytcho Yazadjiev Varna, 2017 Outline 1 Motivation 2 Preliminaries
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationCURVATURE ESTIMATES FOR WEINGARTEN HYPERSURFACES IN RIEMANNIAN MANIFOLDS
CURVATURE ESTIMATES FOR WEINGARTEN HYPERSURFACES IN RIEMANNIAN MANIFOLDS CLAUS GERHARDT Abstract. We prove curvature estimates for general curvature functions. As an application we show the existence of
More informationLocalizing solutions of the Einstein equations
Localizing solutions of the Einstein equations Richard Schoen UC, Irvine and Stanford University - General Relativity: A Celebration of the 100th Anniversary, IHP - November 20, 2015 Plan of Lecture The
More informationTHE BONDI-SACHS FORMALISM JEFF WINICOUR UNIVERSITY OF PITTSBURGH. Scholarpedia 11(12):33528 (2016) with Thomas Mädler
THE BONDI-SACHS FORMALISM JEFF WINICOUR UNIVERSITY OF PITTSBURGH Scholarpedia 11(12):33528 (2016) with Thomas Mädler NULL HYPERSURFACES u = const Normal co-vector @ u is null g @ u @ u =0 Normal vector
More information7 Curvature of a connection
[under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the
More informationQuasi-local mass and isometric embedding
Quasi-local mass and isometric embedding Mu-Tao Wang, Columbia University September 23, 2015, IHP Recent Advances in Mathematical General Relativity Joint work with Po-Ning Chen and Shing-Tung Yau. The
More informationNon-existence of time-periodic dynamics in general relativity
Non-existence of time-periodic dynamics in general relativity Volker Schlue University of Toronto University of Miami, February 2, 2015 Outline 1 General relativity Newtonian mechanics Self-gravitating
More informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More informationA Boundary Value Problem for the Einstein Constraint Equations
A Boundary Value Problem for the Einstein Constraint Equations David Maxwell October 10, 2003 1. Introduction The N-body problem in general relativity concerns the dynamics of an isolated system of N black
More informationSingularities and Causal Structure in General Relativity
Singularities and Causal Structure in General Relativity Alexander Chen February 16, 2011 1 What are Singularities? Among the many profound predictions of Einstein s general relativity, the prediction
More informationDifferential Topology Final Exam With Solutions
Differential Topology Final Exam With Solutions Instructor: W. D. Gillam Date: Friday, May 20, 2016, 13:00 (1) Let X be a subset of R n, Y a subset of R m. Give the definitions of... (a) smooth function
More informationNON-EXISTENCE OF TIME-PERIODIC VACUUM SPACETIMES
NON-EXISTENCE OF TIME-PERIODIC VACUUM SPACETIMES SPYROS ALEXAKIS AND VOLKER SCHLUE Abstract. We prove that smooth asymptotically flat solutions to the Einstein vacuum equations which are assumed to be
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationNotes on Cartan s Method of Moving Frames
Math 553 σιι June 4, 996 Notes on Cartan s Method of Moving Frames Andrejs Treibergs The method of moving frames is a very efficient way to carry out computations on surfaces Chern s Notes give an elementary
More informationDifferential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18
Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results
More informationGeometric inequalities for black holes
Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 3 August, 2012 Einstein equations (vacuum) The spacetime is a four dimensional manifold M with
More informationWELL-POSEDNESS OF THE LAPLACIAN ON MANIFOLDS WITH BOUNDARY AND BOUNDED GEOMETRY
WELL-POSEDNESS OF THE LAPLACIAN ON MANIFOLDS WITH BOUNDARY AND BOUNDED GEOMETRY BERND AMMANN, NADINE GROSSE, AND VICTOR NISTOR Abstract. Let M be a Riemannian manifold with a smooth boundary. The main
More informationFrom the Brunn-Minkowski inequality to a class of Poincaré type inequalities
arxiv:math/0703584v1 [math.fa] 20 Mar 2007 From the Brunn-Minkowski inequality to a class of Poincaré type inequalities Andrea Colesanti Abstract We present an argument which leads from the Brunn-Minkowski
More informationThe volume growth of complete gradient shrinking Ricci solitons
arxiv:0904.0798v [math.dg] Apr 009 The volume growth of complete gradient shrinking Ricci solitons Ovidiu Munteanu Abstract We prove that any gradient shrinking Ricci soliton has at most Euclidean volume
More informationWilliam P. Thurston. The Geometry and Topology of Three-Manifolds
William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed
More informationChapter 7 Curved Spacetime and General Covariance
Chapter 7 Curved Spacetime and General Covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 145 146 CHAPTER 7. CURVED SPACETIME
More informationCurved spacetime and general covariance
Chapter 7 Curved spacetime and general covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 219 220 CHAPTER 7. CURVED SPACETIME
More informationarxiv: v1 [math.dg] 4 Sep 2009
