Foliation of null cones by surfaces of constant spacetime mean curvature near MOTS

F OLIA TION OF NULL CONES BY SURF A CES OF CONST ANT SP A CETIME MEAN CUR V A TURE NEAR MOTS BEN LAMBER T AND JULIAN SCHEUER Abstract. Marginally Outer T rapp ed Surfaces (MOTS) in spacetimes are w ell-known to indicate the existence of black holes. Using flow tec hniques, we pro v e that a neigh b ourho o d of a stable MOTS in a n ull cone may b e foliated by h yp ersurfaces of constant spacetime mean curv ature. W e also pro vide methods to construct prescrib ed spacetime mean curv ature surfaces within n ull cones. 1. Introduction In a spacetime ( ¯ M n +2 , ¯ g ), marginally outer trapp ed surfaces (MOTS) are defined by the prop ert y , that one of the n ull expansions is constan tly zero, indicating that p ossibly after time rev ersion the light ra ys emanating from this surface are not visible from the outside. Under natural assumptions on the spacetime, the famous Hawking Penrose singularit y theorems state that the existence of a MOTS yields the existence of a blac k hole and hence it is of interest to detect whether a spacetime admits them. T o d [ 18 ] suggested using the mean curv ature flo w to find MOTS by flowing h yp ersurfaces in a time-symmetric spacelike ( n + 1)-slice, as the time-symmetry reduces the problem to find minimal surfaces within this slice. In the non time-symmetric case, T o d suggested the nul l me an curvatur e flow within the spacelike time-slice and this strategy was implemen ted b y Bourni–Mo ore [ 2 ], who defined a weak n ull mean curv ature flo w via a level-set approach and found w eak v ersions, so-called gener alise d MOTS . All of the discussed approac hes emplo y curv ature flows within a spacelik e slice of the spacetime, where for the null mean curv ature flo w v ersion the flo w sp eed is induced b y the co dimension 2 geometry of the flo wing surface, namely from the null expansions coming from ¯ L and L , whic h form a normalised null pair of the normal bundle with ¯ g ( L, ¯ L ) = 1. T o mak e this precise, the Gauss equation of a spacelike n -surface Σ ⊂ ¯ M n +2 is given by (1.1) ¯ D X Y = D X Y − χ ( X , Y ) ¯ L − ¯ χ ( X , Y ) L. Then the n ull mean curv ature flo w of Bourni–Mo ore is given b y ∂ t x = − (tr χ ) ν, w e ν is a normal of Σ within the spacelike ( n + 1)-slice. The first smo oth mean curv ature flo w to lo cate MOTS w as inv ented b y Ro esch and sec- ond author [ 17 ], who emplo y ed a flo w within the null h yp ersurface generated b y a spacelik e 2-surface. T o highligh t the crucial difference to the flo w b y Bourni–Mo ore, the ev olution equation is (1.2) ∂ t x = − (tr χ ) ¯ L, Date : March 25, 2026. Key wor ds and phr ases. Null geometry; Spacetime mean curv ature; Null mean curv ature flo w; F oliation; Prescrib ed curv ature. 1 2 BEN LAMBER T AND JULIAN SCHEUER i.e. the flo w mo v es within a null hypersurface. This approach seems more natural as the flo w sp eed is aligned with the flow direction and indeed, under fairly mild assumptions on the spacetime, it is prov ed in [ 17 ], that this flow is able to detect MOTS s mo othly . Subsequently , W olff has found in teresting new prop erties of the flow ( 1.2 ), for example that in the standard n ull cone of Mink o wski space, the induced metrics of the flowing surfaces mov e by Y amab e flo w [ 19 ]. The fact that ( 1.2 ) defines a flo w in null h yp ersurfaces whic h is able to detect MOTS in a spacetime, leads to the follo wing natural question, whic h will be addressed in this pap er: Under which conditions can a neigh b ourho o d of a MOTS in a null h yp ersurface be foliated b y hypersurfaces of constan t sp ac etime me an curvatur e (STCMC)? Here a λ -STCMC hypersurface satisfies |  H | 2 = 2 θ ¯ θ = λ for some constan t λ , and where θ = tr χ, ¯ θ = tr ¯ χ. F oliations b y h yp ersurfaces of constan t mean curv ature in the ends of Riemannian manifolds ha v e b een extensiv ely studied, and are fundamen tal to the famous definition of cen tre of mass in General Relativit y given by Huisk en–Y au [ 12 ], see for example [ 3 , 5 , 10 , 15 ] and references therein (w e do not attempt to include a complete bibliograph y here). In a semi-Riemannian con text, under suitable hypotheses, global foliations by CMC hypersurfaces were shown to exist by Gerhardt [ 7 , 8 , 9 , 6 ]. The search for foliations of initial data sets b y surfaces of constan t spacetime mean curv ature has received some attention in the recen t years, as suc h foliations can also con v enien tly b e used to define centres of mass for isolated systems. This w as first observ ed by Cederbaum–Sako vic h [ 4 ] who pro ved the existence and uniqueness of suc h foliations for ends of asymptotically flat initial data sets under some structural assumptions. Kr¨ onc k e–W olff [ 13 ] extended this to find λ -STCMC foliations on the ends of asymptotically Sc h warzsc hildean light cones. Inspired by this use of STCMC surfaces, Huisk en–W olff [ 11 ] defined an in v erse spacetime mean curv ature flo w and constructed weak solutions. Roughly stated, in this pap er w e answer the question ab ov e b y pro ving the existence of a foliation by STCMC h yp ersurfaces of a null cone near a MOTS, provided the MOTS is stable in the sense of a suitable stability/Jacobi op erator, see Definition 4.1 for details. In the following w e state our main results, but for b etter readabilit y o ccasionally refer to later sections for some precise definitions. Main results. W e state our main theorem. 1.1. Theorem. L et n ≥ 2 and ¯ N b e a nul l c one on a stable MOTS Σ 0 ⊂ ¯ N in a sp ac etime ( ¯ M n +2 , ¯ g ) . Define σ = sup { κ ≥ 0 : Ther e exists a strictly incr e asing smo oth foliation of a futur e r e gion of Σ 0 by stable λ -STCMC hyp ersurfac es with λ ∈ [0 , κ ] } . Then the fol lowing statements hold. (i) σ > 0 . (ii) Either the foliation le aves every c omp act subset of ¯ N , or lim sup λ → σ | A Σ λ | → ∞ , or ther e is a smo oth limit le af Σ σ which is not stable. (iii) The foliation is unique in the sense that for any λ -STCMC surfac e e Σ λ with 0 ≤ λ < σ and e Σ λ ⊂ ∪ 0 ≤ κ<σ Σ κ , ther e holds e Σ λ = Σ λ . FOLIA TION OF NULL CONES BY STCMC SURF ACES 3 R emark. Under reasonable conditions, w e are able to show that curv ature blow up as in part (ii) do esn’t o ccur. See Remark 5.4 for more details. W e briefly explain some terminology used abov e. W e call ¯ N a n ull cone on a MOTS Σ 0 , if Σ 0 is a spacelike co dimension 2 surface of ¯ M n +2 and a null basis { L, ¯ L } of the normal bundle can b e chosen, suc h that ¯ L is future-directed, L is past directed, ( 1.1 ) holds and θ := tr Σ 0 χ = 0 , ¯ θ := tr Σ 0 ¯ χ > 0 . W e giv e a detailed accoun t in section 2 . The MOTS Σ 0 is called stable , if there exists a p ositiv e function f on Σ 0 , such that L Σ 0 f := − ∆ f − 2 τ ( ∇ f ) + f B > 0 , where τ and B are geometric quantities combined from extrinsic and intrinsic geometry of Σ 0 and ¯ N , for details see ( 4.1 ). This notion of stabilit y w as inspired by the stability of a CMC h yp ersurface of Euclidean space, whic h in this setting would be an equiv alen t notion. F urther similar notions of stability of MOTS were discussed in [ 1 , 13 ]. T o understand the statement ab out the foliation and the definition of σ , w e note that our n ull cones without loss of generalit y are of the form ¯ N = [0 , Λ) × S 0 , where for the s -co ordinate s ∈ [0 , Λ), ∂ s is a null v ector and S 0 is a compact base manifold, whic h in case of the ab ov e theorem ma y as well coincide with Σ 0 . By increasing foliation we then mean, that all lea ves Σ λ of the foliation are giv en b y spacelike graphs Σ λ = { ( ω ( z , λ ) , z ) : z ∈ Σ 0 } o v er Σ 0 and ∂ λ ω > 0. R emark. It is possible that the foliation ma y b e extended to mean curv atures b eyond σ . F or example the (rotationally symmetric) STCMC slices of the standard null cone in Sc hw arzsc hild space, when written as a graph s = ω ( z ), ha ve spacetime mean curv ature |  H | 2 ( s ) = s − 2 (1 − 2 M s ). Here the MOTS is at s = 2 M and