Ricci-Bourguignon Flow on Manifolds with Boundary

Hongbing QIU Anqiang ZHU

Abstract The authors consider the short time existence for Ricci-Bourguignon flow on manifolds with boundary. If the initial metric has constant mean curvature and satisfies some compatibility conditions,they show the short time existence of the Ricci-Bourguignon flow with constant mean curvature on the boundary.

Keywords Ricci-Bourguignon flow, Boundary value problem

1 Introduction

The Ricci-Bourguignon flow is

where Ric is the Ricci tensor of the manifold,Ris the Scalar curvature andρis a constant.This flow, which is a generalization of the Ricci flow, was introduced by Bourguignon [2]. For the study of the Ricci-Bourguignon flow, see [4—6, 11]. Catino et. al. [3] proved the short-time existence of solutions to the Ricci Bourguinon flow on closed manifolds.

There are plenty of works on the geometric flows on compact manifolds with boundary.Hamilton [7] showed the short time existence to the harmonic map heat flow from manifolds with Dirichlet, Neumann and mixed boundary by inverse theorem. Shen [13] proved the short time existence of the Ricci flow on compact manifolds with umbilic boundary. Later,Pulemotov[12] obtained a short time existence for Ricci flow on compact manifolds with boundary of constant mean curvature. Gianniostis [9] derived the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary.

Inspired by the previous works, we attempt to study the corresponding existence problems for the Ricci-Bourguignon flow. We obtain the following short time existence for the Ricci-Bourguignon flow on compact manifolds with boundary.

Theorem 1.1Let(M,g0)be a Riemannian manifold with constant mean curvature H0on the boundary. Suppose that μ(t)is a smooth real value function on[0,∞)with μ(0) = 1and g0∈C4+?α(M). In addition on ?M×{0}, the metric g0satisfies the compatibility conditions

where β denotes the tangent direction and n denotes the normal direction with respect to metric g0. Then for ρ

2(M×[0,T))such that the mean curvature H(x,t)satisfies the boundary condition

for all(x,t)∈?M×[0,T]and g(t)converges to g0in the geometric C2+?α(M)sense as t→0.

Remark 1.1Whenρ≡0,the Ricci-Bourguignon flow becomes the usual Ricci flow. Hence the above Theorem 1.1 generalizes a result in [12].

In [12], the WIq-estimate (see [12, Lemma 2.6]) plays a very important role in the proof of the short time existence of the Ricci flow. Therefore it is natural to ask whether a WIqestimate holds for the Ricci Bourguignon flow. However, the case of the Ricci-Bourguignon flow is harder to deal with than the Ricci flow since we now have an additional termRg. And unfortunately we could not apply the theorem in[12]to the Ricci-Bourguignon flow on manifolds with boundary. Instead, we show the short time existence of the DeTurck Ricci-Bourguignon flow by inverse function theorem and obtain the short time existence of the Ricci-Bourguignon flow by DeTurck’s trick (see Section 3 for details). The precise statement of the short time existence for the DeTurck Ricci Bourguignon flow on compact manifolds with boundary is as follows.

Theorem 1.2Let(Mn,g(0))be a Riemannian manifold with boundary. Consider an arbitrary family of background metrics∈C∞(M×[0,∞))that satisfies the zeroth-order com-patibility condition~g(0) =g(x,0). Then for ρ

with the boundary conditions

where W(g,~g)l=Aαβis the second fundamental form on the boundary?M and LW(g,~g)g is the Lie derivative along the vector field W. The solution is C∞on MT??M×{0}, and isif the g(0)satisfies the compatibility conditions(1.2)and μ(0)=1.

The organization of this paper is as follows. In Section 3, we introduce the DeTurck Ricci-Bourguignon flow and show the relationship between the Ricci Bourguinon flow and the DeTurck Ricci Bourguinon flow. In Section 4, the solvability of a linear parabolic initial boundary value problem is obtained. In Section 5, by classic inverse function theorem, we prove the short time existence of the DeTurck Ricci-Bourguignon flow on the compact manifold with boundary.