SOME ESTIMATES OF WANG-YAU QUASILOCAL ENERGY arxiv:0909.0880v1 [math.dg] 4 Sep 2009 PENGZI MIAO 1, LUEN-FAI TAM 2 AND NAQING XIE 3 Abstract. Given a spacelike 2-surface in a spacetime N and a constant
More informationLocal foliations and optimal regularity of Einstein spacetimes
Local foliations and optimal regularity of Einstein spacetimes Bing-Long Chen 1 and Philippe G. LeFloch 2 October 2, 2008 Abstract We investigate the local regularity of pointed spacetimes, that is, timeoriented
More informationRecall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm
Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify
More informationRelative Isoperimetric Inequality Outside Convex Bodies
Relative Isoperimetric Inequality Outside Convex Bodies Mohammad Ghomi (www.math.gatech.edu/ ghomi) Georgia Institute of Technology Atlanta, USA May 25, 2010, Tunis From Carthage to the World Joint work
More informationTheorema Egregium, Intrinsic Curvature
Theorema Egregium, Intrinsic Curvature Cartan s second structural equation dω 12 = K θ 1 θ 2, (1) is, as we have seen, a powerful tool for computing curvature. But it also has far reaching theoretical
More informationWARPED PRODUCTS PETER PETERSEN
WARPED PRODUCTS PETER PETERSEN. Definitions We shall define as few concepts as possible. A tangent vector always has the local coordinate expansion v dx i (v) and a function the differential df f dxi We
More informationScalar curvature and the Thurston norm
Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationON FRACTAL DIMENSION OF INVARIANT SETS
ON FRACTAL DIMENSION OF INVARIANT SETS R. MIRZAIE We give an upper bound for the box dimension of an invariant set of a differentiable function f : U M. Here U is an open subset of a Riemannian manifold
More informationSpacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds
Spacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds Alfonso Romero Departamento de Geometría y Topología Universidad de Granada 18071-Granada Web: http://www.ugr.es/
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationRIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE
RIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE CURTIS PRO AND FREDERICK WILHELM Abstract. If π : M B is a Riemannian Submersion and M has positive sectional curvature, O Neill s Horizontal
More informationTransport Continuity Property
On Riemannian manifolds satisfying the Transport Continuity Property Université de Nice - Sophia Antipolis (Joint work with A. Figalli and C. Villani) I. Statement of the problem Optimal transport on Riemannian
More informationCHAPTER 3. Gauss map. In this chapter we will study the Gauss map of surfaces in R 3.
CHAPTER 3 Gauss map In this chapter we will study the Gauss map of surfaces in R 3. 3.1. Surfaces in R 3 Let S R 3 be a submanifold of dimension 2. Let {U i, ϕ i } be a DS on S. For any p U i we have a
More informationCORRIGENDUM: THE SYMPLECTIC SUM FORMULA FOR GROMOV-WITTEN INVARIANTS
CORRIGENDUM: THE SYMPLECTIC SUM FORMULA FOR GROMOV-WITTEN INVARIANTS ELENY-NICOLETA IONEL AND THOMAS H. PARKER Abstract. We correct an error and an oversight in [IP]. The sign of the curvature in (8.7)
More information4.7 The Levi-Civita connection and parallel transport
Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. April 21, 2009 138 4.7 The Levi-Civita connection and parallel transport In the earlier investigation, characterizing the shortest curves
More informationRiemannian Curvature Functionals: Lecture III
Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationStability and Instability of Extremal Black Holes
Stability and Instability of Extremal Black Holes Stefanos Aretakis Department of Pure Mathematics and Mathematical Statistics, University of Cambridge s.aretakis@dpmms.cam.ac.uk December 13, 2011 MIT
More informationTransparent connections
The abelian case A definition (M, g) is a closed Riemannian manifold, d = dim M. E M is a rank n complex vector bundle with a Hermitian metric (i.e. a U(n)-bundle). is a Hermitian (i.e. metric) connection
More informationHow curvature shapes space
How curvature shapes space Richard Schoen University of California, Irvine - Hopf Lecture, ETH, Zürich - October 30, 2017 The lecture will have three parts: Part 1: Heinz Hopf and Riemannian geometry Part
More informationStability and Instability of Black Holes
Stability and Instability of Black Holes Stefanos Aretakis September 24, 2013 General relativity is a successful theory of gravitation. Objects of study: (4-dimensional) Lorentzian manifolds (M, g) which
More informationAn introduction to General Relativity and the positive mass theorem
An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2 nd, 2007 Wen-ling Huang Department of Mathematics University of
More informationLet F be a foliation of dimension p and codimension q on a smooth manifold of dimension n.