this remains stable until s = 3 M . The strategy of the proof is to construct the lea ves of the foliation b y running curv ature flo ws ( M t ) t> 0 in the n ull cone defined b y ∂ t x =  λ 2 ¯ θ − H  ¯ L, where the me an curvatur e of M t , H = θ | M t is as in ( 1.1 ) and the flow is started from a h yp ersurface M 0 in the future of Σ 0 whic h has strictly larger mean curv ature. This prop erty is crucial to ensure the existence of barriers and the existence of M 0 is guaranteed by the stabilit y of Σ 0 . This λ -family of flows will then satisfy smooth estimates, which are uniform in time and λ . A few further arguments yield the desired foliation. As a side product of our techniques we solve another geometric problem in null hypersur- faces. Crucially , our tec hniques describ ed ab ov e do not seriously dep end on the structure of the forcing term β = λ 2 ¯ θ . Instead w e can allo w v ery general β and this giv es us the opp ortunity to solve the prescribed spacetime mean curv ature problem. W e pro v e the follo wing: 1.2. Theorem. L et n ≥ 2 and ¯ N b e a nul l c one on S 0 . Thr oughout ¯ N , let 0 ≤ c ˚ ¯ χ , c R , C R satisfy (1.3) | ˚ ¯ χ | ≤ c ˚ ¯ χ ¯ θ , 0 ≤ c R ¯ θ 2 ≤ Rc( ¯ L, ¯ L ) ≤ C R ¯ θ 2 . 4 BEN LAMBER T AND JULIAN SCHEUER Then ther e exist c onstants C 0 ( n ) , C 1 ( n ) ≥ 0 , such that the ine quality c ˚ ¯ χ + C R < C 0 + C 1 c R implies the existenc e of an explicit c onstant, D ( n, c R , c ˚ ¯ χ , C R − c R ) which is smo oth in its last thr e e entries, wher e D ( n, c R , 0 , 0) = n 2 (1+ nc R ) 2  ( n − 2)( n − 10) 4 n 2 + n +6 n c R + c 2 R  , such that the fol lowing holds: L et ρ : ¯ N → R b e a smo oth function, such that ρ ≥ |  H | 2 on S 0 and supp ose ther e is a hyp ersurfac e Σ + to the futur e of S 0 with the pr op erty ρ ≤ |  H | 2 on Σ + . If D < 0 , then we additional ly supp ose sup Σ + ω < n 1 + nc R ( e π √ −D − 1) . Then, the flow ˙ x =  ρ 2 ¯ θ − H  ¯ L starting fr om Σ + exists for al l times and c onver ges smo othly to a smo oth pr escrib e d me an curvatur e surfac e Σ with |  H | 2 = ρ. R emark. The case c ˚ ¯ χ = 0 includes all rotationally symmetric n ull cones in rotationally sym- metric spacetimes. Hence the geometric conditions ab o ve ma y b e in terpreted as the assump- tion that the am bien t space is C 2 close to b eing rotationally symmetric. F urthermore, in the ph ysically relev an t cases where n = 2 or n = 10, the condition on sup Σ + ω b ecomes v acuous in the rotationally symmetric situation and allows for large sup Σ + ω if the ambien t space is sufficien tly close to rotationally symmetric. Finally we also note that for null h yp ersurfaces sufficien tly close to rotationally symmetric h yp ersurfaces, increasing c R alw a ys weak ens the h yp otheses. Ac kno wledgmen ts. The authors would like to thank Wilhelm Klingen berg and Durham Univ ersit y for hosting v arious research visits where this w ork w as initiated and completed. The second author w ould lik e to thank the first author and Leeds Univ ersit y for hosting a researc h visit, where parts of this w ork were written. The first author would lik e to thank the second author and Go ethe–Universit¨ at F rankfurt, where the pro of of the first theorem was completed. 2. Basic notions of null geometr y Null h yp ersurfaces. W e discuss special h yp ersurfaces of a Loren tzian manifold ( ¯ M n +2 , ¯ g ), namely those which exhibit an ev erywhere degenerate induced metric. These are mostly referred to as nul l hyp ersurfac es in the literature. They carry some in teresting and counter- in tuitiv e properties. W e denote by ¯ D the Levi-Civita connection of ¯ g and ¯ ∇ denotes the gradien t operator. F or brevit y we will mostly write ⟨· , ·⟩ = ¯ g . Let us first recall, that ev ery smo oth hypersurface ¯ N of a smo oth manifold ¯ M can lo cally b e realised as the level set of a smo oth function. Precisely , let p ∈ ¯ N , then there exists a neighbourho o d U ⊂ ¯ M of p and a function f ∈ C ∞ ( U ), such that ¯ N ∩ U = { f = 0 } . FOLIA TION OF NULL CONES BY STCMC SURF ACES 5 This implies T x ¯ N = k er d f ( x ) and hence ( ¯ ∇ f ( x )) ⊥ = T x ¯ N ∀ x ∈ ¯ N ∩ U . If ¯ N is a n ull hypersurface, the degeneracy of the induced metric ¯ g on ¯ N implies that at ev ery x ∈ ¯ N there exists a nonzero null v ector ¯ L ∈ T x ¯ N with the prop erty ¯ L ⊥ = T x ¯ N . Hence we find ¯ ∇ f ( x ) ∈ T x ¯ N , for otherwise the relation  ¯ ∇ f ( x ) , ¯ L  = 0 w ould imply ¯ L ∈ ker ¯ g , which is imp ossible due to the non-degeneracy of ¯ g . In addition, ¯ ∇ f and ¯ L are linearly dep endent, which follows from the following lemma. 2.1. Lemma. On a ve ctor sp ac e of dimension n + 1 c arrying a non-de gener ate biline ar form ¯ g of signatur e 1 , every subsp ac e W , such that ¯ g | W = 0 , is at most one-dimensional. Pr o of. By Sylv ester’s law of inertia, there exists an n -dimensional subspace Z suc h that ¯ g | Z is p ositive definite. Hence there holds W ∩ Z = { 0 } . How ev er we also ha v e n + 1 ≥ dim( W + Z ) = dim W + n. □ This b ehavior can b e summarised by sa ying that the normal to a null h yp ersurface is also tangent. Under suitable assumptions as discussed later, this equips us with a well- defined, smo oth global nonzero tangent field ¯ L , whic h annihilates the tangent space. F or ¯ X ∈ Σ ∞ ( ¯ N ; T ¯ N ), the latter b eing the space of smo oth sections of the tangent bundle, w e ha v e 0 = ¯ X  ¯ ∇ f , ¯ ∇ f  = 2  ¯ D ¯ X ¯ ∇ f , ¯ ∇ f  = 2 ¯ D 2 f ( ¯ X , ¯ ∇ f ) = 2 ¯ D 2 f ( ¯ ∇ f , ¯ X ) = 2  ¯ X , ¯ D ¯ ∇ f ¯ ∇ f  and it follo ws that ¯ D ¯ ∇ f ¯ ∇ f is also a multiple of ¯ L , which finally translates to ¯ D ¯ L ¯ L = κ ¯ L for some function κ . Thus, the flow generated by ¯ L on ¯ N is a flow of (pre-)geo desics and th us, in fact, the whole null h yp ersurface is ruled (i.e. foliated) b y geo desics. In contrast to the non-degenerate case, the normal v ector ¯ L has no length and th us we cannot normalise it b y division. Instead we use a differen t normalisation, namely ¯ L can b e scaled to mak e κ = 0. This follows from simple reparametrisation of the pregeo desics. F rom no w on w e thus assume that ¯ L has the prop ert y ¯ D ¯ L ¯ L = 0 . The ab ov e flow construction yields the local structure of ¯ N . 2.2. Proposition. Supp ose ¯ M is time-orientable, ¯ N ⊂ ¯ M a nul l hyp ersurfac e, and let S 0 ⊂ ¯ N b e a c omp act sp ac elike hyp ersurfac e. Then ¯ N splits ar ound S 0 , i.e. ther e exists a diffe omor- phism onto an op en subset, (Λ − , Λ + ) × S 0 → N ⊂ ¯ N , wher e we use s ∈ (Λ − , Λ + ) as the c anonic al c o or dinate on this interval. Denoting by ¯ γ the pul lb ack metric on N and by ¯ D ∗ the pul lb ack c onne ction, then ∂ s is a nul l ve ctor and ¯ D ∗ ∂ s ∂ s = ¯ D ¯ L ¯ L = 0 . 6 BEN LAMBER T AND JULIAN SCHEUER F urthermor e, ¯ L c an b e arr ange d futur e-dir e cte d. Pr o of. The splitting follows since on a compact S 0 w e can c ho ose a uniform time in terv al for the geo desic flow, and we simply discard the other part of ¯ N . □ F rom now on, if ¯ N splits around some S 0 , we already simply assume that ¯ N is a pro duct as ab o v e. How ev er, the prop erty κ = 0 still do es not determine ¯ L uniquely , as we hav e one more degree of freedom in c ho osing the initial velocity of the geodesic flo w. This can b e accomplished by requiring that along S 0 w e ha v e ¯ θ := − n X i =1  ¯ D e i e i , ¯ L  ≡ 1 , pro vided this quantit y is nowhere zero for one (and hence any) c hoice of ¯ L , and where ( e i ) 1 ≤ i ≤ n is an orthonormal frame of S 0 . In this case we sa y that ¯ N is lo c al ly a nul l c one ar ound S 0 , meaning that ¯ θ ( s, · ) > 0 for all s sufficiently small, where ¯ θ ( s, · ) is defined analogously via an orthonormal frame of the s -slice { s } × S 0 . Hence in the following, w e simply assume this prop ert y to hold for ¯ N and discard the past of S 0 , i.e. without loss of generality b e the null cone ¯ N iden tified with ¯ N ∼ = [0 , Λ) × S 0 and say that ¯ N is a nul l c one on S 0 . W e then often refer to this splitting as the c anonic al b ackgr ound foliation of the n ull cone. This fixing of ¯ L can b e used to unam biguously define the second fundamen tal form of ¯ N . 