hypersurfaces in Schwarzschild spacetimes (I):

PREPRINT

國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University

www.math.ntu.edu.tw/ ~ mathlib/preprint/2011- 11.pdf

Spacelike spherically symmetric CMC

hypersurfaces in Schwarzschild spacetimes (I):

Construction

Kuo-Wei Lee and Yng-Ing Lee

November 8, 2011

Spacelike spherically symmetric CMC hypersurfaces in Schwarzschild spacetimes (I): Construction

Kuo-Wei Lee and Yng-Ing Lee

Abstract

We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behav- ior near the coordinate singularity r = 2M . Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.

1 Introduction

The Schwarzschild spacetime is the simplest model of a universe containing a star. Its metric is a solution of the vacuum Einstein equations, and is spherically symmetric, asymptotically flat, and Ricci flat. A more remarkable fact is that the Schwarzschild metric is the only spherically symmetric vacuum solution of the Einstein equations.

Spacelike constant mean curvature (S-CMC) hypersurfaces in spacetimes have been con- sidered important and interesting objects in studying the dynamics of spacetime and in general relativity. We refer to [7] for more discussions on the importance of S-CMC hyper- surfaces. From the viewpoint of geometry, aS-CMChypersurface in spacetimes has extremal surface area with fixed enclosed volume [1]. This property is similar to that of a compact

CMC hypersurface in Euclidean spaces.

In this paper, we study spacelike spherically symmetric constant mean curvature (SS- CMC) hypersurfaces in Schwarzschild spacetimes and Kruskal extension. We solve SS-CMC

hypersurfaces in both exterior and interior of the Schwarzschild spacetime, and then analyze their asymptotic behavior, especially at r = 2M. The Kruskal extension is an analytic extension of the Schwarzschild spacetime. When SS-CMC hypersurfaces are mapped to the Kruskal extension, we find relations between SS-CMC hypersurfaces in exterior and interior

2010 Mathematics Subject Classification: Primary 83C15, Secondary 83C05.

such that they can be joined at least C². These statements can be seen in Theorem 1–

5. Furthermore, we get all complete SS-CMC hypersurfaces in the Kruskal extension in Theorem 6.

Our motivation on studying SS-CMChypersurfaces is on one hand that they are easier to deal with and have explicit expressions, and on the other hand that these examples can serve as barrier functions for the general non-symmetric cases. We hope that a deep understanding of these solutions can help us to find right formulation of other general questions in the Schwarzschild spacetime such as Dirichlet problem and etc. After we finished the results in this paper, we found that the problem was also studied by Brill, Cavallo, and Isenberg in [1], and Malec, and ´O Murchadha in [5, 6]. However, the approaches are quite different.

Our viewpoint is purely geometrical and the explicit formula derived in this paper has the advantage on verifying foliation properties conjectured in [5]. This part will appear in a forthcoming paper [4].

The authors want to thank Quo-Shin Chi, Mao-Pei Tsui, and Mu-Tao Wang for their interests and discussions. The first author also like to express his gratitude to Robert Bartnik and Pengzi Miao for helpful suggestions and hospitality when he visited Monash University in 2010. The second author is partially supported by the NSC research grant 99-2115-M-002-008 in Taiwan.

The organization of this paper is as follows. We first give a brief summary of the Schwarzschild spacetime and the Kruskal extension in section 2. A good reference for this part is Wald’s book [8]. In sections 3–5, we study SS-CMChypersurfaces in each region and analyze their asymptotic behavior, especially at r = 2M. How to glue these solutions into complete SS-CMC hypersurfaces and the smoothness properties of the gluing are discussed in section 6.

2 The Kruskal extension

The Schwarzschild spacetime, denoted by S, has a metric ds² = −



1 − 2M r



dt²+ 1 1 − ^2M_r
dr

2+ r²dθ²+ r²sin²θdφ². (1)

We often write h(r) = 1 − ^2Mr . The metric (1) is not defined at r = 0 and r = 2M, and looks singular at both places. But in fact, the Schwarzschild spacetime is nonsingular at r = 2M. It is only a coordinate singularity, which is caused merely by a breakdown of the coordinates. There is a larger spacetime including the Schwarzschild spacetime as a proper subset and it has a smooth metric, especially for points corresponding to r = 2M. Such an analytic extension was obtained by Kruskal in 1960.

Proposition 1. [3, 8] The Schwarzschild metric can be written as ds² =16M²e^−2Mr

r (−dT²+ dX²) + r²dθ² + r²sin²θdφ²

=16M²e^−2Mr

r dUdV + r²dθ²+ r²sin²θdφ², (2) where







(r − 2M)e^2Mr = X²− T² = V U t

2M = ln    

X+ T X− T    

= ln     V U    

. (3)

The metric (2) is nonsingular at r = 2M.

A spacetime diagram for the Kruskal extension is shown in Figure 1. Each point in the Kruskal plane represents a sphere. There is one-to-one and onto correspondence from the region I to the Schwarzschild exterior r > 2M, and from the region II to the Schwarzschild interior 0 < r < 2M. The whole Kruskal extension is the union of regions I, II, I’, and II’, where regions I’ and II’ are exterior and interior of another Schwarzschild spacetime, respectively.

I II

II’

I’

X+ T = 0 X²− T² = −2M X²− T² = −2M X− T = 0

X T

Figure 1: The Kruskal extension of a Schwarzschild spacetime.

