Yongbing Zhang

School of Mathematical Sciencesand Wu Wen-Tsun Key Laboratory of Mathematics,University of Scienceand Technology of China,Hefei230026,China.

Abstract.It w as proved by Graham and Witten in 1999 that conformal invariants of submanifolds can be obtained via volume renormalization of minimal surfaces in conformally compact Einstein manifolds.The conformal invariant of a submanifoldΣis contained in the volume expansion of the minimal surface which is asymptotic toΣ w hen the minimal surface approaches the conformaly inf inity.In the paper we give the explicit expression of Graham-Witten’s conformal invariant for closed four dimensional submanifolds and f ind critical points of the conformal invariant in the case of Euclidean ambient spaces.

Key w ords:Minimal surface,Ad S/CFT,conformal invariant.

1 Introduction

In theintroduction wegive a description of the main result and somerelated background of the paper.The terminologies used in the introduction will be recalled in the next section.

In the paper,we calculate the coeff icientu(4)forn≥3 by using the graphic minimal surface equation ofY.

Theorem 1.1.For n≥3,

Our main aim of thepaper is to calculate theconformal invariant,introduced by Graham and Witten(for each dimensional submanifold)[16],for closed four dimensional submanifolds.Here the conformal invariant for submanifolds is a functional of submanifolds which is invariant under conformal transformations of the metric of the ambient space.Recently,conformal invariants for hypersurfaces,constructed by using thevolume renormalization of solutions to the singular Yamabe problem or general singular volume measures,and related aspects have been extensively studied[11,14,18–22,36].Graham-Witten’s conformal invariants are obtained from the renormalization process of the volume of a minimal surfaceYn+1in a conformally compact Einstein manifold(Xd+1,g+)which is asymptotic to a submanifoldΣnimmersed in the conformal inf inity ofX.

Near the asymptotic boundary,the volume form ofYtakes the form of

wherev(j)arelocally determined functions ofΣ,v(n)=0 fornodd,anddμΣis thevolume form of(Σ,g).As?→0 and fornodd,thevolume

and forneven

where

An area law was proposed by Ryu and Takayanagi[33,34]which holographically identif ies the volume in(1.14)or(1.15)with the entanglement entropy in quantum(conformal)f ield theories.Graham and Witten proved the following

Theorem 1.2([16]).If n isodd,then cn isindependent of thechoiceof special def ining function.If n iseven,then Ln isindependent of thechoiceof special def ining function.

Note that there is the one-to-one correspondence between representative metrics of the conformal class(M,[gconfinf])and special def ining functions.Hencecnfornodd andLnforneven are conformal invariants.cnis called the renormalized volume ofY.Forn=1,an explicit expression of therenormalized volumec1has been obtained by Alexakis and Mazzeo[2].Forn=2,Graham and Witten[16]showed that

Volume renormalization of a conformally compact Einstein manifold had been studied at almost the same time[12,28,29],which can be viewed as the extreme case under the setting of the paper:Σ=MandY=X.The log terms of the volume expansion were calculated explicitly in lower dimensions.For example,in dimension two

2 Minimal submanifolds in conformally compact Einstein manifolds

In the f irst part of this section,referring mainly to[12,16],we give a brief review to the related aspectsof conformally compact Einstein manifolds,and minimal submanifoldsin conformally compact Einstein manifolds.In the second part,we identify the constituent components ofu(4)from the graphic minimal surface equation ofY.

2.1 Minimal surfaces in a conformally compact Einstein manifold

Then

Graham and Witten[16]showed that the graphic minimal surface equation ofYis

M(u)=0,

where

and

By the choice of the local coordinates(xi,yα)onM,we haveu(x,0)=0.Graham and Witten[16]found that atr=0,

which means thatYintersects withMorthogonally;and forn≥1

u=u(2)r2+(even powers)+u(n+1)rn+1+u(n+2)rn+2+···,

and forneven

u=u(2)r2+(even powers)+u(n)rn+wnrn+2logr+u(n+2)rn+2+···,

whereu(k),k

2.2 Coeff icient u(4)in(1.7)and(1.8)for n≥3

Forn≥3,let

where,it follows from(2.10),

It follows from(2.8)that

Note that

and the fact that the coeff icientwin(2.11)can be derived by tracing the coeff icient ofr3in(2.7).First,we have

Lemma 2.1.For theright hand sideof(2.13),wehave

Proof.Note that

Hence

Therefore,

Note that

where in the last equality we used(2.5)and(2.12).Hence

(2.16)then follows from(2.17)and(2.18).

Lemma 2.2.For(2.14),wehave

Proof.It is easy to seethat

It then follows from

that

It is easy to see that

(2.19)then follows from(2.20)and(2.21).

Lemma 2.3.For(2.15),wehave

Proof.It is clear that we have

Hence by using(2.5),weget

On the other hand,it’s easy to see that

(2.22)then follows from(2.23)and(2.24).

Putting(2.16)(2.19)and(2.22)together,we get the following

Proposition 2.1.For n≥3,wehave

Notethatw=u(4),so we have

Proposition 2.2.Let Yn+1,n≥3,bea minimal surfacein(Xd+1,g+).Then

where

3 Proof of Theorem 1.1

Assumen≥3.In this section,we will reformulateQγto get the expression ofu(4),as given by(1.10).Let

It follows from(2.27)that

where

We f irst computeIIα.

Lemma 3.1.Wehave

Proof.LetDrangefrom 1 todso that for?D=?iforD=i≤nand?D=?αforD=α≥n+1.We have

Hence by using(2.5),we get

This completes the proof of the lemma.

We now deal withIα.

Lemma 3.2.Wehave

Proof.LetA,B,C,D={i,α}=1,···,d.LetX⊥denote the normal part ofXand?⊥the covariant differentiation with respect to the normal connection.We have

Then

On the other hand,

Therefore,(3.5)follows from

This completes the proof of the lemma.

Lemma 3.3.Wehave

Proof.Note that

Hence

Note that onΣn

hence we have

It then follows from(2.5)and(2.12)that

That is(3.6).

Lemma 3.4.Wehave

Proof.It follows from(3.2),(3.5)and(3.6)that

Note that

(3.7)then follows from(3.8)and(3.9).

Theorem 3.1.Let Yn+1,n≥3,beaminimal surfacein(Xd+1,g+)and of theform Y=(x,u(x,r),r)near theboundary of X,where

Then wehave

Proof.It follows from(3.1),(3.7)and(3.3)that

Using(2.12),we get(3.10).

Proposition 3.1.Let Y3bea minimal surface in(Xd+1,g+)and of the form Y=(x,u(x,r),r)near theboundary of X,where

Then wehave

4 Proof of Theorem 1.3

and forneven

where

Letn≥4 and

Hence

We rewrite the above as

where

LetA=(aij),B=(bij).Then

Note that

Therefore,

where

C=a+gijaij,D=atr(g?1A)+σ2(g?1A)+b+tr(g?1B).

Then

and the coeff icientv(4)in(4.1)is

Proposition 4.1.Let Yn+1,n≥4,bea minimal surfacein(Xd+1,g+).Then wehave

where

We f irst deal with(4.8).

Lemma 4.1.Wehave

Proof.Note that

hence

As we have shown in(3.4),we have

Therefore,

This completes the proof of the lemma.

Lemma 4.2.Wehave

Proof.Note that

It then follows from(3.5):

and(4.7):

that

Note that

Hence

Recall(3.6):

hence

Then

Recall(3.9):

This completes the proof of the lemma.

It follows from(4.6),(4.10)and(4.9)the following

Proposition 4.2.Let Yn+1,n≥4,be a minimal surface in(Xd+1,g+).The coeff icient v(4)in(4.1)isgiven by

Notice that the second last line in(4.11)vanishes forn=4 and forn≥5 the part involvingΓcancels.Note also that the last line is a divergence which is independent of the choice of local coordinates(xi,yα).By integrating(4.11)over a closed submanifold Σ4and using(2.1)–(2.3),we obtain the expression(1.18)of Graham-Witten’s conformal invariantL4.

5 L4 for closed submanifolds of Euclidean spaces

In this section,we assumen=4,d≥5 and the ambient space is(Md,g)=Rd.It follows from(1.18)that

We now consider the following functional for closed four dimensional submanifoldsF:Σ→Rd

In the sequel we use local coordinates(xi)ofΣwhich is normal at the point of consideration with respect to the induced metric

where

Proof.LetX∈T⊥Σbe a variation f ield.We have

Hence

We then compute

and

Therefore

where

Note that

and

hence

This completes the proof of the proposition.

We search for critical points of L4of the form

Form=1,using Eq.(5.2)we have the critical point of the round sphere and

L4(S4)=24Vol(S4)=64π2.

Form=2 andk1=3,that is to consider S3(1)×S1(r2),in the case we have

whereν1andν2are unit inner normal vector f ields.Eq.(5.2)now reads

hencew ehave the following two solutions of the form(5.5)to(5.1)

The value

In particular

and

Form=2 andk1=2,that is to consider S2(1)×S2(r2),in thecase we have

The equation(5.2)now reads

hencewe have the following solution of the form(5.5)to(5.1)

S2(1)×S2(1).

The value

In particular

Form=3,that is to consider S2(1)×S1(r2)×S1(r3),in the case we have

The value

In particular it is unbounded from above and below,and

Form=4,that is S1(r1)×S1(r2)×S1(r3)×S1(r4),letxi=1

r2i,in the case we have

We have in the case only,modulo conformal equivalence,the following two solutions of the form(5.5)to(5.1)

The value

In particular it is unbounded from above and below,and

Acknowledgments

The author would like to thank Professor Qing Han for his stimulating talk which also brought the work[16]of Graham and Witten to his attention.This work is completed when the author visits Princeton University.He would like to thank the Department of Mathematics for its hospitality.The project is supported by“the Fundamental Research Funds for the Central Universities”and NSFC 12071450.