2.3. Definition. Supp ose ¯ N is a null cone on S 0 . Then we define the se c ond fundamental form of ¯ N b y ¯ χ ( ¯ X , ¯ Y ) =  ¯ D ¯ X ¯ L, ¯ Y  ∀ ¯ X , ¯ Y ∈ Σ ∞ ( ¯ N ; T ¯ N ) . 2.4. R emark. Even if ¯ N is not a n ull cone, w e can define a second fundamental form. How ev er, in this case the ob ject is only defined up to multiplication b y smooth functions on S 0 . Spacelik e graphs. The splitting structure is w ell suited to describ e spacelik e graphs. Sup- p ose that M ⊂ ¯ N is a spacelik e graph given b y M = { x ( z ) = ( ω ( z ) , z ) : z ∈ S 0 } . W e recall the Gauss formula for vector fields X , Y on M and decomp ose the normal part I I ( X , Y ) conv enien tly: Let D b e the Levi-Civita connection of the metric g = x ∗ ¯ g . Then (2.1) ¯ D Y X = D Y X + I I ( X , Y ) = D Y X − h ( X , Y ) ¯ L − ¯ χ ( X , Y ) ν , where the past-directed vector ν complements ¯ L in suc h a wa y that ⟨ ν, ν ⟩ = 0 ,  ν, ¯ L  = 1 , and where we, as commonly done, iden tify X ∈ Σ ∞ ( M ; T M ) with its pushforw ard x ∗ X . W e sa y that { ν, ¯ L } is a nul l p air for M . Note that this terminology even makes sense if M do es not factor through a n ull cone a priori. 2.5. Definition. F or a graph M as given abov e, we define the se c ond fundamental form of M in ¯ N to be the bilinear form h . W e also define H := tr g h as the me an curvatur e of M in ¯ N . FOLIA TION OF NULL CONES BY STCMC SURF ACES 7 The trace of ¯ χ with resp ect to g is more subtle. 2.6. Lemma. L et M and ˜ M b e two sp ac elike gr aphs, which interse ct at a p oint x 0 ∈ ¯ N . Then ther e holds tr g ¯ χ | x 0 = tr ˜ g ¯ χ | x 0 . Pr o of. Here and in the following, for the differential of ω w e adopt the notation ω i := ∂ i ω . Fix a lo cal coordinate frame ( ∂ i ) on S 0 and write x : S 0 → M and ˜ x : S 0 → ˜ M for the em b ed- dings of the graphs. Then x i := x ∗ ∂ i and likewise for ˜ x give rise to co ordinate representations g ij and ˜ g ij . There holds g ij = ⟨ x i , x j ⟩ =  ω i ¯ L + ∂ i , ω j ¯ L + ∂ j  =  ˜ ω i ¯ L + ∂ i , ˜ ω j ¯ L + ∂ j  = ˜ g ij , where we used that ¯ L annihilates ev erything in T ¯ N . In addition there holds ¯ χ ij := ¯ χ ( x i , x j ) = ¯ χ ( ω i ¯ L + ∂ i , ω j ¯ L + ∂ j ) = ¯ χ ( ˜ x i , ˜ x j ) , since (2.2) ¯ χ ( ¯ L, ¯ X ) =  ¯ D ¯ L ¯ L, ¯ X  = 0 ∀ ¯ X ∈ Σ ∞ ( ¯ N ; T ¯ N ) . □ The following definition is hence sensible: 2.7. Definition. (i) Supp ose ¯ N is a null cone S 0 . Then we define the mean curv ature of ¯ N in ¯ M to be the function ¯ θ ∈ C ∞ ( ¯ N ) defined b y ¯ θ ( s, z ) := tr g s ¯ χ, where g s is the induced metric of the spacelik e set { ( s, z ) : z ∈ S 0 } . (ii) F or the constan t graphs { s = const } w e also reserve the notation L for ν , χ for h and θ for H . 2.8. R emark. (i) W e also use the notation L , χ and θ in case that a spacelike submanifold is giv en in absence of a factorising n ull cone. The notation ν , h and H is reserved to distinguish general graphs within a null cone from the resp ectiv e quantities of the co ordinate slices of the background foliation. (ii) Since the quan tit y χ is defined on ev ery leaf of the bac kground foliation, it can be view ed as an elemen t of Σ ∞ ( ¯ N ; T 0 , 2 ¯ N ), via the formula χ ( ¯ X , ¯ Y ) =  ¯ X , ¯ D ¯ Y L  . Where for vector fields on S 0 this tensor is the second fundamen tal of the s -slice { ω ≡ s } , it is left to iden tify its action on the ¯ L directions, (2.3) χ ( ¯ L, ¯ L ) =  ¯ L, ¯ D ¯ L L  = −  ¯ D ¯ L ¯ L, L  = 0 , χ ( ¯ L, ∂ i ) = −  ¯ D ∂ i ¯ L, L  = − ζ ( ∂ i ) , χ ( ∂ i , ¯ L ) =  ∂ i , ¯ D ¯ L L  = −  ¯ D ¯ L ∂ i , L  = −  ¯ D ∂ i ¯ L, L  = − ζ ( ∂ i ) . Here we hav e in tro duced a new linear form. As it is common for higher codimensional sub- manifolds, we ha v e to tak e torsion into accoun t. Contrary to the h ypersurface case, where the deriv ative of the normal is alwa ys tangen t, in higher co dimension there migh t b e comp onen ts 8 BEN LAMBER T AND JULIAN SCHEUER in other normal directions. Hence w e define tw o more quan tities, the torsion of the nul l c one and the torsion of a sp ac elike gr aph M , (2.4) ζ ( ¯ X ) =  ¯ D ¯ X ¯ L, L  ∀ ¯ X ∈ Σ ∞ ( ¯ N ; T ¯ N ) , τ ( ¯ X ) =  ¯ D ¯ X ¯ L, ν  ∀ ¯ X ∈ Σ ∞ ( S 0 ; x ∗ T ¯ N ) . 2.9. R emark. F rom ( 2.1 ) w e immediately obtain that the mean curv ature v ector of a spacelik e graph M ⊂ ¯ M , which factors through a n ull cone M → ¯ N ⊂ ¯ M , is giv en b y  H = − H ¯ L − ¯ θ ν and hence |  H | 2 = 2 H ¯ θ . 2.10. Definition. Within a null cone on S 0 w e call a spacelik e h yp ersurface a MOTS if H = 0. A connection on the lo cal n ull cone. W e hav e already introduced the tensor ¯ χ ∈ Σ ∞ ( ¯ N , T 0 , 2 ¯ N ) and we will introduce sev eral other such tensors later. As we hav e to dif- feren tiate them and as there is no canonical Levi-Civita connection on ¯ N induced from ¯ M due to the degeneracy of the induced metric, we introduce ad ho c a connection whic h mak es use of the canonical bac kground foliation of a null cone. 2.11. Definition. Supp ose ¯ N is a n ull cone on S 0 . Then w e define ˆ D : Σ ∞ ( ¯ N ; T ¯ N ) × Σ ∞ ( ¯ N ; T ¯ N ) → Σ ∞ ( ¯ N ; T ¯ N ) ( ¯ X , ¯ Y ) 7→ ˆ D ¯ X ¯ Y = ¯ D ¯ X ¯ Y −  ¯ D ¯ X ¯ Y , ¯ L  L. This connection is readily extended to arbitrary tensors b y the standard Leibniz rule. Comparison form ulae for spacelik e graphs. Using the parametrisation x ( z ) = ( ω ( z ) , z ) and a lo cal coordinate frame { ∂ i } to S 0 , we hav e x i = ∂ i + ω i ¯ L, whic h immediately implies ⟨ L ◦ x, x i ⟩ = ω i . W e let |·| denote the norm induced on the graphs. There holds (2.5) ν = L + 1 2 |∇ ω | 2 ¯ L − ∇ ω = L − 1 2 |∇ ω | 2 ¯ L − ω i ∂ i , since we immediately observe  ¯ L, ν  = 1, ⟨ ν, x i ⟩ = ω i − ω i = 0 and ⟨ ν, ν ⟩ = 0 as required. Putting these facts together w e obtain (2.6) D x j x i ω = ∂ j ω i − Γ k ij ω k =  ¯ D x j L, x i  +  ν − 1 2 |∇ ω | 2 ¯ L, ¯ D x j x i  = χ ij − h ij + u ¯ χ ij , where we define u := − ⟨ ν, L ⟩ = 1 2 |∇ ω | 2 ≥ 0 , whic h is a geometric function capturing the gradien t of our graphs. W e note that (2.7) τ ( ¯ X ) =  ¯ D ¯ X ¯ L, L + 1 2 |∇ ω | 2 ¯ L − ∇ ω  = ζ ( ¯ X ) − ¯ χ ( ¯ X , ∇ ω ) and also record the follo wing iden tities for later use, where we use ( 2.3 ) and ( 2.5 ), (2.8) ⟨ ν, D ¯ L L ⟩ = ζ ( ∇ ω ) ,  ν, ¯ D ¯ X L  =  u ¯ L − ∇ ω , ¯ D ¯ X L  = − uζ ( ¯ X ) − χ ( ¯ X , ∇ ω ) ∀ ¯ X ∈ Σ ∞ ( ¯ N ; T ¯ N ) . FOLIA TION OF NULL CONES BY STCMC SURF ACES 9 Finally , w e need a lemma that relates the graph function with the connection on ¯ N . 2.12. Lemma. F or T ∈ Σ ∞ ( ¯ N ; T 0 , 1 ¯ N ) ther e holds D x i T ( x j ) = ˆ D x i T ( x j ) + T ( ¯ χ ij ∇ ω − ( u ¯ χ ij + h ij ) ¯ L ) . Pr o of. W e also denote b y T restriction of T to the spacelike graph M = graph ω . D x i T ( x j ) = x i ( T ( x j )) − T ( D x i x j ) = ˆ D x i T ( x j ) + T ( ¯ D x i x j −  ¯ D x i x j , ¯ L  L − D x i x j ) = ˆ D x i T ( x j ) + T ( − ¯ χ ij ν − h ij ¯ L + ¯ χ ij L ) = ˆ D x i T ( x j ) + T ( ¯ χ ij ( L − ν ) − h ij ¯ L ) = ˆ D x i T ( x j ) + T ( ¯ χ ij ( ∇ ω − 1 2 |∇ ω | 2 ¯ L ) − h ij ¯ L ) = ˆ D x i T ( x j ) + T ( ¯ χ ij ∇ ω − ( u ¯ χ ij + h ij ) ¯ L ) . □ W e ha v e obvious similar iden tities for higher order tensors. 