From (3), we know that each r = constant in the Schwarzschild spacetime is a hyperbola in the Kruskal extension, and each t = constant in the Schwarzschild spacetime is two half-lines starting from the origin in the Kruskal extension. Images of r = constant and t= constant under the correspondence are illustrated in Figure 2.

I II

II’

I’

X

T t= 0

t = 0

r= constant < 2M r= 2M, t = ∞ r= 0

t= t0 >0 t= t0 >0

r= 2M, t = −∞

r= 0

r= constant > 2M

Figure 2: Level sets r = constant and t = constant.

I II

II’

I’

L+

L₊

L₊ L₊

L−

L−

L₋ L₋

r = 2M, −∞ < t < ∞

L+ : r = 2M, t = ∞

L− : r = 2M, t = −∞

glued glued

glued glued

r= 0 r = 0

Figure 3: The gluing of Schwarzschild exteriors and interiors.

Now we explain how the Schwarzschild exterior and interior change as they map into the Kruskal extension. The boundary r = 2M, −∞ < t < ∞ of the Schwarzschild exterior and interior blow down to the origin in the Kruskal extension. On the other hand, r = 2M, t =

∞ and r = 2M, t = −∞ blow up to half-lines L⁺ and L− in the Kruskal extension, respectively. The L+ of I is glued to the L+ of II, and the L− of II is glued to the L− of I’, and so on. Moreover, r = 0 is mapped to the hyperbola X²− T² = −2M in the Kruskal

u=−2 u=−1 u=0 u=1 u=2

v=−2 v=−1 v=0 v=1 v=2

v=−2 v=−1 v=0 v=1 v=2 u=2 u=1 u=0 u=−1 u=−2

r t

u=−2 u=−1 u=0 u=1u=2

v=−2 v=−1 v=0 v=1 v=2 v=−2v=−1

v=0v=1 v=2

u=1 u=2 u=−1u=0 u=−2

X T

Figure 4: Null geodesics in the Schwarzschild spacetime and Kruskal extension.

extension. This identification is pictured in Figure 3.

The idea to the construction of the Kruskal extension is using null geodesics. When omitting the spherically symmetric part and solving null geodesics in t-r plane, we can define null coordinates u, v by

u= t − (r + 2M ln |r − 2M|) and v = t + (r + 2M ln |r − 2M|).

These coordinate curves are mapped to ±45^◦ straight lines in the Kruskal extension. Fig- ure 4 presents u = constant and v = constant in the Schwarzschild spacetime and Kruskal extension. Furthermore, we can define null coordinates (U, V ) in the Kruskal extension by

Region I Region II Region I’ Region II’

U e^−4Mu −e^−4Mu −e^−4Mu e^−4Mu

V e^4Mv e^4Mv −e^4Mv −e^4Mv .

Direct computation from (3) gives the relations between (X, T ) and (r, t) as follows:

In region I, X =

√r− 2M(e^r+t4M + e^r−t4M)

2 and T =

√r− 2M(e^r+t4M − e^r−t4M)

2 .

In region II, X =

√2M − r(e^r+t4M − e^r−t4M)

2 and T =

√2M − r(e^r+t4M + e^r−t4M)

2 .

In region I’, X = −

√r− 2M(e^r+t4M + e^r−t4M)

2 and T = −

√r− 2M(e^r+t4M − e^r−t4M)

2 .

In region II’, X = −

√2M − r(e^r+t4M − e^r−t4M)

2 and T = −

√2M − r(e^r+t4M + e^r−t4M)

2 .

In this article, we always take ∂T as future directed timelike vector field. In region I, the vector ∂T points to the direction of increasing t, while in region II it points to the direction of decreasing r. On the other hand, ∂T points to the direction of decreasing t in region I’

and points to the direction of increasing r in region II’.

3 SS-CMC solutions in region I

A vector v is spacelike if hv, vi > 0, null if hv, vi = 0, and timelike if hv, vi < 0. Given a smooth function F on the Schwarzschild spacetime (S, ds²) with ds² as in (1), denote a level set of F by Σ = {x ∈ S | F (x) = constant}, then ∇F is a normal vector field of Σ. If Σ is spacelike, that is, Σ has a positive definite metric induced from (S, ds²), then ∇F forms a timelike normal vector field on Σ. Since

∇F = g^ttFt∂t+ g^rrFr∂r+ g^θθFθ∂θ+ g^φφFφ∂φ

= − 1

h(r)Ft∂t+ h(r)Fr∂r+ 1

r²Fθ∂θ+ 1

r²sin²θFφ∂φ, the spacelike condition on Σ is equivalent to

h∇F, ∇F i < 0 ⇔ − 1

h(r)F_t²+ h(r)F_r²+ 1

r²F_θ²+ 1

r²sin²θF_φ²<0. (4) When Σ is a level set of F and is spacelike, we can without loss of generality assume that ∇F is future directed (or replace F by −F ). That is,

N = ∇F

p−h∇F, ∇F i is future directed unit timelike normal vector field on Σ.

Let {eⁱ}³i=1 be a basis on Σ, then mean curvature of Σ is H = 1

3

3

X

i=1

g^ijhij = 1 3

3

X

i=1

g^ijh∇^eiN, eji = 1 3

3

X

i=1

g^ij

p−h∇F, ∇F ih∇^ei(∇F ), e^ji.