2 Notation

In the following, we use Greek indices for the directions tangent to the boundary andnfor the direction of the inner unit normal vector with respect to the metricg(0). We use T for the symmetric(0,2)tensors onMand T?Mfor the restriction of the bundle T to?M. Let F denote the subbundle of T?Mconsisting of allη∈T?Msuch thatηαβ= 0 forα,β= 1···n?1 andηnn=0. Let F⊥denote the orthogonal complement of F with respect to the metricg(0).PrFis the orthogonal projection on the subbundle F. We usea?bto denote the linear combination of the tensorsaandb.MTdenotesM×[0,T).

3 The DeTurck Ricci-Bourguignon Flow

In this section, we consider the relationship between DeTurck Ricci Bourguignon flow and the Ricci-Bourguignon flow. The DeTurck Ricci Bourguignon flow is

whereW(g(t),t)l=g(t)lrg(t)pq(Γ(g(t))rpq?Γ(~g(t))rpq). In this paper, ~g∈C∞(M×[0,∞)) is a family of smooth background metrics that satisfies the zeroth-order compatibility condition~g(0) =g(x,0). Suppose thatg(t) is a solution to the DeTurck Ricci-Bourguignon flow with boundary condition

SincePrFg(x,t)=0, we haveg(x,t)αn=0. Hence on the boundary, the inverse matrix ofgijis

So the mean curvature is

By the theory of ordinary differential equation, there is a one-parameter transformationφ(t) :M→Msatisfying

with initial conditionφ(0,x)=x. On the boundary, sinceW(g(t))n=0, we haveφ(t):?M→?M. Sinceg(t) is a solution of the DeTurck Ricci-Bourguignon flow,φ?(t)(g(t)) satisfies the Ricci-Bourguignon equation

The mean curvature on the boundary of the metricφ?(g(t)) is

So ifg(t)is a solution to the DeTurck Ricci-Bourguignon flow with the boundary condition(3.2),thenφ(t)?(g(t)) is a solution to the Ricci-Bourguignon flow with constant mean curvature. As in [12], the boundary condition (3.2) is equivalent to

whereζis a symmetric (0,2)-tensor

andζαn(g(x,t))=0, x∈?M, t∈[0,T).

In the following,we only consider the DeTurck Ricci-Bourguignon flow(3.1)with the boundary condition (3.2).

4 A Linear Parabolic PDE with Initial Boundary Value Problem

In this section, we consider the existence of the linearized DeTurck Ricci-Bourguignon flow on manifold with boundary. The main theorem is in the following.

Theorem 4.1Consider the following linear parabolic initial boundary value problem on symmetric2tensors on M,

where F(x,t) ∈(?M×[0,T]), PrF⊥b(x,t) ∈

ProofThe proof is based on Theorem 10.1 in Chapter VII of [10] also see [14]. We only need to show that the boundary conditions satisfy the complementing conditions (see in of the book [10, Chapter VII, p.611]). We fix a point (x0,t0) on the boundary?M× [0,T], and choose a coordinate{xi}such thatgij(x0,t0)=δij, where?α∈Tp?M, α∈{1,···,n?1},and?n(x0,t0) is the inward normal vector. Let L0(g(t)) denote the principle part of the operator L(g(t)),

The principal symbol of the operator L0(g(t)) with coefficients freezing at the point(x0,t0)is

Now we compute the determinant of the principal symbol matrix det L0(x0,t0,iξ,p). Fix a coordinate system

Then the matrix is

Hence we have

whereFis ann×nmatrix

andHis amatrix withH=|ξ|2E. Now we compute the determinant det=det(pE+F)det(pE+H). Obviously,

We can writepE+Fas

whereα=1 ··· 1T, andβ=ξ21?|ξ|2···ξ2n?|ξ|2T.