Trends in Mathematics Information Center for Mathematical Sciences Volume 5, Number 2,December 2002, Pages 59 64 VARIATIONAL PROPERTIES OF HARMONIC RIEMANNIAN FOLIATIONS KYOUNG HEE HAN AND HOBUM KIM Abstract.
More informationarxiv:gr-qc/ v2 14 Apr 2004
TRANSITION FROM BIG CRUNCH TO BIG BANG IN BRANE COSMOLOGY arxiv:gr-qc/0404061v 14 Apr 004 CLAUS GERHARDT Abstract. We consider a brane N = I S 0, where S 0 is an n- dimensional space form, not necessarily
More informationCritical metrics on connected sums of Einstein four-manifolds
Critical metrics on connected sums of Einstein four-manifolds University of Wisconsin, Madison March 22, 2013 Kyoto Einstein manifolds Einstein-Hilbert functional in dimension 4: R(g) = V ol(g) 1/2 R g
More information1 Curvature of submanifolds of Euclidean space
Curvature of submanifolds of Euclidean space by Min Ru, University of Houston 1 Curvature of submanifolds of Euclidean space Submanifold in R N : A C k submanifold M of dimension n in R N means that for
More informationRough solutions of the Einstein-vacuum equations
Annals of Mathematics, 6 005, 43 93 Rough solutions of the Einstein-vacuum equations By Sergiu Klainerman and Igor Rodnianski To Y. Choquet-Bruhat in honour of the 50 th anniversary of her fundamental
More informationExercise 1 (Formula for connection 1-forms) Using the first structure equation, show that
1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral
More informationClassical differential geometry of two-dimensional surfaces
Classical differential geometry of two-dimensional surfaces 1 Basic definitions This section gives an overview of the basic notions of differential geometry for twodimensional surfaces. It follows mainly
More informationSymplectic critical surfaces in Kähler surfaces
Symplectic critical surfaces in Kähler surfaces Jiayu Li ( Joint work with X. Han) ICTP-UNESCO and AMSS-CAS November, 2008 Symplectic surfaces Let M be a compact Kähler surface, let ω be the Kähler form.
More informationDEVELOPMENT OF MORSE THEORY
DEVELOPMENT OF MORSE THEORY MATTHEW STEED Abstract. In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined
More information3 Parallel transport and geodesics
3 Parallel transport and geodesics 3.1 Differentiation along a curve As a prelude to parallel transport we consider another form of differentiation: differentiation along a curve. A curve is a parametrized
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More informationWarped Products. by Peter Petersen. We shall de ne as few concepts as possible. A tangent vector always has the local coordinate expansion
Warped Products by Peter Petersen De nitions We shall de ne as few concepts as possible. A tangent vector always has the local coordinate expansion a function the di erential v = dx i (v) df = f dxi We
More informationPoisson Cohomology of SU(2)-Covariant Necklace Poisson Structures on S 2
Journal of Nonlinear Mathematical Physics Volume 9, Number 3 2002, 347 356 Article Poisson Cohomology of SU2-Covariant Necklace Poisson Structures on S 2 Dmitry ROYTENBERG Department of Mathematics, Pennsylvania
More informationDeforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary
Deforming conformal metrics with negative Bakry-Émery Ricci Tensor on manifolds with boundary Weimin Sheng (Joint with Li-Xia Yuan) Zhejiang University IMS, NUS, 8-12 Dec 2014 1 / 50 Outline 1 Prescribing
More informationTakao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,...
J. Korean Math. Soc. 40 (2003), No. 4, pp. 667 680 HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES Takao Akahori Abstract. The mirror conjecture means originally the deep relation between complex
More informationLecture 5: Hodge theorem
Lecture 5: Hodge theorem Jonathan Evans 4th October 2010 Jonathan Evans () Lecture 5: Hodge theorem 4th October 2010 1 / 15 Jonathan Evans () Lecture 5: Hodge theorem 4th October 2010 2 / 15 The aim of
More informationThe oblique derivative problem for general elliptic systems in Lipschitz domains
M. MITREA The oblique derivative problem for general elliptic systems in Lipschitz domains Let M be a smooth, oriented, connected, compact, boundaryless manifold of real dimension m, and let T M and T
More informationON THE RADIUS OF INJECTIVITY OF NULL HYPERSURFACES
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 21, Number 3, July 2008, Pages 775 795 S 0894-0347(08)00592-4 Article electronically published on March 18, 2008 ON THE RADIUS OF INJECTIVITY OF NULL
More informationMATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY
MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract
More informationChapter 14. Basics of The Differential Geometry of Surfaces. Introduction. Parameterized Surfaces. The First... Home Page. Title Page.
Chapter 14 Basics of The Differential Geometry of Surfaces Page 649 of 681 14.1. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate
More information