3. Evolution equa tions 3.1. Ev olution equations for general speeds. W e study general ev olutions of the form (3.1) ˙ x ( t, z ) = f ( t, z ) ¯ L ( x ( t, z )) , where x : [0 , T ∗ ) × S 0 → ¯ N is a flo w of graphs and a dot indicates the time deriv ativ e. 3.1. Lemma. Supp ose that x satisfies ( 3.1 ) in a nul l c one on S 0 . Then the evolution of ω is ˙ ω − ∆ ω = f − θ + H − u ¯ θ . L et e ω b e any smo oth function on ¯ N with ¯ L ( e ω ) = 0 . If x satisfies ( 3.1 ) then ˙ ˜ ω − ∆ ˜ ω = − g ij ˆ D ∂ i ∂ j ˜ ω − ω i ( ¯ θ δ k i − 2 ¯ χ k i ) ˜ ω k . Pr o of. The ev olution of ω follo ws immediately from ( 2.6 ). W e ha v e ˙ ˜ ω = 0. On the other hand, using Lemma 2.12 , w e ha v e D x i x j ˜ ω = ˆ D x i x j ˜ ω + ¯ χ ij ω k ˜ ω k . Note that ˆ D ¯ L ¯ L = 0 and ˆ D ∂ i ¯ L = ¯ χ k i ∂ k + ζ i ¯ L, so ˆ D ¯ L ¯ L ˜ ω = 0 and ˆ D ∂ i ¯ L ˜ ω = ˆ D ¯ L∂ i ˜ ω = − ¯ χ k i ˜ ω k . Therefore, ∆ ˜ ω = g ij D ij ˜ ω = g ij ( ˆ D ∂ i ∂ j ˜ ω − ¯ χ k i ˜ ω k ω j − ¯ χ k j ˜ ω k ω i + ¯ χ ij ω k ˜ ω k ) = g ij ˆ D ∂ i ∂ j ˜ ω + ω i ( ¯ θ δ k i − 2 ¯ χ k i ) ˜ ω k . □ 10 BEN LAMBER T AND JULIAN SCHEUER 3.2. Lemma. Supp ose that x satisfies ( 3.1 ) in a nul l c one on S 0 . Then ¯ D ˙ x ν = −∇ f − f τ k x k and (3.2) ˙ u = ⟨∇ f , L ⟩ − f ¯ χ ( ∇ ω , ∇ ω ) . Pr o of. W e ha v e  ¯ D ˙ x ν, x i  = −  ν, ¯ D ˙ x x i  = −  ν, ¯ D x i ˙ x  = −  ν, ¯ D x i ( f ¯ L )  = − d f ( x i ) − f τ i ,  ¯ D ˙ x ν, ¯ L  = −  ν, ¯ D ˙ x ¯ L  = 0 and 0 = d dt ⟨ ν, ν ⟩ =  ¯ D ˙ x ν, ν  . The first statemen t no w follows. F or ˙ u w e compute ˙ u = D ∇ f + f τ k x k , L E −  ν, ¯ D ˙ x L  = ⟨∇ f , L ⟩ + f τ ( ∇ ω ) − f ζ ( ∇ ω ) = ⟨∇ f , L ⟩ − f ¯ χ ( ∇ ω, ∇ ω ) , where we used ( 2.3 ), ( 2.7 ) and ( 2.8 ). □ F or the following ev olution equations we need to clarify our conv ention on the curv ature tensor. W e use the one from [ 16 ], R ( X , Y ) Z = D Y ( D X Z ) − D X ( D Y Z ) + D [ X,Y ] Z, whic h in co ordinates reads R m lk j x m = R ( x k , x j )( x l ) . W e obtain the Gauss equation, [ 16 , p. 100, Thm. 5] R m lk j = ¯ R m lk j + ¯ g (I I kl , I I m j ) − ¯ g (I I j l , I I m k ) = ¯ R m lk j + ¯ χ kl h m j + h kl ¯ χ m j − ¯ χ j l h m k − h j l ¯ χ m k , where I I is the full second fundamental form. 3.3. Lemma. Supp ose that x satisfies ( 3.1 ) in a nul l c one on S 0 . Then ˙ g ij = 2 f ¯ χ ij , ˙ h ij = − D ij f − d f ( x j ) τ i − d f ( x i ) τ j + f ( ¯ χ k j h ki − D i τ ( x j ) − τ i τ j −  ¯ R ( x i , ¯ L ) x j , ν  ) and (3.3) ˙ H = − ∆ f − 2 τ ( ∇ f ) − f ( ¯ χ ij h ij + D i τ i + | τ | 2 + Rc( ¯ L, ν ) +  ¯ R ( ν, ¯ L ) ν, ¯ L  ) . Pr o of. W e compute ˙ g ij =  ¯ D ˙ x x i , x j  +  x i , ¯ D ˙ x x j  =  ¯ D x i ( f ¯ L ) , x j  +  x i , ¯ D x j ( f ¯ L )  = 2 f ¯ χ ij . FOLIA TION OF NULL CONES BY STCMC SURF ACES 11 Using Lemma 3.2 , we compute ˙ h ij = − ∂ t  ¯ D x i x j , ν  = −  ¯ D ˙ x ( ¯ D x i x j ) , ν  −  ¯ D x i x j , ¯ D ˙ x ν  = −  ¯ D x i ( ¯ D x j ˙ x ) + ¯ R ( x i , ˙ x ) x j , ν  −  ¯ D x i x j , ¯ D ˙ x ν  = − D ¯ D x i ( d f ( x j ) ¯ L + f ( ¯ χ k j x k + τ j ¯ L )) , ν E −  ¯ R ( x i , ˙ x ) x j , ν  −  ¯ D x i x j , ¯ D ˙ x ν  = − ∂ i ( d f ( x j )) − d f ( x j ) τ i − d f ( x i ) τ j − f D ¯ D x i ( ¯ χ k j x k + τ j ¯ L ) , ν E −  ¯ R ( x i , ˙ x ) x j , ν  −  ¯ D x i x j , ¯ D ˙ x ν  = − ∂ i ( d f ( x j )) − d f ( x j ) τ i − d f ( x i ) τ j + f ¯ χ k j h ki − f ∂ i ( τ j ) − f τ i τ j −  ¯ R ( x i , ˙ x ) x j , ν  −  ¯ D x i x j , ¯ D ˙ x ν  = − D ij f − d f ( x j ) τ i − d f ( x i ) τ j + f ¯ χ k j h ki − f D i τ j − f τ i τ j − f  ¯ R ( x i , ¯ L ) x j , ν  , whic h giv es the second equation. T o take the trace, we note that  ¯ R ( x i , ¯ L ) x i , ν  = Rc( ¯ L, ν ) − 1 2  ¯ R ( ¯ L + ν, ¯ L )( ¯ L + ν ) , ν  + 1 2  ¯ R ( ¯ L − ν, ¯ L )( ¯ L − ν ) , ν  = Rc( ¯ L, ν ) − 1 2  ¯ R ( ν, ¯ L )( ¯ L + ν ) , ν  − 1 2  ¯ R ( ν, ¯ L )( ¯ L − ν ) , ν  = Rc( ¯ L, ν ) −  ¯ R ( ν, ¯ L ) ¯ L, ν  . Using this iden tit y , along with the ev olution of g ij and h ij , we see ˙ H = − ∆ f − 2 τ ( D f ) + f ( − ¯ χ ij h ij − D i τ i − | τ | 2 − Rc( ¯ L, ν ) −  ¯ R ( ν, ¯ L ) ν, ¯ L  ) . □ 3.2. Ev olution equations for PMCF. W e define the Prescribed Mean Curv ature Flow (PMCF) to b e (3.4) ˙ x = ( β − H ) ¯ L where β : ¯ N → R is a smo oth function. W e start with a corollary of Lemma 3.1 . 3.4. Lemma. Under ( 3.4 ) , the evolution of ω is ˙ ω − ∆ ω = β − θ − u ¯ θ . F or e ω any smo oth function on ¯ N with ¯ L ( e ω ) = 0 , the fol lowing evolution e quation holds: ˙ ˜ ω − ∆ ˜ ω = − g ij ˆ D ∂ i ∂ j ˜ ω − ω i ( ¯ θ δ k i − 2 ¯ χ k i ) ˜ ω k . Our pro of is based on C 1 -estimates. Hence w e need ev olution equations up to that order. 3.5. Lemma. Under ( 3.4 ) , the evolution of u is ˙ u − ∆ u = β i ω i − β ¯ χ ij ω i ω j − u ¯ θ k ω k + 2 ζ j u j − g ij ( ˆ D ∇ ω χ ) ij − 4 u ¯ χ kj ζ j ω k − 2 χ kj ζ j ω k − ¯ θ u ¯ χ ij ω i ω j − ¯ θ χ ij ω i ω j + 2 u ¯ χ lj ¯ χ l k ω j ω k − 2 ¯ χ ij ω i u j − | D 2 ω | 2 − ¯ R i lk i ω l ω k . Pr o of. F or this pro of, if w e furnish the functions H , β , u or ω b y indices, we mean co v ariant differen tiation with resp ect to D , e.g. ω ij dx i dx j = D x j ( ω i dx i ) dx j . 12 BEN LAMBER T AND JULIAN SCHEUER All co ordinates are taken with resp ect to ( x i ) 1 ≤ i ≤ n and lifting of indices happ ens with resp ect to g . Using equation ( 3.2 ), w e obtain ˙ u = ⟨∇ ( β − H ) , L ⟩ − ( β − H ) ¯ χ ( ∇ ω , ∇ ω ) = − H i ω i + β i ω i − ( β − H ) ¯ χ ( ∇ ω , ∇ ω ) . W e recall 2 u = ω k ω k and compute with the help of ( 2.6 ), u i = ω ki ω k = ω k ( u ¯ χ ik + χ ik − h ik ) . W e obtain u ij = g mi ω m kj ω k + ω ki ω k j = g mi ω m j k ω k + g mi R m lk j ω l ω k + ω ki ω k j = ω ij k ω k + ( ¯ χ lk h ij + h lk ¯ χ ij − ¯ χ j l h ik − h j l ¯ χ ik ) ω l ω k + g mi ¯ R m lk j ω l ω k + ω ki ω k j . T racing with resp ect to g giv es ∆ u = (∆ ω ) k ω k + H ¯ χ ij ω i ω j + ω k h l k ( ¯ θ ω l − 2 ¯ χ lj ω j ) + | D 2 ω | 2 + ¯ R i lk i ω l ω k = (∆ ω ) k ω k + H ¯ χ ij ω i ω j + ( uω k ¯ χ l k + ω k χ l k − u l )( ¯ θ ω l − 2 ¯ χ lj ω j ) + | D 2 ω | 2 + ¯ R i lk i ω l ω k = (∆ ω ) k ω k + H ¯ χ ij ω i ω j + ¯ θ u ¯ χ ij ω i ω j + ¯ θ χ ij ω i ω j − ¯ θ u i ω i − 2 ¯ χ lj ω j ( uω k ¯ χ l k + ω k χ l k ) + 2 ¯ χ ij ω i u j + | D 2 ω | 2 + ¯ R i lk i ω l ω k . W e further expand the first term with the help of Lemma 2.12 , ( 2.3 ) and ( 2.6 ): (∆ ω ) k ω k = ( u ¯ θ + g ij χ ij − H ) k ω k = ¯ θ u k ω k + u ¯ θ k ω k + g ij ( ˆ D ∇ ω χ ) ij + 2 χ ( ¯ χ j k ∇ ω − ( u ¯ χ j k + h j k ) ¯ L, x j ) ω k − H k ω k = ¯ θ u k ω k + u ¯ θ k ω k + g ij ( ˆ D ∇ ω χ ) ij + 2 ω k ω l χ j l ¯ χ j k + 2 ω k ( u ¯ χ kj + h kj ) ζ j − H k ω k = ¯ θ u k ω k + u ¯ θ k ω k + g ij ( ˆ D ∇ ω χ ) ij + 2 ω k ω l χ j l ¯ χ j k + 2(2 uω k ¯ χ kj + ω k χ kj − u j ) ζ j − H k ω k = ¯ θ u k ω k + u ¯ θ k ω k − 2 ζ j u j + g ij ( ˆ D ∇ ω χ ) ij + 2 χ j l ¯ χ j k ω k ω l + 4 u ¯ χ kj ζ j ω k + 2 χ kj ζ j ω k − H k ω k and combining these equalities we get ∆ u = u ¯ θ k ω k − 2 ζ j u j + g ij ( ˆ D ∇ ω χ ) ij + 4 u ¯ χ kj ζ j ω k + 2 χ kj ζ j ω k − H k ω k + H ¯ χ ij ω i ω j + ¯ θ u ¯ χ ij ω i ω j + ¯ θ χ ij ω i ω j − 2 u ¯ χ lj ¯ χ l k ω j ω k + 2 ¯ χ ij ω i u j + | D 2 ω | 2 + ¯ R i lk i ω l ω k . The claim follo ws from com bining with Lemma 3.2 and cancellation of terms in v olving H . □ W e also recall Ra yc haudh uri’s optical equation, whic h we need in the sequel: 3.6. Prop osition. On ¯ N , ¯ L ¯ θ = −| ˚ ¯ χ | 2 − 1 n ¯ θ 2 − Rc( ¯ L, ¯ L ) . FOLIA TION OF NULL CONES BY STCMC SURF ACES 13 Pr o of. ¯ L ¯ θ = ¯ L ( g ij  ¯ D ∂ i ¯ L, ∂ j  ) = − g ik (  ¯ D ¯ L x k , x l  +  x k , ¯ D ¯ L x l  ) g lj  ¯ D ∂ i ¯ L, ∂ j  ) + g ij  ¯ D ¯ L ¯ D ∂ i ¯ L, ∂ j  + g ij  ¯ D ∂ i ¯ L, ¯ D ∂ j ¯ L  = −| ¯ χ | 2 + g ij  ¯ R ( ∂ i , ¯ L ) ¯ L, ∂ j  = −| ¯ χ | 2 − Rc( ¯ L, ¯ L ) − 1 2  ¯ R ( ¯ L + L, ¯ L ) ¯ L, ¯ L + L  + 1 2  ¯ R ( ¯ L − L, ¯ L ) ¯ L, ¯ L − L  = −| ˚ ¯ χ | 2 − 1 n ¯ θ 2 − Rc( ¯ L, ¯ L ) . □ 3.7. Corollary . The evolution of u satisfies the estimate ˙ u − ∆ u = 2 ζ i u i − 2 ¯ χ ij ω i u j − | D 2 ω | 2 + O ( u 3 2 ) + 4 n 2 ¯ θ 2 u 2 − 2 u 2 Rc( ¯ L, ¯ L ) + 2 u 2 | ˚ ¯ χ | 2 + ( 4 n − 1) ¯ θ u ˚ ¯ χ kj ω k ω j + 2 u ˚ ¯ χ lj ˚ ¯ χ l k ω j ω k , wher e O ( r ) is char acterise d by the estimate |O ( r ) | ≤ C (1 + | β | + | ¯ Lβ | + | dβ | )(1 + | r | ) wher e C is a c onstant which is b ounde d while the flow r emains in any c omp act set. Pr o of. W e group all terms in the evolution of u = 1 2 |∇ ω | 2 according to their order. Noting that x i = ∂ i + ω i ¯ L , there holds for ev ery function f on the graph M ∇ f = g ij f j x i = f i ∂ i + f i ω i ¯ L, and in particular for the graph function ω itself we get ∇ ω = ω i ∂ i + 2 u ¯ L. Then there holds, using ( 2.2 ), ( 2.3 ) and ( 2.4 ), β i ω i − 2 χ kj ζ j ω k = ∇ ω ( β ) − 2 χ kj ζ j ω k = O ( u ) , − β ¯ χ ij ω i ω j = O ( u ) . Then we hav e − u ¯ θ k ω k = − u ∇ ω ( ¯ θ ) = − 2 u 2 ¯ L ( ¯ θ ) + O ( u 3 2 ) . The next relev ant terms are − g ij ( ˆ D ∇ ω χ ) ij − 4 u ¯ χ kj ζ j ω k = − g ij ( ω k ˆ D ∂ k χ )( x i , x j ) − 2 ug ij ( ˆ D ¯ L χ )( x i , x j ) + O ( u 3 2 ) = O ( u 3 2 ) , since ˆ D ¯ L χ ( ¯ L, ¯ L ) = 0, due to ( 2.3 ), and where w e also used ( 2.4 ). The next tw o second order terms are − ¯ θ u ¯ χ ij ω i ω j + 2 u ¯ χ lj ¯ χ l k ω j ω k = − ¯ θ u ( ˚ ¯ χ ij + ¯ θ n g ij ) ω i ω j + 2 u ( ˚ ¯ χ lj + ¯ θ n g lj )( ˚ ¯ χ l k + ¯ θ n δ l k ) ω j ω k = ( 4 n 2 − 2 n ) ¯ θ 2 u 2 + ( 4 n − 1) ¯ θ u ˚ ¯ χ kj ω k ω j + 2 u ˚ ¯ χ lj ˚ ¯ χ l k ω j ω k . W e con tin ue, using ( 2.3 ), − ¯ θ χ ij ω i ω j = − ¯ θ ω i ω j χ ( ∂ i , ∂ j ) + 4 u ¯ θ ω i ζ i = O ( u 3 2 ) . 