Note that the vectorαis an eigenvector ofpE+F,

LetV={γ∈Rn, γT·β=0}.For anyγ∈V, we have

Note that the dimension ofVisn?1, andα /∈V. Hence the eigenvalues of matrixpE+Farep+|ξ|2with multiplicityn?1 andp+(1 ?2(n?1)ρ)|ξ|2with multiplicity 1. The determinant ofpE+Fis

Combining (4.9) and (4.13), we have

The roots ofL0(x0,t0;iξ,p) = 0 arep= ?|ξ|2andp= ?(1 ?2(n?1)ρ)|ξ|2. The matrix differential operator Lx,t,is parabolic ifρ

Now, we compute the adjoint matrix of L0(x0,t0;iξ,p) which is denoted by

Since

the inverse is

Obviously (pE+H)?1=We compute the inverse of the matrixpE+F. Since

we suppose

wherekis a constant to be determined.

Since

we have

Note thatGcan be written as

where

Since the vectorα=1 ··· 1Tis a eigenvector of the matrixpE+F, we have

Combining the above, we have

The adjoint matrix is

Next we compute the boundary differential operator matrixBy the definition of the subbundleF, the boundary conditionPrF(u)(x0,t0)=0 is equivalent to

In local coordinate, the conditioncan be expressed as

and

So the indices of the boundary equations in[10,Theorem 10.1]areσαn=?2, σnn=?1, σαβ=?1. Hence the principal symbols of the boundary differential operator at (x0,t0) are

and the matrix of the principal symbol of the boundary operator is

whereX=C1iξnEis ann×nmatrix,Z=C1iξnEismatrix ,Y=Eis a(n?1)×(n?1)type matrix andDenoteζ=(ξ1,···,ξn?1,0)∈Tx0?M,τ=ξnand (0,···,0,1) =νx0. Consider the polynomialL0(x0,t0;i(ζ+τν),p) as a function ofτon the whole complex plane. It has positive imaginary rootsτ=with multiplicity?1 andτ=with multiplicity 1. Denote

Now we prove that the row of the matrix B0(x0,t0;iξ,p)·is independent moduloL+(x0,t0;ζ,p,τ) with respect toτ. We observe that the independence of the row of the matrix B0(x0,t0;iξ,p)·(x0,t0;iξ,p)modulois equivalent to the independence of the row of the matrix

Ifτ=we have

Since Re(p)≥?δ|ζ|2for some 0<δ 0, we have the row of the matrix

Now we prove the rows of the matrix

belongs to the setV1=span{α}.

Ifτ=the nonzero solution of the linear equation (4.21) belongs to the setV2={γ∈Rn, βT·γ=0}.ObviouslyV1∩V2=0. So the row of the matrix

SinceX=iC1τE, the row of the matrix

Based on the above analysis, we conclude that

are linearly independent modulo the polynomialL+as a polynomial inτif the vectorζand the numberpsatisfy

where 0<δ

By [12, Chapter VII, Theorem 10.1], the linear parabolic initial boundary value problem has a unique solutionuij∈(M×[0,T]) and satisfies the following estimate

ifF(x,t) andb(x,t) satisfy the necessary compatible conditions.

5 A Boundary Value Problem for the DeTurck Ricci-Bourguignon Flow

In this section, we use inverse function theorem to prove the short time existence of the initial boundary value problem of the DeTurck Ricci-Bourguignon flow. Firstly recall the inverse function theorem (see [1, Chapter 3])

Theorem 5.1(see [1, 8])Assume thatE :U?B1→B2is a continuous differential map,where Bi, i= 1,2are Banach spaces and U is an open set in B1. If there is a continuous linear operator A:B2→B1such thatE′(x0)A= idB2, then there is a C1map g from the neighborhood of y0=E(x0)to the neighborhood of x0such thatE(g(y))=y.

In this section, we denote

and

LetB2=B3×B4.As closed linear subsets of Banach spaces,B1andB2are Banach spaces.

Now we apply the inverse function theorem to the operator

where E(h(x,t)) =+E(g0?tE(g(0),0)+h(x,t),B(g0?tE(g(0),0)+h(x,t)) andUis a neighbourhood of 0 inB1.

On the boundary?M, we have

sincegαn(x,0)=0 andW|t=0=0. So under the condition (1.2), we have

forx∈?M. Hence under the conditions of Theorem 1.2, the range of the map E is actually inB2and E is well defined.