3.1 SS-CMC solutions in region I

We start to study SS-CMC solutions in the Schwarzschild exterior which maps to the region I in the Kruskal extension.

Proposition 2. SupposeΣ¹ = (f1(r), r, θ, φ) is aSS-CMChypersurface in the Schwarzschild exterior. Then the mean curvature equation is

f₁^′′+ 1

h− (f1^′)²h  2h r +h^′

2

 +h^′

h



f₁^′ − 3H 1

h − (f1^′)²h

³₂

= 0,

where h(r) = 1 − ^2M_r and H is the mean curvature. The explicit expression of f₁^′ can be derived as

f₁^′(r; H, c1) = l₁(r; H, c1) h(r)

s 1

1 + l²₁(r; H, c1), where l₁(r; H, c1) = 1 ph(r)

Hr+ c₁ r²



for some constant c1, and the integration gives f1(r; H, c1,c¯1) =

Z r r1

l1(r; H, c1) h(r)

s 1

1 + l²₁(r; H, c1)dr + ¯c1, (5) where c¯₁ is a constant and r₁ ∈ (2M, ∞) is fixed.

Proof. Take F (t, r, θ, φ) = −t + f¹(r) and Σ¹ becomes a level set of F . In addition, ∇F =

1

h(r)∂t+ f₁^′(r)h(r)∂r is future directed because it points to the direction of increasing t. The spacelike condition (4) is equivalent to

− 1

h(r) + (f₁^′(r))²h(r) < 0 ⇔ |f1^′(r)h(r)| < 1. (6) Thus the future directed unit timelike normal vector can be expressed as

e₄ =

 1

h(r), h(r)f₁^′(r), 0, 0 q 1

h(r) − (f1^′(r))²h(r)

. (7)

There is a canonical orthonormal frame on Σ¹ e₁ = (0, 0, 1, 0)

r , e₂ = (0, 0, 0, 1)

rsin θ , and e3 = (f₁^′(r), 1, 0, 0) q 1

h(r) − (f1^′(r))²h(r)

. (8)

The second fundamental form of Σ¹ can be calculated directly, and we have

h₁₁= 1

1

h − (f1^′)²h
¹₂ hf₁^′

r , h₂₂ = 1

1

h − (f1^′)²h
¹₂ hf₁^′

r ,

h33= 1

1

h − (f1^′)²h
¹₂

 1

1

h − (f1^′)²h



f₁^′′+h^′f₁^′ h



+h^′f₁^′ 2

 ,

and hij = 0 for i 6= j. Hence the mean curvature equation becomes f₁^′′+ 1

h− (f1^′)²h  2h r +h^′

2

 +h^′

h



f₁^′ − 3H 1

h − (f1^′)²h

³₂

= 0, (9)

which is a second order ordinary differential equation.

To solve f₁(r), we define sin(η(r)) = f₁^′(r)h(r). The spacelike condition (6) implies that the change of variable is meaningful, and we can choose the range of η in −^π₂,^π₂
. Equation (9) becomes

(tan η)^′+ 2 r + h^′

2h



tan η − 3H 1 h¹²



= 0 ⇒ tan η = 1 ph(r)



Hr+ c1

r²

 , where c1 is a constant. We write l1(r; H, c1) = √¹

h(r) Hr+^c_r¹2
= tan η for convenience. On the other hand, since sin η = f₁^′h, it gives tan η = √ ^f1′h

1−(f1^′h)². Therefore, f₁^′h

p1 − (f1^′h)² =l1 ⇒ f1^′ = l₁ h

s 1

1 + l²₁ and f₁(r; H, c₁,¯c₁) =

Z r r1

l₁(r; H, c₁) h(r)

s 1

1 + l₁²(r; H, c1)dr + ¯c₁, where ¯c1 is a constant and r1 ∈ (2M, ∞) is a fixed number.

Here are some remarks on the SS-CMC solutions in (5).

Remark 1. We can choose r₁ satisfying r₁+ 2M ln |r1− 2M| = 0.

Remark 2. The sign of l₁(r) is the same as the sign of f₁^′(r), and the condition for l₁(r)T 0 is equivalent to Hr³+ c1 T 0. So f1^′(r) changes sign at most once. More explicitly, we have

(a) If H > 0 and c1 ≥ −8M³H, then f1(r) is increasing on r > 2M.

(b) If H > 0 and c1 <−8M³H, then f1(r) is decreasing on

2M, ^−c_H^1
1₃

, and increasing on

−c1

H

¹₃ ,∞

. Function f1(r) has a unique minimum at r = ^−c_H^1
1₃ . (c) If H < 0 and c₁ ≤ −8M³H, then f₁(r) is decreasing on r > 2M.

(d) If H < 0 and c1 >−8M³H, then f1(r) is increasing on

2M, ^−c_H^1
1₃

, and decreasing on

−c1

H

¹₃ ,∞

. Function f₁(r) has a unique maximum at r = ^−c_H^1
1₃ . Remark 3. The second fundamental form of Σ¹ with basis (8) satisfies

h₁₁ = h22= H + c₁

r³, h₃₃= H − 2c1

r³ , and h11, h₂₂, h₃₃→ H as r → ∞.

In particular, if c1 = 0, then h11 = h22 = h33= H. We call this hypersurface umbilical slice.