14 BEN LAMBER T AND JULIAN SCHEUER F or the final term inv olving the Riemann tensor, we compute b y completing the basis of T ¯ M using 1 √ 2 ( ¯ L ± L ),  − ¯ R ( x j , ∇ ω ) x j , ∇ ω  = − Rc( ∇ ω , ∇ ω ) + 1 2  ¯ R ( ¯ L + L, ∇ ω )( ¯ L + L ) , ∇ ω  − 1 2  ¯ R ( ¯ L − L, ∇ ω )( ¯ L − L ) , ∇ ω  = − Rc( ∇ ω , ∇ ω ) +  ¯ R ( L, ∇ ω ) ¯ L, ∇ ω  +  ¯ R ( ¯ L, ∇ ω ) L, ∇ ω  = − Rc( ∇ ω , ∇ ω ) + 2  ¯ R ( L, ∇ ω ) ¯ L, ∇ ω  = − 4 u 2 Rc( ¯ L, ¯ L ) − 2 uω i Rc( ∂ i , ¯ L ) − ω i ω j Rc( ∂ i , ∂ j ) + 2 ω i ω j  ¯ R ( L, ∂ i ) ¯ L, ∂ j  + 4 ω j u  ¯ R ( L, ¯ L ) ¯ L, ∂ j  = − 4 u 2 Rc( ¯ L, ¯ L ) + O ( u 3 2 ) . Plugging everything together and also using Prop osition 3.6 gives the result. □ W e will make use of test functions to obtain estimates for u . The following lemma reduces requiremen ts on the test function to an ordinary differential inequalit y . 3.8. Lemma. Supp ose that ϕ = uµ − 2 ( ω ) for some µ : R → R + . Then, at any p ositive maxi- mum of ϕ , ˙ ϕ − ∆ ϕ ≤ 4 µϕ 2 [Φ( x ) µ + Ψ µ ′ + µ ′′ ] + C µ O ( ϕ 3 2 ) wher e Φ = 1 n 2 ¯ θ 2 − 1 2 Rc( ¯ L, ¯ L ) + 1 2 | ˚ ¯ χ | 2 + 4 − n 4 n ¯ θ ˚ ¯ χ kj ω k ω j u − 1 + 1 2 ˚ ¯ χ lj ˚ ¯ χ l k ω j ω k u − 1 , Ψ = n − 4 2 n ¯ θ − ˚ ¯ χ ij ω i ω j u − 1 . and C µ = C µ ( | µ | C 1 , inf µ ) . F or ˜ ω as in L emma 3.4 , define ˜ ϕ = uµ − 2 ( ω − ˜ ω ) , then at any maximum of ˜ ϕ , ˙ ˜ ϕ − ∆ ˜ ϕ ≤ 4 µ ˜ ϕ 2 [Φ( x ) µ + Ψ µ ′ + µ ′′ ] + ˜ C µ e O ( ˜ ϕ 3 2 ) wher e ˜ C µ = ˜ C µ ( | µ | C 2 , inf µ ) and e O ( r ) which is char acterise d by the estimate | e O ( r ) | ≤ e C (1 + | β | + | ¯ Lβ | + | dβ | )(1 + | r | ) wher e e C is a c onstant which is a c onstant which is b ounde d while the flow r emains in any c omp act set, but also may dep end on ˆ D -derivatives of e ω up to se c ond or der. Pr o of. The function ϕ = uρ ( ω ) satisfies ˙ ϕ − ∆ ϕ = ρ ( ˙ u − ∆ u ) + uρ ′ ( ω )( ˙ ω − ∆ ω ) − uρ ′′ ( ω ) | D ω | 2 − 2 ⟨∇ u, ∇ ρ ⟩ ≤ ρu [2 ζ i (log u ) i − 2 ¯ χ ij ω i (log u ) j − u − 1 | D 2 ω | 2 + O ( u 1 2 ) + 4Φ( x ) u + ρ − 1 ρ ′ ( ω )( β − θ − u ¯ θ ) − 2 ρ − 1 ρ ′′ ( ω ) u − 2 ρ − 1 ρ ′ ⟨∇ log u, ∇ ω ⟩ ] . A t a maxim um w e hav e D log u = − ρ − 1 ρ ′ D ω . Additionally w e ha v e | D u | 2 = | ω i ω k i x k | 2 ≤ | D ω | 2 | D 2 ω | 2 = 2 u | D 2 ω | 2 , so at a maximum, u − 1 | D 2 ω | 2 ≥ 1 2 | D log u | 2 = 1 2 ρ − 2 ( ρ ′ ) 2 | D ω | 2 = ρ − 2 ( ρ ′ ) 2 u. FOLIA TION OF NULL CONES BY STCMC SURF ACES 15 Hence, at a maximum, ˙ ϕ − ∆ ϕ ≤ ρu [ − 2 ρ − 1 ρ ′ ζ i ω i + 2 ρ − 1 ρ ′ ¯ χ ij ω i ω j − ρ − 2 ( ρ ′ ) 2 u + O ( u 1 2 ) + 4Φ( x ) u − ρ − 1 ρ ′ ( ω ) ¯ θ u − 2 ρ − 1 ρ ′′ ( ω ) u + 2 ρ − 2 ( ρ ′ ) 2 | D ω | 2 + ρ − 1 ρ ′ ( β − θ )] = ρu 2 [4Φ( x ) + ρ − 1 ρ ′  4 − n n ¯ θ + 2 ˚ ¯ χ ij ω i ω j u − 1  − 2 ρ − 1 ρ ′′ + 3 ρ − 2 ( ρ ′ ) 2 ] + O ( u 3 2 )( ρ + | ρ ′ | ) . W e no w set µ = 1 √ ρ and note that − 2 µ ′ = ρ ′ ρ 3 2 = µρ − 1 ρ ′ , 4 µ ′′ = − 2 ρ ′′ ρ 3 2 + 3 ( ρ ′ ) 2 ρ 5 2 = µ (3 ρ − 2 ( ρ ′ ) 2 − 2 ρ − 1 ρ ′′ ) , so ˙ ϕ − ∆ ϕ ≤ ρµ − 1 u 2 [4Φ( x ) µ − 2  4 − n n ¯ θ + 2 ˚ ¯ χ ij ω i ω j u − 1  µ ′ + 4 µ ′′ ] + O ( u 3 2 )( µ − 2 + | µ ′ | µ − 3 ) = 4 µϕ 2 [Φ( x ) µ + Ψ µ ′ + µ ′′ ] + O ( u 3 2 )( µ − 2 + | µ ′ | µ − 3 ) . The claim no w follo ws as O ( u 3 2 ) ≤ (1 + µ 3 ) O ( ϕ 3 2 ). W e no w rep eat the ab o ve computation but with ˜ ϕ = uρ ( ω − ˜ ω ). Lemma 3.4 implies ˙ ˜ ω − ∆ ˜ ω = e O ( u 1 2 ) , and hence for Ω = ω − ˜ ω we obtain ˙ Ω − ∆Ω = ˙ ω − ∆ ω + e O ( u 1 2 ) . A t a maximum of ˜ ϕ , D log u = − ρ − 1 ρ ′ D Ω, and so using the previous evolution and Corol- lary 3.7 , ˙ ˜ ϕ − ∆ ˜ ϕ = ρ ( ˙ u − ∆ u ) + uρ ′ (Ω)( ˙ Ω − ∆Ω) − uρ ′′ (Ω) |∇ Ω | 2 − 2 ⟨∇ u, ∇ ρ ⟩ ≤ ρu [2 ζ i (log u ) i − 2 ¯ χ ij ω i (log u ) j − u − 1 | D 2 ω | 2 + O ( u 1 2 ) + 4Φ( x ) u + ρ − 1 ρ ′ ( β − θ − u ¯ θ + e O ( u 1 2 )) − ρ − 1 ρ ′′ |∇ Ω | 2 − 2 ρ − 1 ρ ′ ⟨∇ log u, ∇ Ω ⟩ ] = ρu [ − 2 ρ − 1 ρ ′ ζ i Ω i + 2 ρ ′ ρ − 1 ¯ χ ij ω i Ω j − u − 1 | D 2 ω | 2 + 4Φ( x ) u + ρ − 1 ρ ′ ( β − θ − u ¯ θ ) − ρ − 1 ρ ′′ |∇ Ω | 2 + 2 ρ − 2 ( ρ ′ ) 2 |∇ Ω | 2 + (1 + ρ − 1 | ρ ′ | ) e O ( u 1 2 )] = ρu [2 ρ − 1 ρ ′ ¯ χ ij ω i ω j − u − 1 | D 2 ω | 2 + 4Φ( x ) u − ρ − 1 ρ ′ ¯ θ u − ρ − 1 ρ ′′ |∇ Ω | 2 + 2 ρ − 2 ( ρ ′ ) 2 |∇ Ω | 2 + (1 + ρ − 1 | ρ ′ | ) e O ( u 1 2 )] . As |∇ Ω | 2 = |∇ ω | 2 + 2 ω i ˜ ω i + ˜ ω i g ij ˜ ω j = |∇ ω | 2 + e O ( u 1 2 ) , and u − 1 | D 2 ω | 2 ≥ 1 2 ρ − 2 ( ρ ′ ) 2 |∇ Ω | 2 = ρ − 2 ( ρ ′ ) 2 u + ρ − 2 ( ρ ′ ) 2 e O ( u 1 2 ) , 16 BEN LAMBER T AND JULIAN SCHEUER w e see that ˙ ˜ ϕ − ∆ ˜ ϕ ≤ ρu [2 ρ ′ ρ − 1 ¯ χ ij ω i ω j − ρ − 2 ( ρ ′ ) 2 u + 4Φ( x ) u − ρ − 1 ρ ′ ¯ θ u − 2 ρ − 1 ρ ′′ u + 4 ρ − 2 ( ρ ′ ) 2 u + (1 + ρ − 1 | ρ ′′ | + ρ − 2 | ρ ′ | 2 + ρ − 1 | ρ ′ | ) e O ( u 1 2 )] = ρu 2  4Φ( x ) + ρ ′ ρ − 1  4 − n n ¯ θ + 2 ˚ ¯ χ ij ω i ω j u − 1  − 2 ρ − 1 ρ ′′ + 3 ρ − 2 ( ρ ′ ) 2  + ( ρ + | ρ ′ | + | ρ ′′ | + ρ − 1 | ρ ′ | 2 ) e O ( u 3 2 ) . Substituting µ = 1 √ ρ , we hav e ρ ′ = − 2 µ ′ µ 3 , ( ρ ′ ) 2 ρ = 4( µ ′ ) 2 µ 4 , ρ ′′ = − 2 µ ′′ µ 3 + 6 ( µ ′ ) 2 µ and so ˙ ˜ ϕ − ∆ ˜ ϕ ≤ 4 µ ˜ ϕ 2 [Φ( x ) µ + Ψ µ ′ + µ ′′ ] +  µ − 2 + | µ ′ | µ − 3 + | µ ′ | 2 µ − 4 + | µ ′ | 2 µ − 1 + | µ ′′ | µ − 3  e O ( u 3 2 ) . The claim no w follo ws as previously . □ Finally , our conv ergence results rely on monotonic mo vemen t of the flo w, hence w e require the following evolution whic h immediately follo ws from ( 3.3 ). 3.9. Lemma. Under ( 3.4 ) , the evolution of the sp e e d f = β − H is given by (3.5) ˙ f = ∆ f + 2 τ ( ∇ f ) + f ( ¯ L ( β ) + ¯ χ ij h ij + D i τ i + | τ | 2 + Rc( ¯ L, ν ) +  ¯ R ( ν, ¯ L ) ν, ¯ L  ) . 4. Sp acetime CMC folia tions near a MOTS and pr oof of Theorem 1.1 W e now demonstrate that there exists a foliation under suitable assumptions. First we recall the definition of a sp ac etime c onstant me an curvatur e surfac e (STCMC), which has b een studied recently in several pap ers, e.g. [ 4 , 11 ]. 4.1. Definition. A hypersurface Σ of ¯ N is a λ -STCMC surface if on Σ, |  H | 2 = 2 H ¯ θ = λ. W e sa y a smo oth foliation made up of graphs Σ ξ := { ( ω ( z , ξ ) , z ) : z ∈ Σ } ab o ve Σ is strictly increasing if ω ξ > 0 everywhere. W e sa y that a λ -STCMC surface is stable if there is a smo oth function f > 0 on Σ with L Σ f > 0 where L Σ f = − ∆ f − 2 τ ( ∇ f ) + f B and (4.1) B := − ˚ ¯ χ ij h ij − λ 2 ( ¯ θ − 2 | ˚ ¯ χ | 2 + 2 n + ¯ θ − 2 Rc( ¯ L, ¯ L )) − D i τ i − | τ | 2 − Rc( ¯ L, ν ) − ¯ g ( ¯ R ( ν, ¯ L ) ν, ¯ L )) . W e observ e the follo wing lemma. 4.2. Lemma. A c omp act λ -STCMC hyp ersurfac e Σ is stable if and only if ther e is lo c al ly a smo oth strictly incr e asing foliation of hyp ersurfac es Σ ξ ab ove Σ (expr esse d as gr aphs ω ( z , ξ ) ) which have ∂ ∂ ξ    ξ =0 |  H ξ | 2 > 0 . FOLIA TION OF NULL CONES BY STCMC SURF ACES 17 Pr o of. By equation ( 3.3 ) and Prop osition 3.6 , for a general v ariation ˙ x = f ¯ L we hav e d dt |  H | 2 = 2 f H ¯ L ( ¯ θ ) + 2 ¯ θ ˙ H = 2 ¯ θ [ − ∆ f − 2 τ ( ∇ f ) + f B ] where B = − ¯ χ ij h ij − H ¯ θ − 1 | ˚ ¯ χ | 2 − H 1 n ¯ θ − D i τ i − | τ | 2 − Rc( ¯ L, ν ) − ¯ g ( ¯ R ( ν, ¯ L ) ν, ¯ L ) − H ¯ θ − 1 Rc( ¯ L, ¯ L )) = − ˚ ¯ χ ij h ij − H ( ¯ θ − 1 | ˚ ¯ χ | 2 + 2 n ¯ θ + ¯ θ − 1 Rc( ¯ L, ¯ L )) − D i τ i − | τ | 2 − Rc( ¯ L, ν ) − ¯ g ( ¯ R ( ν, ¯ L ) ν, ¯ L )) . Giv en a strictly increasing foliation of h yp ersurfaces ab ov e Σ then we may set f = ω ξ | ξ =0 > 0. Then, using the abov e (swapping the parameter t with ξ ), 0 < ∂ ∂ ξ    ξ =0 |  H ξ | 2 = 2 ¯ θ [ − ∆ f − 2 τ ( ∇ f ) + f B ] so Σ is stable. On the other hand, giv en a stable λ -STCMC h ypersurface Σ with graph function ω Σ ( z ), then we ma y define a strictly increasing foliation by ω ( z , ξ ) = ω Σ ( z ) + ξ f ( z ). This has ω ξ = f so by the abov e computation ∂ ∂ ξ    ξ =0 |  H ξ | 2 > 0. □ R emark. (i) Clearly , a sufficien t condition for stabilit y is the condition B > 0. In this case w e may simply take f = 1. In practice, this is easier to verify . (ii) In general to hav e a positive function satisfying the ab o ve, the maxim um principle implies that it is necessary that B > 0 somewhere (for the function to hav e a p ositive minimum), but in general B > 0 will not b e a necessary condition. Our metho d for pro ving Theorem 1.1 is to flow to λ -STCMC h yp ersurfaces b y the flo w (4.2) ˙ x = ( λ 2 ¯ θ − 1 − H ) ¯ L, whic h is a sp ecial case of ( 3.4 ) with β = λ 2 ¯ θ − 1 . W e define the pr escription of the flow to b e the constant λ . W e b egin with several simple consequences of the maximum principle. Supp ose that Σ 1 and Σ 2 are defined by graph functions ω 1 and ω 2 . W e sa y Σ 2 is ab ov e Σ 1 if ω 2 ≥ ω 1 . W e sa y Σ 2 is strictly ab ov e Σ 1 if ω 2 > ω 1 . 4.3. Lemma. Supp ose that Σ 1 and Σ 2 have me an curvatur e ve ctors  H 1 and  H 2 r esp e ctively, so that Σ 2 is ab ove Σ 1 and, c onsider e d as functions in the gr aphic al p ar ametrisations, |  H 2 | 2 ≥ |  H 1 | 2 . Then either Σ 1 = Σ 2 or Σ 1 and Σ 2 ar e disjoint. Pr o of. F rom ( 2.6 ), the mean curv ature is related to the graph function b y |  H ( z ) | 2 = 2 ¯ θ ( ω ( z ) , z ) θ ( ω ( z ) , z ) + |∇ ω ( z ) | 2 ¯ θ 2 ( ω ( z ) , z ) − 2 ¯ θ ( ω ( z ) , z )∆ ω ( z )] = − b ( e D ω , ω , z ) − a ij ( ω , z ) e D 2 ij ω , where we write e D for the Levi-Civita connection on S 0 , w e note that b is some smo oth function and a ij = 2 ¯ θ ( ω ( z ) , z ) g ij ( ω , z ) is p ositive definite. 18 BEN LAMBER T AND JULIAN SCHEUER W e compute 0 ≤ |  H 2 | 2 − |  H 1 | 2 = − [ b ( ˜ D ω 2 , ω 2 , z ) − b ( ˜ D ω 1 , ω 1 , z ) + a ij ( ω 2 , z ) ˜ D ij ω 2 − a ij ( ω 1 , z ) ˜ D ij ω 1 ] = −E ( ω 2 − ω 1 ) where E = ˆ a ij ˜ D ij + ˆ b i ˜ D i + ˆ c and ˆ a ij = a ij ( ω 2 , z ) ˆ b i = ˆ 1 0 ∂ b ∂ p i ( τ e D ω 2 + (1 − τ ) e D ω 1 , τ ω 2 + (1 − τ ) ω 1 , z ) dτ ˆ c = ˆ 1 0 ∂ b ∂ ω ( τ e D ω 2 + (1 − τ ) e D ω 1 , τ ω 2 + (1 − τ ) ω 1 , z ) dτ + e D ij ω 1 ˆ 1 0 ∂ a ij ∂ ω ( τ ω 2 + (1 − τ ) ω 1 , z ) dτ . Set Ω = ω 2 − ω 1 . Hence, Ω ≥ 0 and for a uniformly elliptic operator, E Ω ≤ 0. The Lemma no w follo ws from the strong maxim um principle. □ 4.4. Lemma. Supp ose that for i ∈ { 1 , 2 } , x i satisfies ( 4.2 ) with pr escription λ i , for λ 1 ≤ λ 2 . If the c orr esp onding gr aph of x 2 is ab ove the one of x 1 at the initial time, then this pr op erty holds at al l later times. If λ 1 < λ 2 then the gr aph of x 2 is strictly ab ove the one of x 1 at al l p ositive times. Pr o of. As seen in Lemma 3.4 , for a flo w x = graph ω satisfying ( 4.2 ) with prescription λ , the graph function ω satisfies ˙ ω = λ 2 ¯ θ − θ − u ¯ θ = F ( ˜ D 2 ω , ˜ D ω , ω , · ) , where ˜ D is the connection on S 0 . Let ω i , i = 1 , 2, b e the graph functions corresp onding to x i , then the function ω 2 − ω 1 satisfies a linear equation with lo cally b ounded co efficien ts, as can be seen from a computation similar to the one in Lemma 4.3 . The first statement follows from the standard parab olic maximum principle, as for example in [ 14 , Lemma 2.3]. F or the second statemen t, note that we already kno w ω 2 ≥ ω 1 ev erywhere. A t a h yp othetical p oin t ( t, z ), at which Ω is zero, there holds 0 ≥ ∂ t ( ω 2 − ω 1 )( t, z ) ≥ λ 2 2 ¯ θ − 1 ( ω 2 , · ) − λ 1 2 ¯ θ − 1 ( ω 1 , · ) − θ ( ω 2 , · ) + θ ( ω 1 , · ) − 1 2 |∇ ω 2 | 2 ¯ θ ( ω 2 , · ) + 1 2 |∇ ω 1 | 2 ¯ θ ( ω 1 , · ) = 1 2 ( λ 2 − λ 1 ) ¯ θ − 1 ( ω 2 , · ) > 0 , whic h is a con tradiction. The claim no w follows. □ W e now pro vide a lo cal uniqueness of λ -STCMCs which are sufficiently close to a stable κ -STCMC. 4.5. Lemma. Supp ose that b Σ is a stable κ -STCMC with gr aph function b ω and a smo oth p ositive function f b e given for which L b Σ f > 0 . Then ther e is a c onstant ϵ = ϵ ( ¯ N , b Σ , f ) > 0 such that for any λ ≥ κ and any λ -STCMC hyp ersurfac e Σ with gr aph function ω , which lies ab ove b Σ and satisfies | b ω − ω | C 2 < ϵ , is unique. FOLIA TION OF NULL CONES BY STCMC SURF ACES 19 Pr o of. F or f as in the statemen t, w e consider the stability op erator on the p erturb ed manifolds Σ b ω + φf whose graphs are given by b ω + φf for some smo oth φ : b Σ → R . Then L Σ b ω + φf f = Q f ( · , φ, D φ, D 2 φ ) for some smooth Q f where Q f ( · , 0 , 0 , 0) > 0. By compac tness and con tin uit y there exists a δ > 0 such that for any ω with | b ω − ω | C 2 < δ we hav e L Σ ω f > 0 and so (4.3) d dξ    ξ =0 |  H Σ ω + ξ f | 2 > 0 . Supp ose there are tw o solutions ω 1 and ω 2 ab o ve b Σ both of whic h ha v e constan t spacetime mean curv ature λ ≥ κ and | b ω − ω i | C 2 < ϵ := δ 16(1+ | f | C 2 )(1+ | f − 1 | C 2 ) for i = 1 , 2. Supp ose for a contradiction that 0 < M := max(( ω 2 − ω 1 ) f − 1 ) . Then, w e kno w that M < δ 8 | f | C 2 and w e define the function ω ( · , ξ ) := ω 1 + ξ f for ξ ∈ [0 , M ]. W e note that | ω ( · , ξ ) − b ω | C 2 < | ω 1 − b ω | C 2 + ξ | f | C 2 ≤ δ 16 + δ 8 ≤ δ 4 . In particular, Σ ω ( · ,M ) has spacetime mean curv ature |  H ω ( · ,ξ ) | 2 = λ + ˆ M 0 d dξ |  H ω ( · ,ξ ) | 2 dξ > λ, where we used ( 4.3 ). No w ω ( · , M ) touc hes ω 2 from ab o ve, which is imp ossible by Lemma 4.3 . Hence M ≤ 0. A similar argument interc hanging ω 1 and ω 2 implies that ω 1 = ω 2 . □ Next, we demonstrate that small oscillation λ -STCMCs may alw a ys b e pro duced by the flo w. 4.6. Prop osition. Supp ose b Σ is a κ -STCMC with gr aph function b ω . Then ther e exists a δ = δ ( ¯ N , b Σ) > 0 such that the fol lowing holds: L et e Σ b e a hyp ersurfac e ab ove b Σ with e κ := min e Σ |  H | 2 ∈ ( κ, κ + 1) and gr aph function e ω wher e | b ω − e ω | C 0 < δ . F or λ ∈ [ κ, e κ ] , let x λ ( t, z ) = x ( t, z , λ ) b e the solutions to ( 4.2 ) with pr escription λ starting fr om e Σ . Then: (i) F or every existenc e time t , z ∈ S 0 and λ ∈ [ κ, e κ ] , the flow satisfies the a priori estimates | ω ( t, z , λ ) − b ω ( z ) | < δ and u ( t, z , λ ) < C ( ¯ N , b Σ , e Σ) . (ii) The solutions x λ exist for al l times t ∈ [0 , ∞ ) and ar e uniformly smo oth in t, z . (iii) The c orr esp onding gr aph functions ω ar e monotonic al ly de cr e asing in time. (iv) The c orr esp onding gr aph functions ω ar e monotonic al ly incr e asing in λ . (v) As t → ∞ , e ach flow ω ( t, · , λ ) c onver ges to a smo oth λ -STCMC surfac e given by b ω λ . The function b ω ( · , λ ) := b ω λ is monotonic al ly incr e asing in λ and has uniform smo oth estimates in z which ar e indep endent of λ . 20 BEN LAMBER T AND JULIAN SCHEUER Pr o of. The key to proving the ab ov e is in showing the a priori estimates given in bullet p oint (i). Applying the maximum principle to the evolution of f = β − H given in ( 3.5 ), we see that for all times that the flow exists, f ≤ 0. Hence the flowing graph functions ω ( t, z , λ ) are non-increasing in time. F urthermore, we observ e 0 < ω ( t, z , λ ) − b ω ( z ) < δ : The upp er estimate follows b y applying Lemma 4.4 , as the flows x λ ha v e λ > κ for all times t and any c hoice of δ . The lo wer estimate follows as otherwise there w ould exist a time of first touching t which would con tradict Lemma 4.3 . Firstly , w e pick δ 0 small enough so that { ( s, z ) : z ∈ S 0 , s = b ω ( z ) + r, r ∈ [0 , δ 0 ] } ⊂ ¯ N , and consider δ ≤ δ 0 . As λ ∈ [ κ, κ + 1), b y compactness, the prescription function β = λ 2 ¯ θ − 1 and its deriv atives are bounded, sp ecifically , | β | + | dβ | + | ¯ Lβ | < C 0 ( ¯ N , b Σ , κ ) . In Lemma 3.8 we therefore observe that b y compactness there is a C 1 = C 1 ( ¯ N , b Σ) (but indep enden t of λ ) so that the functions Φ and Ψ satisfy | Φ | ≤ c ( n )( ¯ θ 2 + | Rc( ¯ L, ¯ L ) | + | ˚ ¯ χ | 2 + ¯ θ | ˚ ¯ χ | ) < C 1 | Ψ | ≤ c ( n )( ¯ θ + | ˚ ¯ χ | ) < C 1 . W e consider the test function µ ( s ) = 1 − a 2 s 2 for s ∈ [0 , a − 1 ) where a > 0 will b e determined later. W e ha ve µ ≤ 1 , 0 ≥ µ ′ ≥ − 2 a, µ ′′ = − 2 a 2 , so Φ( x ) µ + Ψ( x ) µ ′ + µ ′′ ≤ C 1 + 2 aC 1 − 2 a 2 ≤ C 1 + C 2 1 − a 2 . Set a = p C 1 + C 2 1 + 1 and choose δ = min { 1 2 a − 1 , δ 0 } . Due to the b ounds on ω − b ω , this means that µ is smooth and p ositive on the flowing surface with uniform b ounds aw a y from zero and infinit y . Then for ˜ ϕ = uµ − 2 ( ω − b ω ) w e hav e ˙ ˜ ϕ − ∆ ˜ ϕ ≤ C (1 + ˜ ϕ 3 2 ) − 3 ˜ ϕ 2 . Hence ˜ ϕ has no increasing maxima for ˜ ϕ large enough, and so w e hav e the claimed gradient estimate and w e ha v e completed the claim in part (i). W e observe that part (i) implies uniform in ( t, λ ) C 1 -estimates of ω ( t, · , λ ). Applying PDE theory , the flow exists for all time, and for all k there is a C k , uniform in λ suc h that | ω ( · , · , λ ) | C k + α ; k + α 2 ([0 , ∞ ) ×S 0 ) < C k . P art (ii) no w follo ws. As noted earlier, b y the maxim um principle, f = β − H ≤ 0 for all the time. Hence the flo ws mo v e monotonically as stated in (iii). F urthermore, Lemma 4.4 implies that if λ 1 < λ 2 then the flow x λ 2 is ab ov e x λ 1 at all times, so the flows are monotonically increasing in λ , from which (iv) follo ws. F or eac h fixed λ , the flow is monotonically decreasing in time and bounded b elow, so there m ust b e a limit as t → ∞ . By Arzela–Ascoli this limit must b e smo oth and the flow m ust con v erge uniformly smoothly (b y uniform estimates and interpolation). This limit m ust b e a stationary p oin t as otherwise the flow cannot con v erge. Hence x ( · , t, λ ) con v erges to a STCMC FOLIA TION OF NULL CONES BY STCMC SURF ACES 21 surface b Σ λ . Finally , due to the monotonicit y of the flow, λ 1 < λ 2 implies that b Σ λ 2 is ab ov e b Σ λ 1 , completing (v). □ 4.7. Prop osition. Supp ose that b Σ ⊂ ¯ N is a stable κ -STCMC hyp ersurfac e. Then ther e exists an α > 0 such that ther e exists a c ontinuous λ -STCMC foliation of a futur e side d neighb ourho o d of b Σ for λ ∈ [ κ, κ + α ) . Pr o of. Let b Σ = graph b ω and supp ose that γ = min { ϵ, δ } where ϵ is as in Lemma 4.5 and δ is as in Prop osition 4.6 . F or ξ ∈ (0 , γ ) to b e determined, define ˜ ω = b ω + ξ f | f | C 0 . By diminishing γ further, on e Σ, there holds κ < |  H | 2 < κ + 1 , due to the pro of of Lemma 4.5 . W e note, | ˜ ω | C 3 ≤ | b ω | C 3 + γ | f | C 3 | f | − 1 C 0 . F urthermore | ˜ ω − b ω | C 0 < δ , so, applying Proposition 4.6 , w e obtain a family of STCMC solutions b ω ( · , λ ) for λ ∈ [ κ, ˜ κ ] such that | b ω ( · , λ ) | C 2 ,α < C ( b ω , γ , f ). F urthermore, as the b ω ( · , λ ) are b ounded b et w een b ω and ˜ ω , | b ω ( · , λ ) − b ω | C 0 ≤ ξ . Therefore by in terp olation, w e ma y choose ξ small enough so that | b ω ( · , λ ) − b ω | C 2 < ϵ. W e no w need to c heck that there can b e no “gaps” b etw een the manifolds pro duced in this w a y for λ small enough. Fix λ ∈ ( κ, ˜ κ ). Then b y monotonicity of b ω ( · , λ ) there are limits ω ± := lim i →∞ ω ( · , λ ± i − 1 ) . These limits are smo oth b y Arzela-Ascoli, and the conv ergence is smo oth b y interpolation and uniform estimates. Hence they are b oth λ -STCMC surfaces, eac h of whic h is C 2 ϵ -close to b ω , and so b y Lemma 4.5 , they are b ω λ . □ Pr o of of The or em 1.1 . Prop osition 4.7 implies that giv en a stable MOTS there is a p ositive s > 0 such that the foliation exists with λ -STCMC lea v es Σ λ for λ ∈ [0 , s ]. W e ha v e to show that the foliation is smo oth in the sense that λ 7→ ω ( · , λ ) is smooth. W e emplo y the implicit function theorem as in [ 9 , p. 305]. There, a smo oth op erator G ( λ, b ω ) = |  H | 2 ( b ω ) − λ is defined, and from the pro of of Lemma 4.2 w e observ e ∂ b ω G ( b ω , τ ) = 2 ¯ θ L Σ b ω , where Σ = graph( b ω ). W e note that the op erator L Σ is not self-adjoin t. How ev er, as discussed in detail in [ 1 , Sec. 4, Def. 5.1 and Def. 5.2], under the condition of stabilit y it has a strictly p ositiv e smallest real eigenv alue. Hence ∂ b ω G ( ω , λ ) is inv ertible and the implicit function theorem shows that the assignment λ 7→ b ω ( · , λ ) is smo oth and increasing. The foliation is also strictly increasing in the sense of our definition because taking the deriv ative with resp ect to λ , w e obtain 0 = ∂ λ G ( λ, b ω λ ) = ∂ λ |  H λ | 2 − 1 = 2 ¯ θ λ L b Σ λ ∂ λ b ω λ − 1 < 0 at a zero minimum of ∂ λ ω λ . Hence those zeros can not occur and we conclude σ > 0. 22 BEN LAMBER T AND JULIAN SCHEUER If | A Σ λ | < C , then by elliptic regularity theory , all higher deriv atives are uniformly b ounded. Hence we may take a limit to get a σ -STCMC surface Σ σ . If Σ σ is stable, we may apply Prop osition 4.7 to see that σ was not maximal. Finally , supp ose that ˜ Σ λ is a smo oth λ -STCMC surface contained in the foliated set for some λ ∈ [0 , σ ). In particular, there is a highest leaf of the foliation with mean curv ature λ high and a lo w est leaf of the foliation with mean curv ature λ low , whic h Σ λ in tersects. Lemma 4.3 implies that if ˜ Σ λ is not a leaf of the foliation, then λ high < λ and λ < λ low , whic h is imp ossible as λ low ≤ λ high . Hence ˜ Σ λ is a leaf of the foliation. □ 5. The prescribed mean cur v a ture pr oblem and proof of Theorem 1.2 The pro of of Theorem 1.2 pro ceeds by pro viding C 1 -estimates, from which ev erything else follo ws from parab olic regularity as in section 4 . First of all we note that from the maxim um principle, the h yp ersurfaces S 0 and Σ + are barriers for the flo w. The key to the C 1 -estimates is evident from the evolution equation of ϕ in Lemma 3.8 . A sufficient ingredien t for obtaining a b ound on u from this lemma is to find a p ositive test function µ = µ ( ω ) with the prop ert y (5.1) Φ µ ( ω ) + Ψ µ ′ ( ω ) + µ ′′ ( ω ) < 0 . F rom the structure of this ODE it is eviden t that finding a p ositive solution can p otentially b e hamp ered b y µ ′′ ha ving to b e very negativ e. Hence, in some cases, the allo w ed range for ω has to b e restricted. W e giv e the details in the following and pro v e that under the curren t conditions, the required test functions can be found. Before w e can do so, we ha v e to control ¯ θ on ¯ N giv en the v alidit y of ( 1.3 ). 5.1. Lemma. Supp ose that e quation ( 1.3 ) holds on ¯ N . Then for al l s ∈ [0 , Λ) , ( n − 1 + C R + c 2 ˚ ¯ χ ) − 1 ( n − 1 + C R + c 2 ˚ ¯ χ ) − 1 + s ≤ ¯ θ ( s, · ) ≤ ( n − 1 + c R ) − 1 ( n − 1 + c R ) − 1 + s . Pr o of. By the Rayc haudh uri equations in Prop osition 3.6 w e know that − ( n − 1 + C R + c 2 ˚ ¯ χ ) ¯ θ 2 ≤ ∂ s ( ¯ θ ) = ¯ L ( ¯ θ ) = − 1 n ¯ θ 2 − | ˚ ¯ χ | 2 − Rc( ¯ L, ¯ L ) ≤ − ( n − 1 + c R ) ¯ θ 2 . Hence, ( n − 1 + c R ) ≤ ∂ s ( ¯ θ − 1 ) ≤ ( n − 1 + C R + c 2 ˚ ¯ χ ) . As ¯ θ (0 , · ) = 1, b y integrating w e obtain ( n − 1 + C R + c 2 ˚ ¯ χ ) − 1 ( n − 1 + C R + c 2 ˚ ¯ χ ) − 1 + s ≤ ¯ θ ( s, · ) ≤ ( n − 1 + c R ) − 1 ( n − 1 + c R ) − 1 + s . □ Giv en this con trol on ¯ θ , we can solv e ( 5.1 ). 