he second fundamental formh0ij,1≤i,j≤n−1 of Σρwith respect to the normalen=∂∂ρis given by(1.3)ωni=n−1∑j=1h0ijωj.

he second fun-damental formhijof Σρwith respect tods2is given by(1.6)hij=u−1h0ij.

Theorem 2.1.The initial value problem (2.1) has a unique solutionuonΣ0×[0,∞)such that(a)u(z) = 1 +m0ρn−2+vwherem0is a constant andvsatisfies|v|=Oρ1−n
and|∇0v|=O(ρ−n);(b)The metricds2=u2dr2+gris asymptotically flat in the sense of (2.23) with scalarcurvatureR≡0outsideΣ0;(c)The ADM massmADMofds2is given byc(n)mADM= (n−1)ωn−1m0= limr→∞ZΣrH0(1−u−1)dσr= limr→∞ZΣr(H0−H)dσr,for some positive constantc(n), whereH0andHare the mean curvatures ofσrwith respect to the Euclidean metric andds2respectively.

pontaneous collapse theories

Forsimplicity, let us assume that the boundary of Ω has only one component.Letι: Σ :=∂Ω→Rnbe its isometric embedding. Letν:ι(Σ)→Sn−1be the outer unit normal. Sinceι(Σ) is assumed to be a strictly convexhypersurface inRnthere is a smooth family of embeddingsF: Σ×[0,∞]→RnwhereFt(σ) =F(σ, t) =ι(σ) +tν(ι(σ)).Note thatFt(Σ) are the ‘outer’ distance surfaces ofι(Σ). IfˆΩ denotes thebounded domain enclosed byι(Σ), then{Ft(Σ)}t≥0foliatesRn\ˆΩ and theEuclidean metric on this set can be written asG=dt2+gt,wheregtis the first fundamental form of the embeddingFt: Σ→Rn.

we use the dilationinvariance of weighted H ̈older norms together with suitable curvature conditionsto obtain uniform bounds of solutions to the initial value problem (1) with initialconditionu−1(1 +ǫ,·) on [1 +ǫ,∞). By Arzela-Ascoli Theorem, there exists aweak solution to (1) withu−1(1,·) = 0 (Theorem 2). S

We introduce the scaling transformation ̃u(t) =√tt+ 1u(t+ 1) wheret∈(0,∞).

∇0and∇20are the gradient and Hessianoperator of the Euclidean metric respectively. If we writeu2dr2+gr=∑i,jgijdzidzj.Then direct computations show (see the computations in (2.24), (2.27) below, for example):(2.23)|gij−δij|+ρ|∇0gij|+ρ2|∇20gij|≤Cρ2−n.By the result in [B1], the ADM mass of the metricds2=u2dr2+gris well defined, becausethe scalar curvature ofds2is zero outside a compact set.

the scalar curvatureRofds2is given byR= (1−u−2)Rρ+u−2n−1∑i,jR0ijij+ 2n−1∑i=1Rnini= (1−u−2)Rρ+u−2R0−2u−1∆ρu+ 2u−3∂u∂ρH0whereR0is the scalar curvature ofNwith respect tods20andRρis the scalar curvatureof Σρwith the induced metric.

Theorem 1.1.LetMn1andMn2be hypersurfaces ofNn+1that are tan-gent atpand let0be a unitary vector that is normal toMn1atp. SupposethatMn1remains aboveMn2in a neighborhood ofpwith respect to0. De-note byH1r(x)andH2r(x)ther-mean curvature atx2WofMn1andMn2,respectively. Assume that, for somer,1rn, we haveH2r(x)H1r(x)in a neighborhood of zero; ifr2, assume also that2(0), the principal cur-vature vector ofM2at zero, belongs tor. ThenMn1andMn2coincide in aneighborhood ofp

LetMn1andMn2be hypersurfaces ofNn+1that are tangentatp, i.e., which satisfyTpM1=TpM2. Fix a unitary vector0that is normaltoMn1atp. We say thatMn1remains aboveMn2in a neighborhood ofpwith respect to0if, when we parametrizeMn1andMn2by'1and'2asin (1.1), the corresponding functions1and2satisfy1(x)2(x) in aneighborhood of zero.