Remark 4. The graphs of f₁(r; H, c₁,¯c₁) for ¯c₁ ∈ R gives a foliation in the Schwarzschild exterior.

3.2 Asymptotic behavior of SS-CMC solutions in region I

We analyze the asymptotic behavior of SS-CMC solutions f1(r) in this subsection. Here we omit the dependency of f1 on H, c1, ¯c₁ when there is no confusion.

Proposition 3. For a SS-CMC hypersurface Σ¹ = (f1(r), r, θ, φ), we have lim

r→∞f₁^′(r) = 1 if H > 0; lim

r→∞f₁^′(r) = −1 if H < 0, and lim_r→∞f₁^′(r) = 0 if H = 0. Furthermore, Σ¹ is asymptotically null for H 6= 0 as r → ∞, and Σ¹ is asymptotically to some constant slice (t = t0, r, θ, φ) for H = 0 as r → ∞.

Proof. Since

r→∞lim f₁^′(r) = lim

r→∞

l1

h(r)

s 1

1 + l²₁ = lim

r→∞

Hr+_r^c12

1 − ^2Mr

q

1 −^2Mr + Hr +_r^c12

2, the limit is 0 if H = 0, and is _|H|^H if H 6= 0.

We compute

h∇F, ∇F i = − 1

h(r) + h(r)(f₁^′(r))² = −1

h(r)(1 + l₁²) = −1 h(r) + Hr + ^c_r¹2

2, (10) and have lim

r→∞h∇F, ∇F i = 0 if H 6= 0.

Proposition 4. For aSS-CMChypersurfaceΣ¹ = (f1(r; H, c1,c¯₁), r, θ, φ) in the Schwarzschild exterior, the following conclusions hold:

(a) If c1 <−8M³H, then f₁^′(r) < 0 near r = 2M, and f₁^′(r) is of order O((r − 2M)⁻¹).

It implies that lim

r→2M⁺f₁(r) = ∞.

(b) If c1 = −8M³H, then H · f1^′(r) > 0, and f₁^′(r) is of order O((r − 2M)⁻¹²). It implies that lim

r→2M⁺f₁(r) is finite.

(c) If c1 >−8M³H, then f₁^′(r) > 0 near r = 2M, and f₁^′(r) is of order O((r − 2M)⁻¹).

It implies that lim

r→2M⁺f1(r) = −∞.

When c₁ 6= −8M³H, the curve(f₁(r), r) in (t, r) spacetime is bounded by two null geodesics near r = 2M. For all c1 ∈ R, the spacelike condition is preserved as r → 2M⁺.

Proof. From (10), we know that if c1 6= −8M³H, then lim

r→2M⁺h∇F, ∇F i = (^{2M H+−1}4M 2^c1 )² <0, and if c1 = −8M³H, then

r→2Mlim⁺h∇F, ∇F i = lim

r→2M⁺

−1

r−2M r +_H

(r−2M )(r²+2M r+4M²) r²

2 = −∞.

Hence the spacelike condition is preserved as r → 2M⁺ for all c1 ∈ R.

Now we prove the asymptotic behavior of f1(r).

If c1 <−8M³H, then f₁^′(r) < 0 (and thus l1(r) < 0) on r ∈ (2M, 2M + δ¹) for some δ1 >0.

Therefore, on (2M, 2M + δ₁) by the Taylor’s theorem, we have

f₁^′ = l1

h

s 1

1 + l₁² = 1

−h s

1 − 1 1 + l²₁

≈ 1

−h 1 − 1 2

 1

1 + l²₁



− 1 8

 1

1 + l₁²

2

− 3 16

 1

1 + l₁²

3

− · · ·

!

≈ 1

−h +1 2

1

h+ Hr +^c_r¹2

2 + 1 8

h

h+ Hr +_r^c12

22 + · · ·

= 1

−h + remainder terms.

Remainder terms can be bounded above by ¹

2(^Hr+c1r2)², so f₁^′(r) is of order O((r − 2M)⁻¹).

Furthermore, we have

1

−h(r) ≤ f^′(r) ≤ 1

−h(r) + 1

2 Hr +_r^c12

2 (11)

on (2M, 2M + δ1). We integrate inequalities (11) and get Z r1

r − x

x− 2Mdx ≤ Z r1

r

f₁^′(x)dx ≤ Z r1

r − x

x− 2M + 1

2 Hx +_x^c12

2

! dx.

The integral Rr1

r

1

2(^Hx+c1x2)²dx is finite, and we denote it by C1. It follows that

− (r¹ + 2M ln(r1− 2M)) + (r + 2M ln(r − 2M))

≤ f¹(r1) − f¹(r)

≤ − (r¹ + 2M ln(r1− 2M)) + (r + 2M ln(r − 2M)) + C¹

⇒ −(r + 2M ln(r − 2M)) + C²− C¹ ≤ f¹(r) ≤ −(r + 2M ln(r − 2M)) + C², where C2 = f1(r1) + (r1 + 2M ln(r1− 2M)). Hence the curve t = f¹(r) is bounded by two null geodesics t + (r + 2M ln(r − 2M)) = C²− C¹ and t + (r + 2M ln(r − 2M)) = C² near r= 2M.

If c1 = −8M³H, then

l1 =

 r

r− 2M

¹₂

 Hr³− 8M³H r²



= H r − 2M r

¹₂

 r²+ 2Mr + 4M² r

 .