5.2. Lemma. Supp ose the validity of ( 1.3 ) on ¯ N = [0 , Λ) × S 0 and supp ose that we ar e given an interval [ a, b ] ⊂ [0 , Λ) . A dditional ly, supp ose that one of the fol lowing two c ases hold: (1) Either n ≤ 6 and (5.2) c ˚ ¯ χ ≤ c R 2 + 6 − n 4 n , FOLIA TION OF NULL CONES BY STCMC SURF ACES 23 (2) or n ≥ 7 and (5.3) 2 c ˚ ¯ χ + c 2 ˚ ¯ χ + C R ≤ n − 6 2 n . Then ther e is an explicit c onstant D ( n, c R , c ˚ ¯ χ , C R − c R ) which is smo oth in its last thr e e entries wher e D ( n, c R , 0 , 0) = n 2 (1+ nc R ) 2  ( n − 2)( n − 10) 4 n 2 + n +6 n c R + c 2 R  , such that the fol lowing holds: • If D ≥ 0 then ther e is a p ositive solution µ : [ a, b ] → R to ( 5.1 ) . • If D < 0 and b < n (1+ nc R )  e π √ −D − 1  , then ther e exists a p ositive solution µ : [ a, b ] → R to ( 5.1 ) . Pr o of. Our aim will be to solv e ( 5.1 ) by comparing this with solutions of the Euler-Cauc hy equations, using our estimates on ¯ θ from Lemma 5.1 to estimate the co efficients in ( 5.1 ). If Φ < 0 then it suffices to pick µ = 1. Hence, from now on we assume that there are points with Φ ≥ 0. Set W := ( n − 1 + c R ) − 1 = n 1 + nc R , Z := ( n − 1 + C R + c 2 ˚ ¯ χ ) − 1 = n 1 + n ( C R + c 2 ˚ ¯ χ ) ≤ W so that from Lemma 5.1 we see that (5.4) Z ≤ ( s + Z ) ¯ θ ≤ ( s + W ) ¯ θ ≤ W. Supp ose first that n ≤ 6. W e define the constants B 1 := ( W  n − 4 2 n + 2 c ˚ ¯ χ  if n − 4 2 n + 2 c ˚ ¯ χ > 0 Z  n − 4 2 n + 2 c ˚ ¯ χ  otherwise , A δ := W 2  n − 2 − c R 2 + 3 2 c 2 ˚ ¯ χ + | 4 − n | 2 n c ˚ ¯ χ  + δ, where δ ≥ 0 will b e c hosen later. Note that A δ has b een chosen so that Φ ≤ A δ − δ ( s + W ) 2 , so as Φ ≥ 0, we see that A δ ≥ δ . W e will shortly find µ = µ ( s ) > 0 which solves (5.5) 0 = A δ ( s + W ) 2 µ + B 1 s + W µ ′ + µ ′′ . Giv en suc h µ , b y our c hoice of A δ and the p ositivit y of µ , w e then estimate (5.6) Φ µ + Ψ µ ′ + µ ′′ ≤ A δ − δ ( s + W ) 2 µ + Ψ µ ′ + µ ′′ = (( s + W )Ψ − B 1 ) µ ′ s + W − δ µ ( s + W ) 2 . Note that b y our choice of B 1 and ( 5.4 ), ( s + W )Ψ − B 1 ≤ ( n − 4 2 n + 2 c ˚ ¯ χ )( s + W ) ¯ θ − B 1 ≤ 0 . Hence, our aim is to show that w e ma y find a solution µ to ( 5.5 ) with δ > 0, µ > 0 and µ ′ > 0, so that the right hand side of ( 5.6 ) is negativ e. W e note that b y our assumption ( 5.2 ) B 1 ≤ ( n − 1 + c R ) W = 1 . W e set D 1 ,δ = ( B 1 − 1) 2 − 4 A δ then: 24 BEN LAMBER T AND JULIAN SCHEUER • If D 1 ,δ ≥ 0, let µ : [0 , ∞ ) → R b e given by µ ( s ) = ( s + W ) 1 2 (1 − B 1 + √ D 1 ,δ ) whic h solv es ( 5.5 ) and satisfies the required prop erties µ > 0 and µ ′ > 0, b ecause B 1 ≤ 1. • If D 1 ,δ < 0, then for some η ∈ (0 , π 2 ) define I 1 ,δ,η = [0 , W ( e ( π − 2 η ) / √ − D 1 ,δ − 1)) and let µ : I 1 ,δ,η → R b e giv en b y µ ( s ) = ( s + W ) 1 2 (1 − B 1 ) sin  √ − D 1 ,δ 2 log( W − 1 s + 1) + η  , whic h solv es ( 5.5 ) and, as B 1 ≤ 1, has b oth µ > 0 and µ ′ > 0 on I 1 ,δ,η . W e no w choose δ > 0 to ensure the strict inequality on the claimed in terv al. W e define D = lim δ → 0 D 1 ,δ = ( B 1 − 1) 2 − 4 A 0 and note that D 1 ,δ is monotonically decreasing in δ . • If D > 0, pic k δ > 0 sufficiently small that D 1 ,δ > 0, then µ as described ab ov e will solv e ( 5.1 ). • If D ≤ 0, then by a suitable choice of δ w e can achiev e W ( e π / √ − D 1 ,δ − 1) > b . As the ends of the in terv al v ary con tinuously with η , we ma y pic k η > 0 small enough so that [0 , b ] ⊂ I 1 ,δ,η . Giv en these choices, µ restricted to [ a, b ] with D 1 ,δ < 0 as abov e satisfies ( 5.1 ) as required. No w supp ose that n ≥ 7. Our aim is to follow an identical pro cess with A δ , W and Z defined as ab ov e, but we will replace B 1 with B 2 := Z  n − 4 2 n − 2 c ˚ ¯ χ  . Similarly to the previous cases we will shortly choose µ > 0 to b e a solution of (5.7) 0 = A δ ( s + W ) 2 µ + B 2 s + W µ ′ + µ ′′ , so that, as in ( 5.6 ), Φ µ + Ψ µ ′ + µ ′′ ≤ (( s + W )Ψ − B 2 ) µ ′ s + W − δ µ ( s + W ) 2 . Note that ( 5.3 ) implies that 2 c ˚ ¯ χ ≤ n − 4 2 n . Hence using equation ( 5.4 ), ( s + W )Ψ − B 2 ≥ ( n − 4 2 n − 2 c ˚ ¯ χ )( s + W ) ¯ θ − B 2 ≥  n − 4 2 n − 2 c ˚ ¯ χ  Z − B 2 = 0 . Therefore this time, we searc h for solutions of ( 5.7 ) with δ > 0, µ > 0 and µ ′ < 0 to render the right hand side of ( 5.6 ) negative. W e note that ( 5.3 ) implies B 2 ≥ Z  1 n + c 2 ˚ ¯ χ + C R  = 1 . W e set D 2 ,δ = ( B 2 − 1) 2 − 4 A δ then: FOLIA TION OF NULL CONES BY STCMC SURF ACES 25 • If D 2 ,δ ≥ 0, let µ : [0 , ∞ ) → R b e given by µ ( s ) = ( s + W ) 1 2 (1 − B 2 + √ D 2 ,δ ) , whic h s olv es ( 5.5 ) and satisfies the required prop erties µ > 0 and µ ′ < 0. • If D 2 ,δ < 0, then for some η ∈ (0 , π 2 ) define I 2 ,δ,η = [0 , W ( e ( π − 2 η ) / √ − D 1 ,δ − 1)) and let µ : I 2 ,δ,η → R b e giv en b y µ ( s ) = ( s + W ) 1 2 (1 − B 2 ) cos  √ − D 1 ,δ 2 log( W − 1 s + 1) + η  , whic h s olv es ( 5.5 ) and, as B 2 ≥ 1, has b oth µ > 0 and µ ′ < 0 on the stated in terv al. As in the previous case, set D := lim δ → 0 D 2 ,δ = ( B 2 − 1) 2 − 4 A 0 and note that D 2 ,δ is monotonically decreasing in δ . Picking δ, η > 0, a contin uit y argument as in the case n ≤ 6 completes the pro of. Finally , we chec k that in either of the cases, D has the claimed form. Note that W , Z , B 1 , B 2 , A 0 are all smooth functions of c R , C R − c R and c ˚ ¯ χ (for 0 ≤ c R , C R − c R , c ˚ ¯ χ ), and hence, D is smo oth in each of its last three entries. If C R − c R = c ˚ ¯ χ = 0 then W = Z = n 1+ nc R , B 1 = B 2 = n − 4 2 n W and A 0 = W 2 ( n − 2 − c R 2 ). Hence W − 2 D = ( W − 1 B 1 − W − 1 ) 2 − 4 W − 2 A 0 = ( n − 6 2 n − c R ) 2 − 4 n − 2 + 2 c R = ( n − 2)( n − 10) 4 n 2 + 6 + n n c R + c 2 R , as claimed. □ W e pro ceed with the C 1 -estimates. 5.3. Lemma. Under the assumptions of The or em 1.2 , the flow ( 3.4 ) with β = ρ/ (2 ¯ θ ) satis- fies uniform C 1 -estimates during the evolution and, in turn, satisfies smo oth estimates and c onver ges to a solution to |  H | 2 = ρ. Pr o of. W e pic k C 0 ( n ) and C 1 ( n ) so that conditions ( 5.2 ) and ( 5.3 ) in Lemma 5.2 hold. By the maximum principle, the flow remains b etw een S 0 and Σ + . The assumption on Σ + is precisely there to ensure that, using Lemma 5.2 , we can build a test function ϕ = uµ − 2 ( ω ) as in Lemma 3.8 , which ensures a strictly negative sign on the highest order term in the ev olution equation of ϕ , which in turn yields a gradien t b ound on the flowing manifolds. Here w e also crucially use that the flo w remains in the region b et ween S 0 and Σ + . Using the C 1 - b ounds, standard b o otstrapping gives smo oth estimates and, from the monotonicity of the flo w, con v erges to a stationary limit of the flow. □ 5.4. R emark. Supp ose that S 0 is a MOTS. Then for c ˚ ¯ χ , c R , C R , D as in Theorem 1.2 . Define M = ( Λ if D ≥ 0 min n n 1+ nc R ( e π √ −D − 1) , Λ o otherwise. Then, either • the foliation of Theorem 1.1 lea ves ev ery compact subset of S 0 × [0 , M ), or • there is a smo oth unstable σ -STCMC Σ σ ⊂ S 0 × [0 , M ) and the region b etw een S 0 and Σ σ is smo othly foliated b y STCMC h yp ersurfaces. 26 BEN LAMBER T AND JULIAN SCHEUER Pr o of. In the construction, replace Prop osition 4.6 (i) with the following gradien t estimate, whic h depends only on c ˚ ¯ χ , C R , C R − c R , D , but is independent of initial data. As in the previous Lemma, we can build a test function ϕ = uµ − 2 ( ω ) as in Lemma 3.8 , with evolution given b y ˙ ϕ − ∆ ϕ = − 2 cϕ 2 + C ( ϕ 3 2 + 1) . for some constan ts c, C > 0 dep ending only on c ˚ ¯ χ , C R , C R − c R , D . W e start by estimating ϕ for small times. By ODE comparison, w e hav e ϕ ≤ Θ( t ) where ˙ Θ ≤ − 2 c Θ 2 + C (Θ 3 2 + 1) ≤ − c Θ 2 + C and Θ(0) = max M 0 ϕ . Define ˜ Θ := max  0 , Θ − 2 q C 2 c  so that when the deriv ative exists, ∂ t ˜ Θ ≤ ( − 2 c Θ 2 + C , if Θ > 2 q C 2 c 0 otherwise ≤ ( − c Θ 2 , if Θ > 2 q C 2 c 0 otherwise ≤ ( − c ˜ Θ 2 , if Θ > 2 q C 2 c 0 otherwise = − c ˜ Θ 2 , so solving this we may estimate ϕ ( · , t ) ≤ 2 r C 2 c + 1 ct , indep enden tly of initial data. Therefore, we ha v e an estimate on ϕ at time t = 1 which is indep enden t of initial data. Applying the maximum principle b ey ond this p oin t implies a b ound on ϕ dep ending only on c ˚ ¯ χ , C R , C R − c R , D for t ≥ 1. Standard PDE estimates imply C k estimates on the flow. Hence the limit surfaces Σ λ satisfy uniform smo oth estimates indep enden tly of initial data. Hence if the foliation doesn’t leav e ev ery compact set of S 0 × [0 , M ), then all leav es satisfy uniform smo oth b ounds. 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[17] Henri Ro esch and Julian Scheuer, Me an curvatur e flow in nul l hyp ersurfac es and the dete ction of MOTS , Comm un. Math. Phys. 390 (2022), no. 3, 1149–1173. [18] Paul T o d, L o oking for mar ginal ly tr app e d surfaces , Class. Quantum Grav. 8 (1991), no. 5, 115–118. [19] Markus W olff, R ic ci flow on surfac es along the standar d lightc one in the 3 + 1 -Minkowski sp ac etime , Calc. V ar. P artial Differ. Equ. 62 (2023), no. 3, art. 90. School of Ma thema tics, The University of Leeds, Leeds, LS2 9JT, UK Email addr ess : b.s.lambert@leeds.ac.uk Goethe-Universit ¨ at Frankfur t am Main, Institut f ¨ ur Ma thema tik, Rober t-Ma yer-Str. 10, 60325 Frankfur t, Germany Email addr ess : scheuer@math.uni-frankfurt.de