Lemma 4.2.The functionm(r) =ZΣrH0(1−u−1)dσris nonincreasing inr, whereH0is the mean curvature ofΣrinRn.

we can solve (2.1)with initial valueu−10= 0. In fact, by Lemma 2.2,u0satisfies:1−exp−Zr0ψ(s)ds−12≤u0(x,r)≤1−exp−Zr0φ(s)ds−12.This means that Σ0is a minimal surface with respect to the asymptotically flat metricu2dr2+gr.

solve (2.1) and show that the metricds2=u2dr2+gris asymptotically flatoutside Σ0. We will also compute the mass ofds2.

Let Σ0be a compact strictly convex hypersurface inRn,Xbe the position vector ofa point on Σ0, and letNbe the unit outward normal of Σ0atX. Let Σrbe the convexhypersurface described byY=X+rN, withr≥0. The Euclidean space outside Σ0canbe represented by(Σ0×(0,∞),dr2+gr)wheregris the induced metric on Σr. Consider the following initial value problem(2.1)2H0∂u∂r= 2u2∆ru+ (u−u3)Rron Σ0×[0,∞)u(x,0) =u0(x)whereu0(x)>0 is a smooth function on Σ0,H0andRrare the mean curvature and scalarcurvature of Σrrespectively, and ∆ris the Laplacian operator on Σr.

u2dρ2+gρhas the scalar curvatureR, if and onlyifusatisfies(1.10)H0∂u∂ρ=u2∆ρu+12(u−u3)Rρ−12uR0+u32R.

Given a functionRonN, we want to find the equation forusuch that(1.2)ds2=u2dρ2+gρhas scalar curvatureR.

Let Σ be a smooth compact manifold without boundary with dimensionn−1 and letN= [a,∞)×Σ equipped with a Riemannian metric of the form(1.1)ds20=dρ2+gρfor a point (ρ,x)∈N. Heregρis the induced metric on Σρwhich is the level surfaceρ=constant

By (4), we haveddtZΣ×{t}(Hη−Hu)dσt!=ZΣ×{t}(η−1−u−1)H21+K(η−u)−12(η−1−u−1)(H21+|h1|2)dσt.(9)By the Gauss equation and the assumption thatRic(gη) = 0, we have(10)2K=H2η−|hη|2=η−2(H21−|h1|2).Therefore, it follows from (9) and (10) that(11)ddtZΣ×{t}(Hη−Hu)dσt!=−ZΣ×{t}K(η−u)2u−1dσt≤0,where we also used the assumption thatK >0.

Assumption:The scalar curvatureR(gt) =: 2Kofgtand the meancurvatureH1of the leaves Σ×{t}with respect tog1are everywhere positive.Proposition 2(cf. [2], [23], [22]).Under the above assumption, given anypositive functionu0onΣ×{0}, there is a smooth positive functionuonΣ×[0, t0]such that the scalar curvatureR(gu)ofguis identically zero andu|t=0=u0.

Lemma 2.3.(2.1) has a unique solutionufor allrwhich satisfies the estimates in Lemma2.2.

Let Σ0be a smooth compact strictly convex hypersurface inRn. Letrbe the distance function from Σ0. Then the metric on the exteriorNof Σ0is given bydr2+gr, wheregris the induced metric on Σr, which is the hypersurface with distancerfrom Σ0. The functionuwith prescribed scalar curvatureR= 0 is given by2H0∂u∂r= 2u2∆ru+ (u−u3)RrwhereH0is the mean curvature of Σr,Rris the scalar curvature of ΣrandR0is the scalarcurvature of Σrwith the induced metric fromRnand ∆ris the Laplacian on Σr.

θ dμ ≥ p 16 π | Σ |

GIBBONS-PENROSE INEQUALITY

θ dμ ≥ p 16 π | Σ |