Direct computation gives

f₁^′ = H

 r

r− 2M

¹₂ 

r(r²+ 2Mr + 4M²)²

r³+ H²(r − 2M)(r²+ 2Mr + 4M²)²

¹₂ ,

and thus f₁^′ is of order O((r − 2M)⁻¹²).

If c₁ >−8M³H, then both f₁^′(r) and l₁(r) are positive on (2M, 2M + δ₂) for some δ₂ >0.

By the Taylor’s theorem, we have

f₁^′ = 1 h

s

1 − 1

1 + l²₁ ≈ 1 h −1

2

1

h+ Hr +^c_r¹2

2− 1 8

h

h+ Hr + ^c_r¹2

22 − · · · . The remainder terms are greater than ⁻¹

2(^Hr+r2^c1)² on (2M, 2M + δ2). This implies 1

h(r) − 1

2 Hr +_r^c12

2 ≤ f1^′(r) ≤ 1 h(r). We integrate the above inequalities and get

Z r1

r

1

h(x) − 1

2 Hx +_x^c12

2

! dx ≤

Z r1

r

f₁^′(x)dx ≤ Z r1

r

1 h(x)dx.

The integral Rr1

r

1

2(^Hx+c1x2)²dx is finite, and we denote it by C3. It follows that r1+ 2M ln(r1− 2M) − (r + 2M ln(r − 2M)) − C³

≤ f1(r₁) − f1(r)

≤ r¹+ 2M ln(r1− 2M) − (r + 2M ln(r − 2M))

⇒ (r + 2M ln(r − 2M)) + C⁴ ≤ f¹(r) ≤ (r + 2M ln(r − 2M)) + C³+ C4,

where C4 = f1(r1) − (r¹+ 2M ln(r1− 2M)). Hence the curve t = f¹(r) is bounded by two null geodesics t − (r + 2M ln(r − 2M)) = C⁴ and t − (r + 2M ln(r − 2M)) = C³ + C4 near r= 2M.

Figure 5 pictures SS-CMC hypersurfaces in the Schwarzschild exterior and their images in region I of the Kruskal extension.

4 SS-CMC solutions in region II

We consider SS-CMC hypersurfaces in the Schwarzschild interior in this section.

c1 <−8M³H c₁ = −8M³H c1 >−8M³H

r t

r= 0 r= 0

r= 2M r= 2M

c₁ <−8M³H c1 = −8M³H c₁ >−8M³H

Figure 5: SS-CMC hypersurfaces in Schwarzschild exterior and region I.

4.1 Cylindrical hypersurfaces r = constant

Notice that h(r) = 1 −^2Mr <0 on 0 < r < 2M, so in this region r-direction is timelike and t-direction is spacelike. Furthermore, −∂^r is future directed. We can assume that aSS-CMC

hypersurface is written as (t, g(t), θ, φ) for some function r = g(t).

Proposition 5. [5] Each constant slice r = r0, r₀ ∈ (0, 2M) is a ^SS-CMC hypersurface with H(r0) = 2r0− 3M

3pr₀³(2M − r⁰). These hypersurfaces are called cylindrical hypersurfaces.

Cylindrical hypersurfaces are known in [5]. Here we give a simple proof for completeness.

Proof. Choose e4 = (0, −p−h(r), 0, 0) to be a future directed unit timelike normal vector, and there is a canonical orthonormal frame

e₁ = (0, 0, 1, 0)

r , e₂ = (0, 0, 0, 1)

rsin θ , e₃ = (1, 0, 0, 0) p−h(r)

on constant slices. Since ∇^∂t∂r = _2h(r)^h′(r)∂t,∇^V∂r = ^V_r for V ∈ T^(p,q){p} × S², we have

h11 = h∇^e1e4, e1i = −p−h(r)h∇^e1∂r, e1i = −p−h(r)

r ,

h₂₂ = h∇^e2e₄, e₂i = −p−h(r)h∇^e2∂r, e₂i = −p−h(r)

r ,

h₃₃ = h∇^e3e₄, e₃i = −1

p−h(r)h∇^∂t∂r, ∂ti = h^′(r) 2p−h(r),

and hij = 0 for i 6= j. Hence the mean curvature is

H = 1 3

−2p−h(r)

r + h^′(r) 2p−h(r)

!

= 1

3p−h(r)

 2h(r)

r + h^′(r) 2



= 2r − 3M 3pr³(2M − r), which is a constant for each fixed r ∈ (0, 2M).

The following corollary is an easy consequence of Proposition 5.

Corollary 1. Cylindrical hypersurfaces r = r0, r₀ ∈ (0, 2M) have the following properties.

(a) If r0 ∈ 0,³₂M
, then H(r0) < 0 and lim

r→0⁺H(r) = −∞.

(b) If r₀ ∈ ³₂M,2M
, then H(r0) > 0 and lim

r→2M⁻H(r) = ∞.

(c) If r0 = ³₂M , then the cylindrical hypersurface is a maximal hypersurface.

4.2 Noncylindrical SS-CMC hypersurfaces

For r = g(t) 6= constant, we consider its inverse function, and denote t = f²(r) whenever it is defined. Since f2(r) is obtained from the inverse function, we have f₂^′(r) 6= 0 and will allow f₂^′(r) = ∞ or −∞.

Proposition 6. Suppose Σ² = (f2(r), r, θ, φ) is a SS-CMC hypersurface in Schwarzschild interior. Then f₂^′ can be derived as

f₂^′ =













 1

−h s

l²₂

l²₂− 1 if f₂^′(r) > 0 1

h s

l²₂

l²₂− 1 if f₂^′(r) < 0,

where l₂(r; H, c₂) = 1 p−h(r)

−Hr − c2

r²

 .

The function l₂ should satisfy l₂ >1, which implies c2 < 0 when H > 0 and c2 < −8M³H when H <0. The integration of f₂^′ gives

f₂^∗(r; H, c₂,¯c₂) = Z r

r2

1

−h(r) s

l²₂(r; H, c₂)

l₂²(r; H, c2) − 1dr + ¯c₂, or (12) f₂^∗∗(r; H, c2,¯c^′₂) =

Z r r^′₂

1 h(r)

s

l²₂(r; H, c2)

l²₂(r; H, c₂) − 1dr + ¯c^′₂ (13) according to the sign of f₂^′(r), where ¯c2,c¯^′₂ are constants, andr2, r₂^′ are points in the domain of f₂^∗(r) and f₂^∗∗(r), respectively.

Remark 5. In this article, when we write f2(r), it means both f₂^∗(r) and f₂^∗∗(r).

Proof. First we consider the case f₂^′(r) > 0. Denote F (t, r, θ, φ) = −t + f²(r), we have

∇F = h(r)¹ ∂t+ f₂^′(r)h(r)∂r is future directed because it is in the direction of decreasing r.

The spacelike condition (4) is equivalent to

− 1

h(r) + (f₂^′(r))²h(r) < 0 ⇔ (f2^′(r)h(r))² >1. (14) Hence future directed timelike normal vector is

e₄ =

 1

h(r), h(r)f₂^′(r), 0, 0 q 1

h(r) − (f2^′(r))²h(r) ,

which has the same expression as (7), and we can take a canonical orthonormal frame on Σ² with the same expressions as (8). Therefore, the mean curvature equation will be

f₂^′′+ 1

h− (f2^′)²h  2h r +h^′

2

 +h^′

h



f₂^′ − 3H 1

h − (f2^′)²h

³₂

= 0. (15)

To solve f₂(r), from (14) we can make change of variable by sec(η(r)) = f₂^′(r)h(r). Since h(r) = 1 − ^2Mr < 0 on 0 < r < 2M, we can choose the range of η to be ^π₂, π
. Then equation (15) becomes

(csc η)^′+ 2 r + h^′

2h



csc η + 3H 1

(−h)¹² = 0 ⇒ csc η = 1 p−h(r)

−Hr − c₂ r²



, (16)

where c2 is a constant. When writing l2(r; H, c2) = √ ¹

−h(r) −Hr − ^cr²²
= csc η, we have

f₂^′ = sec η

h = 1

−hq

1 −csc¹²η

= 1

−h s

l²₂ l²₂− 1. We remark that l₂ = csc η > 1 because η ∈ ^π₂, π
.

For the case f₂^′(r) < 0, we choose F (t, r, θ, φ) = t − f²(r) such that ∇F = −_h(r)¹ ∂t− f₂^′(r)h(r)∂r is future directed. The spacelike condition is the same as (14), but future directed timelike normal vector is

e₄ =

−h(r)¹ ,−f2^′(r)h(r), 0, 0 q 1

h(r) − (f2^′(r))²h(r) .

There is a canonical orthonormal frame e₁ = (0, 0, 1, 0)

r , e₂ = (0, 0, 0, 1)

rsin θ , and e3 = (−f2^′(r), −1, 0, 0) q 1

h(r) − (f2^′(r))²h(r)

on Σ² such that it has the same orientation as the case of f₂^′(r) > 0. The second fundamental form of Σ² in (S, ds²) are

h₁₁= − 1

1

h − (f2^′)²h
¹₂ hf₂^′

r , h₂₂ = − 1

1

h − (f2^′)²h
¹₂ hf₂^′

r ,

h33= 1

1

h − (f2^′)²h
¹₂

 −1

1

h − (f2^′)²h



f₂^′′+h^′f₂^′ h



− h^′f₂^′ 2

 ,

and hij = 0 for i 6= j. Hence the mean curvature equation becomes

f₂^′′+ 1

h − (f2^′)²h  2h r +h^′

2

 +h^′

h



f₂^′ + 3H 1

h− (f2^′)²h

³₂

= 0. (17)

From (14), we can change variable by sec(η(r)) = f₂^′(r)h(r), and the range of η can be chosen as 0,^π₂
because h(r) = 1 −^2M_r <0 on 0 < r < 2M. Then (17) becomes

(csc η)^′+ 2 r + h^′

2h



csc η + 3H 1

(−h)¹² = 0 ⇒ csc η = 1 p−h(r)

−Hr − c2

r²

 , which has the same expression as (16). Set l2(r; H, c2) = √ ¹

−h(r) −Hr − ^cr²²
= csc η, then we have

f₂^′(r; H, c2) = 1 h(r)

s

l₂²(r; H, c2) l²₂(r; H, c2) − 1. We remark that l₂ = csc η > 1 because η ∈ (0, ^π₂).

4.3 Domain of SS-CMC solutions in region II

The condition l2(r) > 1 will put restrictions on the domain of f2(r). We have l₂(r) = 1

p−h(r)

−Hr − c₂ r²

>1 ⇒ −Hr³− r³²(2M − r)¹² > c₂.

Define a function kH(r) on (0, 2M) by

kH(r) = −Hr³− r³²(2M − r)¹², (18) then the domain of f₂(r) will be

{r ∈ (0, 2M)|k^H(r) > c2} ∪ {r ∈ (0, 2M)|k^H(r) = c2 and f2(r) is finite}.

Now we analyze the function kH(r) to determine the set.

Proposition 7. Consider kH(r) as in (18), then kH(r) has a unique minimum point at r= rH, where rH is determined by 3Hr

3 2

H(2M − r^H)¹² = 2rH − 3M.

Proof. We differentiate kH(r) to get

k_H^′ (r) = −r¹² (2M − r)¹²

3Hr²³(2M − r)¹² + 3M − 2r

. (19)

Denote ¯kH(r) = 3Hr³²(2M − r)¹², then

¯k_H^′ (r) = 3Hr¹²

(2M − r)¹²(3M − 2r).

It implies ¯k_H^′ (³₂M) = 0, and ¯kH(r) is monotone on (0,³₂M) and (³₂M,2M). Furthermore,

¯kH(r) and the function p(r) = 2r − 3M intersect at r = r^H. (See Figure 6.) That is, 3Hr

3 2

H(2M − r^H)²¹ = 2rH − 3M and k^′H(rH) = 0, so rH is the critical point of kH(r).

3 2M

3

2M 2M

2M

rH

rH

rH

c2

c2

c2

c2

c₂ c₂

t t

t t

r r

r r

¯kH(r)

¯kH(r)

kH(r)

kH(r)

(a) H > 0 (b) H < 0

Figure 6: Graphs of ¯kH(r), p(r) = 2r − 3M, k^H(r), and horizontal lines l(r) = c2.

Proposition 8. Denote cH = min

r∈(0,2M )kH(r) = kH(rH), where kH(r) is as in (18), and rH

is as in Proposition 7. There are three types of noncylindrical SS-CMC hypersurfaces Σ² = (f2(r), r, θ, φ) according to the value of c2, where f₂(r) = f₂^∗(r; H, c2,¯c2) or f₂^∗∗(r; H, c2,c¯^′₂).

(a) If c2 < cH, then f2(r) is defined on (0, 2M).

(b) If c2 = cH, then f₂(r) is defined on (0, rH) ∪ (r^H,2M).

(c) If cH < c₂ <max(0, −8M³H), then f₂(r) is defined on (0, r^′] or [r^′′,2M) for some r^′ and r^′′, which depend on H and c2. When we take r2 = r₂^′ = r^′(or r^′′) and ¯c2 = ¯c^′₂ in (12) and (13), Σ² = (f₂^∗(r; H, c2,¯c₂) ∪ f2^∗∗(r; H, c2,c¯^′₂), r, θ, φ) is a complete ^SS-CMC hypersurface in the Schwarzschild interior.

Proof.

(a) If c2 < cH, then l2(r) > 1 for all r ∈ (0, 2M), which implies f²(r) is defined on (0, 2M).

(b) If c2 = cH, then f2(r) is defined on (0, rH) ∪ (r^H,2M). We need to check the behavior of f2(r) as r → r^H. First, we know lim

r→rH

f₂^′(r) = ∞ or −∞ because l²(r) = 1. Next, noting that c2 = −HrH³ − r

3 2

H(2M − r^H)²¹ and f₂^′(r) = −Hr³− c²

h(r)p(−Hr³− c²)²+ r³(r − 2M) or −Hr³− c²

−h(r)p(−Hr³− c²)²+ r³(r − 2M), we expand (−Hr³− c²)²+ r³(r − 2M) in the power of (r − r^H) to attain

p(−Hr³− c2)²+ r³(r − 2M)

= r

P₁(r; rH)(r − r^H)²+ 2r²_H

−3Hr

3 2

H(2M − r^H)¹² + 2rH − 3M

(r − r^H)

=p

P1(r; rH)(r − r^H)²,

where P1(r; rH) is a polynomial. The last equality holds because rH is the critical point of kH(r). Thus f₂^′(r) ∼ O((r − r^H)⁻¹) and lim

r→rH

f2(r) = ∞ or −∞. That is, the domain of f2(r) is (0, rH) ∪ (r^H,2M).

(c) If cH < c₂ <max(0, −8M³H), then f₂(r) is defined on (0, r^′) or (r^′′,2M). Here we only discuss the case at r^′ because the case at r^′′ is similar. First, we know lim

r→r^′f₂^′(r) = ∞ or −∞. Next, since c² = −H(r^′)³− (r^′)³²(2M − r^′)¹² is not a critical value of kH(r), the expansion of (−Hr³− c²)²+ r³(r − 2M) in the power of (r − r^′) becomes

p(−Hr³− c²)²+ r³(r − 2M)

= r

P₂(r; r^′)(r − r^′)²+ 2(r^′)²

−3H(r^′)³²(2M − r^′)¹² + 2r^′− 3M

(r − r^′),

where P2(r; r^′) is a polynomial, and −3H(r^′)³²(2M − r^′)¹² + 2r^′− 3M 6= 0. It implies f₂^′(r) ∼ O((r − r^′)⁻¹²), and lim

r→r^′f2(r) is a finite value. Domain of f2(r) can be ex- tended to r = r^′. When taking r2 = r^′₂ = r^′ and ¯c₂ = ¯c^′₂, we have f₂^∗(r^′; H, c2,¯c2) = f₂^∗∗(r^′; H, c2,¯c^′₂) = ¯c2 and Σ² = (f₂^∗(r; H, c2,¯c2) ∪ f2^∗∗(r; H, c2,¯c^′₂), r, θ, φ) is a complete

SS-CMChypersurface in the Schwarzschild interior.

Proposition 9. In case (c) of Proposition 8, the SS-CMC hypersurface Σ² is C^∞.

Proof. It suffices to check the smoothness of Σ² at the joint point, and here we show the case of r2 = r₂^′ = r^′. The case of r2 = r^′₂ = r^′′ is similar. Noting that ¯c2 = ¯c^′₂ and r < r^′, we have f^∗(r) ≤ ¯c², f^∗∗(r) ≥ ¯c², and f^∗(r^′) = f^∗∗(r^′) = ¯c₂. Hence when rewrite the surface as a graph of r = g(t), we have g(¯c2) = r^′ and its inverse corresponds to t = f₂^∗(r) for t ≤ ¯c² and to t = f₂^∗∗(r) for t ≥ ¯c². Direct computation gives

g^(2k+1)(t) =

( Pk

i=0Ak,i(l²₂− 1)ⁱ⁺¹² if t < ¯c₂ (−1)^2k+1Pk

i=0Ak,i(l₂²− 1)ⁱ⁺¹² if t > ¯c₂, g^(2k)(t) =

( Pk

i=0Bk,i(l²₂− 1)ⁱ if t < ¯c₂ (−1)^2kPk

i=0Bk,i(l₂²− 1)ⁱ if t > ¯c₂,

where Ak,i and Bk,i are functions of h, l2 and their derivatives with respective to r. As t→ ¯c², we have r → r^′ and lim

r→r^′l²₂− 1 = 0, it implies that lim

t→¯c⁻₂

g^(2k+1)(t) = lim

t→¯c⁺₂

g^(2k+1)(t) = 0 and lim

t→¯c⁻₂

g^(2k)(t) = lim

t→¯c⁺₂

g^(2k)(t) = Bk,0. Hence Σ² is smooth.

4.4 Asymptotic behavior of SS-CMC solutions in region II

Next, we discuss the asymptotic behavior ofSS-CMChypersurfaces in Schwarzschild interior that will be needed in section 6.

Proposition 10. For a SS-CMChypersurface Σ² = (f₂(r; H, c₂,¯c₂), r, θ, φ) in Schwarzschild interior withc2 <−8M³H, we have f₂^′(r) is of order O((2M −r)⁻¹) as r → 2M⁻. It implies that lim

r→2M⁻f₂(r) = ∞ or −∞, and the curve (f²(r), r) in (t, r) plane is bounded by two null geodesics as r→ 2M⁻. Furthermore, the spacelike condition is preserved as r→ 2M⁻. Proof. Since c₂ < −8M³H, f₂(r) is defined on (2M − δ3,2M) for some δ₃ > 0. We only need to consider the case f₂^′(r) > 0 because of symmetry. On one hand, since

f₂^′(r) = 1 (−h)q

1 −(−Hr^{r3(2M −r)3}−c2)²

≥ 1

−h,

we have

Z r r2

f₂^′(x)dx ≥ −(x + 2M ln(2M − x))|^x=rx=r2

⇒ f2(r) ≥f2(r₂) + (r₂+ 2M ln(2M − r2) − (r + 2M ln(2M − r))

= − (r + 2M ln(2M − r)) + C⁵,

where C5 = f2(r2) + r2 + 2M ln(2M − r²). The curve t = f2(r) is bounded below by the null geodesic t + (r + 2M ln(2M − r)) = C⁵ near r = 2M.

On the other hand, because _l¹2

2 is very small near r = 2M, by Taylor’s expansion we get s

1 − 1

l₂² ≈ 1 −1 2

 1 l²₂



− 1 8

 1 l²₂

2

− · · · ≥ 1 − 1 l²₂



− 1 l²₂

2

− · · ·

= 1 − 1

l²₂− 1 = 1 − −h

−Hr − r^c22

2

− (−h) There is a constant C₆ >0 such that C₆

−Hr − ^cr²²

2

− 2(−h)

>1 on (2M − δ4,2M), a subset of (2M − δ³,2M). That is, we have

1

−Hr − ^cr²²

2

− (−h) < C₆ 1 + C6(−h) and

s 1 − 1

l²₂ ≥ 1 − C6(−h)

1 + C₆(−h) = 1 1 + C₆(−h). Thus

f₂^′(r) = 1 (−h)q

1 −l¹²₂

≤ 1

(−h)(1 + C6(−h)) = 1

−h + C6, which integrates to

Z r r2

f₂^′(x)dx ≤ Z r

r2

 1

−h(x) + C6

 dx.

Hence

f₂(r) ≤ f2(r₂) − (x + 2M ln(2M − r))|^x=rx=r2 + C₆(r − r2)

= −(r + 2M ln(2M − r)) + C⁵+ C7,

where C5 = f2(r2) + (r2+ 2M ln(2M − r²)) and C7 = C6(2M − r²). The curve t = f2(r) is bounded above by the null geodesic t + (r + 2M ln(2M − r)) = C⁵+ C7 near r = 2M.