Assume thatTis so small thatg0?tE(g0,0)is a metric onMfort∈[0,T). We also assume that for allh(x,t)∈U,g(t)=g0?tE(g(0),0)+h(x,t) is a metric onM. Now we prove that there is a bounded linear operator

such thatDE(0)°A=id,that is for any (f(x,t),b(x,t)) ∈B2, there is only oneu∈B1, such that

and

We compute the linearization of the Deturck Ricci-Bourguignon flow atg(t)=g0?tE(g(0),0).Denotegλ(t)=g(t)+λu(t),λ∈(??,?). The linearized operator ?DE(g(t),t) is

Here Δ and ?are Laplace operator and covariant differential operator respectively with respect to the metricg(t).M1andM2are smooth functions.

Next we compute the linearization of the boundary operatorB(g(t),t). Recall the boundary operator is

whereζis a (0,2)-tensor

andζαn(g(x,t),t)=0, x∈?M, t∈[0,T).

By computation, the linearization of the boundary operator is

whereM3is a smooth function.

Now we consider the solvability of linear equation

with boundary condition

and initial condition

whereb(x,t) ∈B4,f(x,t) ∈B3. From the conditionsu(x,0) = 0 andf(x,0) = 0, we haveSinceb(x,0) = 0, ?tuαn(x,0) = 0 =?tbαn(x,0), x∈?M,the necessary compatibility conditions forregularity are satisfied on?M×{t= 0}. Sinceg(t) =g(0)?tE(g(0)) ∈the regularity assumptions about the coefficients in Theorem 4.1 are satisfied. By Theorem 4.1, there is only one solutionu(x,t)∈B1with

In the following, we verify that the map E is continuously differentiable.

Lemma 5.1Fori=1,2,

ProofBy computation, we have

whereM1(g(t),~g(t))is a smooth function ofg,?g,~g,?~gandM2is smooth function ofg,?g,?2g,~g,?~g,?2~g. SinceDE(g(t),t)vijis a linear operator, we can write

wherea,b,care smooth functions. So we have

Sincea,b,care smooth function, we have

Lemma 5.2Ifi=1,2, for the boundary operator B,‖(DB(g1,t)?DB(g2,t))v

ProofBy the definition of the boundary operator, we have

Hence

and

whereMis a smooth function. So we have

Similarly, we have

We now prove the short time existence of the Deturck Ricci Bourguignon flow on manifold with boundary. Letg(t)=g0?tE(g0,0). We have

whereLW(g0,~g(0))g0=0. By choosing smooth

where

is a symmetric 2 tensor, we have

On the boundary?M×[0,T], we also have fori,j=1,···,n?1 ori=j=n,

Obviously,B1

μ(t)2g0,tαβ=0.As forB1μ(t)2g0,t

nn, we have

Forα=1,···,n?1,

Hence

Since the boundary operatorB(g(t)) is continuously differential, we have the estimate

Similarly, we have

So for any?>0, we can choose a small 0

By classic inverse function theorem, there is anh(x,t) ∈B1, such thatg(x,t) =g(x,0)?tE(g(x,0),0)+h(x,t) satisfies

and

Now we have the local existence for the DeTurck Ricci-Bourguignon flow inBy standard interior regularity and boundary regularity estimate for the strictly parabolic type PDE systems, we obtain the following theorem.

Theorem 5.2Let g(t) ∈be a solution to the DeTurck Ricci-Bourguignonflow with boundary value(3.2). Let l=k+α. Then the following hold:(1) (Interior regularity)Suppose thatThen g(x,t)∈(M°×(0,T]).

(2) (Boundary regularity)If μ(t) ∈[0,T])and the data g0,μ(t),~g(t)satisfy the necessary compatibility conditions at ?M×{0},then g(x,t)∈

(3) (Boundary regularity for positive time)If μ(t)∈[0,T]), then g(x,t)∈for any0<δ

Sincethe DeTurck vector fieldWis inif

By Theorem 5.2, the DeTurck vector fieldW(g(t),t) ∈C∞(MT??M×0) if ~g(x,t) andμ(t) are smooth. By the differentiability property of the flow, we can obtain a unique flowφtfort>0, which is smooth onM×(0,T] andC1onM×[0,T], satisfying

AcknowledgementThe deepest gratitude goes to the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially.