STABILITY OF NON COMPACT STEADY AND EXPANDING GRADIENT RICCI SOLITONS

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ALIX DERUELLE

Abstract. We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the linear stability of such gradient Ricci solitons.

1. Introduction

Once a smooth manifoldM without boundary is fixed, the Ricci flow, introduced by Hamil- ton, can be considered as a dynamical system on the space of metrics ofM modulo the action of diffeomorphisms and homotheties. The fixed points of such dynamical system are called Ricci solitons. By definition, these solutions look likeτ(t)φ^∗_tgwhere τ(t) is a positive scaling factor, where (φ_t)_tis a one parameter family of diffeomorphisms andgis a fixed metric. If this family of diffeomorphisms is generated by a gradient vector field, we call these fixed points gradient Ricci solitons. Finally, if one plugs this family of metrics into the Ricci flow equation and evaluates this expression at a fixed time, one gets a static equation.

More precisely, a gradient Ricci soliton is a triplet (M, g,∇f) where (M, g) is a Riemannian manifold andf :M →R is a smooth function (called the potential function) satisfying

1

2L^∇f(g) = Ric(g) +g 2,

where ∈ {−1,0,1}. We will focus on the steady (= 0) and the expanding (= 1) cases.

We assume that (M, g) is complete, this suffices to ensure the completeness of the vector field

∇f : [Zha09].

Let us fix a gradient Ricci soliton (M, g0,∇^g⁰f0). In this paper, we study the stability of such gradient Ricci soliton under Ricci flow, i.e.







∂_tg=−2 Ric(g(t)) on M×(0,+∞), g(0) =g₀+h,

whereh is a symmetric 2-tensor onM (denoted by h∈S²T^∗M) such thatg(0) is a metric.

As we want to prove convergence to the fixed background metric g0, it is more convenient to consider the following modified (-)Ricci flow (MRF) :







∂_tg=−2 Ric(g(t))−g(t) +L^∇^g⁰^f0(g(t)) on M×(0,+∞), g(0) =g₀+h.

As a first remark, the (MRF) flow is equivalent with the usual Ricci flow : see lemma A.5.

Secondly, there is a vast amount of literature concerning the stability of Einstein metrics

1

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(constant potential function) under the Ricci flow both on compact and non compact man- ifolds. Rather than making an exhaustive list of the existing work, we prefer to discuss the differences between our approach and the other approaches shared by different works. For instance, in the compact Ricci-flat case [HM14] and the shrinking gradient Ricci soliton case [Ach12], [Kro15], the authors consider the previous modified Ricci flow with∇^g(t)f(t) instead of the fixed vector field ∇^g⁰f0 where f(t) is a smooth function depending on time implicitly produced by a minimizing procedure involving the relevant functional (Perelman’s energy or entropy). As we consider non Einstein steady and expanding Ricci solitons, they are necessarily non compact : in that setting, the corresponding functionals are not even defined.

For that reason, we decide to follow the strategy of [SSS11] on the stability of non compact Hyperbolic spaces. In the spirit of the so called DeTurck’s trick, as the (MRF) is a degenerate parabolic equation, we first need to consider the following modified (-)Ricci harmonic map heat flow (MRHF) :







∂_tg=−2 Ric(g(t))−g(t) +L^∇^g⁰^f0(g(t)) +L^V(g(t)) on M×(0,+∞) g(0) =g₀+h

whereV is a vector field defined in coordinates by : V_i :=g_ik(Γ(g)^k_rs−Γ(g₀)^k_rs)g^rs,or globally, V(g) := div_g₀g−1

2∇^g⁰tr_g₀g. (1)

As in [SSS11], we consider perturbations that are close in theL^∞norm to the fixed background gradient Ricci soliton. In order to state our first result, we need (to recall) several definitions.

Definition 1.1. • The weighted laplacian operator associated to a gradient Ricci soliton (M, g,∇f) is ∆_f := ∆ +∇f.

• The weighted Lichnerowicz operator associated to a gradient Ricci soliton (M, g,∇f) is L := −∆_f −2 Rm(g)∗, where (Rm(g)∗h)_ij := Rm(g)_ikljh_kl, for a symmetric 2- tensor h.

• A gradient Ricci soliton (M, g,∇f) is strictly stable if there exists a positive λ such that

Z

M

hLh, hidµ_f = Z

M

|∇h|²− h2 Rm(g)∗h, hidµ_f ≥λ Z

M

|h|²dµ_f, (2) for any symmetric 2-tensors h with compact support, where dµ_f :=e^fdµg, dµg being the Riemannian measure associated to the metric g.

This notion of stability is legitimated by the fact that the linearization of the (MRHF) is exactly the Lichnerowicz operator : proposition 2.1 in section 2.

Finally, we will consider the following convexity assumptions on the geometry of the fixed background gradient Ricci soliton (called under the same name (H)) :

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• If (M, g,∇f) is a steady gradient Ricci soliton then (H) consists in Ric≥0, lim

+∞R= 0 sup

M

|Rm(g)|<+∞.

• If (M, g,∇f) is an expanding gradient Ricci soliton then (H) consists in sup

M

|Rm(g)|<+∞, lim inf

x→+∞

f(x) rp(x)² >0, wherer_p denotes the distance function to a fixed point p∈M. Our main theorem is

Theorem 1.2. Let (M, g0,∇^g⁰f0) be a strictly stable expanding or steady gradient Ricci soliton satisfying(H). Then it is dynamically stable.

More precisely, if g is a continuous metric such that kg−g₀k_L2

f0

:=

Z

M

|g−g₀|²_g

0dµ_f₀ 1/2

=:I

is finite, there exists β := β(n, k₀, λ, I) > 0 such that the following holds, where k₀ :=

sup_M|Rm(g0)|. If g satisfies kg−g0k_L^∞ ≤ β, then there exists an immortal solution to the modified Ricci harmonic map heat flow converging exponentially fast to g₀ in any global C^k norm for k∈N.

Then, we are able to go back to the (MRF) :

Theorem 1.3. Let (g(t))_t∈[0,+∞) be the solution to (MRHF) obtained in theorem 1.2. Then there exists a one parameter family of diffeomorphisms (ψ_t)_t∈[0,+∞) of M that satisfies ψ₀ = Id_M and such that (ψ_t^∗g(t))_t∈[0,+∞) is a smooth solution to (MRF) with the same initial condition. Moreover, there exists a diffeomorphism ψ∞ of M such that

ψ_t→ψ∞, ψ^∗_tg(t)→ψ_∞^∗ g₀,

as t tends to +∞, where the convergence is in globalC^k norms, for anyk∈N.

Remark 1.4. If one is interested in weighted L^p_f

0(M, S²T^∗M) estimates with p≥2, where L^p_f

0(M, S²T^∗M) :=

h∈S²T^∗M | Z

M

|h|^pe^f⁰dµ_g₀ <+∞

,

one can actually prove theorem 1.2 in this setting in the case of gradient steady Ricci solitons.

In fact, such approach has been investigated by Bamler [Bam15] for symmetric spaces : the main tools coming from harmonic analysis are the Hardy-Littlewood maximal local operator together with the Hardy-Littlewood maximal inequality which holds for weights as in the case of gradient steady Ricci solitons, since the potential function has bounded mean oscillation.

Nonetheless, this approach is not applicable in the case of expanding gradient Ricci solitons because the potential function has not bounded mean oscillation...

In a second part, we investigate sufficient geometric assumptions that implyλ-stability, the assumption (H) being quite reasonable to assume.

We consider positively curved gradient Ricci solitons first. This choice is actually made with respect to the existing examples. Before stating the results, we describe the strategy to prove λ-stability. This is nothing more than proving that the bottom of the spectrum of the Lichnerowicz operator acting on symmetric two tensors that are square integrable

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with respect to the weighted measure dµ_f is positive. It is well-known that the spectrum can be decomposed into two parts : the essential spectrum and the discrete spectrum. We first analyse the essential spectrum by studying the bottom of the spectrum of the weighted laplacian acting on square integrable (in the weighted sense) functions (propositions 6.5 and 6.11). Then we investigate the non existence of non trivial eigenfunctions ofLassociated to a non positive eigenvalue by proving a priori exponential decay in the spirit of Agmon [Agm82]

and Simon [Sim75] : theorems 6.10 and 6.15.

• Concerning gradient Ricci steady solitons, the only known one with positive curvature is the Bryant soliton in dimension greater than 2. It behaves like a paraboloid. We are able to prove the following :

Theorem 1.5. Let(M, g,∇f)be a steady gradient Ricci soliton with positive sectional curvature with

lim+∞R= 0 ; lim inf

t→+∞e^αtinf

f=tRic>0,

for some α∈[0,1). Then the bottom of the spectrum of L is positive.

Remark 1.6. Note that the curvature of the cigar soliton onR², discovered by Hamil- ton, decays precisely as e^−f. Moreover, the author has shown that this decay is essentially sharp among nonnegatively curved steady gradient Ricci solitons[Der12]. In that sense, the previous theorem can be seen as a gap theorem in terms of curvature decay. Nonetheless, even if theorem 1.5 discards the cigar soliton, Ma and Witt have proved its stability[MW11a] using the special structure of the Ricci flow in dimension 2.

We observe that this curvature condition is satisfied by the Bryant soliton too since in this case, lim inft→+∞t²inff=tRic>0. Hence,

Corollary 1.7. The Bryant soliton is dynamically stable.

• Concerning expanding gradient Ricci solitons, Bryant, in unpublished notes, has also built a one parameter family of rotational symmetric examples on Rⁿ for n ≥ 3 [Section 5, Chap. 1, [CCG⁺07]] asymptotic to the cone (C(Sⁿ⁻¹), dr²+ (cr)²g_Sⁿ⁻¹) withc∈(0,+∞). These examples have positive curvature if and only ifc∈(0,1). In that setting, we are able to prove the following :

Theorem 1.8. Let (M, g,∇f) be an expanding gradient Ricci soliton with positive sectional curvature. Then the bottom of the spectrum of L is positive. In particular, it is dynamically stable.

In dimension 3, we are able to only assume positive scalar curvature thanks to an estimate due to Anderson and Chow [AC05]. More precisely,

Theorem 1.9. Let (M, g,∇f) be a 3-dimensional expanding gradient Ricci soliton satisfying (H) with positive scalar curvature with

lim inf

x→+∞e^αf(x)R(x)>0,

for some α∈ [0,1). Then the bottom of the spectrum of L is positive. In particular, it is dynamically stable.

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Remark 1.10. Chih-Wei Chen and the author proved that all expanding gradient Ricci solitons with quadratic curvature decay are asymptotically conical : the asymptotic cone is a metric cone whose section is a smooth compact manifold endowed with a C^1,α metric. See [CD] for more details. For instance, the assumption of theorem 1.9 on the asymptotical be- haviour of the scalar curvature is satisfied if the section of the asymptotic cone is a smooth positively curved sphere with curvature greater than 1 : in that case, lim infx→+∞r²_p(x)R(x) is the minimum of the scalar curvature restricted to the section which is positive.

Remark 1.11. According to Siepmann [Sie13], the Ricci curvature decays as e^−αf for any α∈[0,1) in case the asymptotic cone of an expanding gradient Ricci soliton is Ricci flat. In dimension3, being Ricci flat is equivalent to being flat. Therefore, the extra condition on the scalar curvature in theorem 1.9 essentially amounts to discard the flat case.

What about the stability of the Bryant expanding gradient Ricci solitons with negative curvature, i.e. the ones whose asymptotic cones are (C(Sⁿ⁻¹), dr² + (cr)²g_Sⁿ⁻¹) with c ∈ (1,+∞) ?

Actually, it seems that the stability of expanders is intimately related to rigidity : Chodosh [Cho14] proved that any positively curved expanding gradient Ricci solitons asymptotic to (C(Sⁿ⁻¹), dr²+ (cr)²g_Sⁿ⁻¹) withc∈(0,1) are rotationally symmetric and theorem 1.8 implies their stability. The principal ingredient here is the maximum principle for symmetric 2-tensors due to Hamilton : this does not hold in case of negative curvature. Nonetheless, we are able to prove that if the anglec is close enough to 1, then it is stable, the tool employed here is a Bochner method which can be seen as the counterpart of pointwise estimates due to maximum principles. There is no doubt that this method should lead to rigidity as well.

Theorem 1.12. Let (M, g,∇f) be an expanding gradient Ricci soliton satisfying (H) such that

infM R+n

2 −2 sup

S²T^∗M

σ(Rm(g))>0, sup

S²T^∗M

σ(Rm(g)) := sup{hRm(g)∗h, hi_g | h∈S²T^∗M, |h|_g = 1}

= sup{Rm(g)_ikljh_ijh_kl | h∈S²T^∗M, |h|_g = 1}.

Then the bottom of the spectrum of L is positive. In particular, it is dynamically stable.

Remark 1.13. Note that this condition is satisfied if the curvature operator is bounded by a sufficiently small constant, only depending on the dimension n and this is not a vacuous condition since Ricci expanders are normalized by their very definition. Indeed, the Bryant examples described above furnish expanding gradient Ricci solitons with arbitrary small cur- vatures when the parameter (angle) c goes to 1. This is in sharp contrast with the shrinking case : Munteanu and Wang [MW11c] have shown that if the Ricci curvature of a shrinking gradient Ricci soliton is small enough then it is isometric to Euclidean space.

Remark 1.14. One can ask if there is a finite threshold depending on the angle c >1 of the Bryant expanding gradient Ricci solitons concerning their rigidity/stability ?

Acknowledgements. The author would like to thank Gilles Carron for his comments on a first draft. This paper benefited from numerous conversations with Peter Topping, Panagiotis Gianniotis and Felix Schulze. Finally, the author would like to thank the referee for pointing out a certain number of inaccuracies in an earlier version of this paper.

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The author is supported by the EPSRC on a Programme Grant entitled ‘Singularities of Geometric Partial Differential Equations’ (reference number EP/K00865X/1).

2. The linear case

The purpose of this section is to give some feeling to the reader with these flows.

We start by computing the linearization of the modified Ricci Harmonic map heat flow : Proposition 2.1. Let (M, g,∇f) be a gradient Ricci soliton, i.e.

Ric +g

2 =∇²f,

with∈ {−1,0,1}. Then the linearized operator of the (MRHF) is the weighted Lichnerowicz operator, i.e.

D_g₀ −2 Ric(g)−g+L^∇^g⁰^f0(g) +LV(g)(g)

(h) = ∆_f₀h+ 2 Rm(g₀)∗h, for any symmetric2-tensorh∈S²T^∗M.

Proof. The linearization of the Ricci curvature atg₀ given for instance in [Chap. 2, [CLN06]]

is

D_g₀(−2 Ric)(h) = ∆_g₀h+ 2 Rm(g₀)∗h−Sym(Ric(g₀)⊗h) +∇^g⁰^,2trh−Ldivg0h(g0),

forh∈S²T^∗M and where, in coordinates, Sym(Ric(g0)⊗h)ij := Ric(g0)_ikh_kj+ Ric(g0)_jkh_ki. On the other hand, ifh∈S²T^∗M,

LV(g0+h)(g0+h) =LV(h)(g0) +LV(h)(h), that is,

D_g₀ LV(·)(·)

(h) =LV(h)(g₀).

Therefore,

Dg0 −2 Ric(g)−g+L^∇^g⁰f0(g) +LV(g)(g)

(h) = ∆g0h+ 2 Rm(g0)∗h−Sym(Ric(g0)⊗h) +∇^g⁰^,2trh−Ldivg0h(g₀) +LV(h)(g₀) +L^∇^g⁰^f0(h)−h.

Now, by the very definition ofV(g) given in (1), one has Dg0 −2 Ric(g)−g+L^∇^g⁰f0(g) +LV(g)(g)

(h) = ∆g0h+ 2 Rm(g0)∗h−Sym(Ric(g0)⊗h) +L^∇^g⁰^f0(h)−h.

Finally, since

L^∇^g⁰f0(h) =∇^g_∇⁰g0f0h+ 1

2Sym(L^∇^g⁰f0(g0)⊗h),

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and since (M, g₀,∇^g⁰f₀) is a gradient Ricci soliton, i.e. L^∇^g⁰^f0(g₀) = 2 Ric(g₀) +g₀, D_g₀ −2 Ric(g)−g+L^∇^g⁰^f0(g) +LV(g)(g)

(h) = ∆_g₀h+ 2 Rm(g₀)∗h−Sym(Ric(g₀)⊗h) +∇^g_∇⁰g0f0h+1

2Sym(L^∇^g⁰^f0(g0)⊗h)−h

= ∆f0h+ 2 Rm(g0)∗h−Sym(Ric(g0)⊗h) +1

2Sym(L^∇^g⁰^f0(g₀)⊗h)−h

= ∆_f₀h+ 2 Rm(g₀)∗h.

Hence the result.

We continue with a general computation concerning the evolution of the weighted divergence of a solution to the linearized case of (MRHF).

Theorem 2.2. Let (M, g,∇f) be a gradient Ricci soliton, i.e.

Ric +g

2 =∇²f,

with∈ {−1,0,1}. Let h be a symmetric 2-tensor satisfying the heat equation ∂th = ∆fh+ 2 Rm∗h. Then the vector field div_f(h) := div_g(h) +h(∇^gf) satisfies

∂t−∆_f + 2

(div_f(h)) = 0.

Proof. First, note that

∂t(h(∇f))i = (∂th_ik)∇_kf = ∆_fh_ik∇_kf + 2(Rm∗h)_ik∇_kf

= ∆(h_ik∇_kf) +∇_jh_ik∇_kf∇_jf + 2(Rm∗h)_ik∇_kf

−2∇_lhik∇_l∇_kf−hik∆(∇_kf)

= ∆(h(∇f)i) +∇_j(h(∇f)_i)∇_jf −h_ik∇_j∇_kf∇_jf + 2(Rm∗h)_ik∇_kf

−2∇_lh_ik∇_l∇_kf−h_ik∆(∇_kf)

= ∆f(h(∇f)i)−

2h(∇f)i−h(Ric(∇f))i+ 2(Rm∗h)_ik∇_kf

−2∇_lh_ik∇_l∇_kf−h_ik(∇_k∆f + Ric(∇f)_k)

= ∆_f(h(∇f)_i)−

2h(∇f)_i−2h(Ric(∇f))_i+ 2(Rm∗h)_ik∇_kf

−divh−2∇_lh_ikRic_lk−h(∇R)_i, Hence, by the soliton identities A.1,

∂_t(h(∇f)) = ∆_f(h(∇f))−

2h(∇f) + 2(Rm∗h)(∇f)−divh−2∇h∗Ric, where (Rm∗h)(∇f)_i:= Rmijlkhjl∇_kf and (∇h∗Ric)i=∇_lhikRiclk.

Now,

∂_tdivh= div(∆_fh+ 2 Rm∗h).

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First of all,

div(∇_∇fh) = ∇_i(∇_kf∇_kh·i) =∇_i∇_kf∇_kh·i+∇_kf∇_i∇_kh·i

= divh

2 + Ric_ik∇_kh·i+∇_∇f(divh)− ∇_kf(Rm^p_ik·h_pi+ Rm^p_ikihp·)

= divh

2 + Ric∗∇h+∇_∇f(divh) +h(Ric(∇f))−(Rm∗h)(∇f) Also,

div ∆h = ∇_i∇_k∇_kh·i = ∆(divh)− ∇_k(Rm^p_ik·hpi+ Rm^p_ikihp·) + [∇_i,∇_k]∇_kh·i

= ∆(divh) +∇_k(Ric_kphp·−(Rm∗h)_k·) + [∇_i,∇_k]∇_kh·i

= ∆(divh) +h(div Ric) + Ric∗∇h−div(Rm∗h) + [∇_i,∇_k]∇_kh·i. Note that

[∇_i,∇_k]∇_kh·i = −Rm^p_ikk∇_ph·i−Rm^p_iki∇_kh·p−Rm^p_ik·∇_kh_pi

= −Ricip∇_ph·i+ Ric_kp∇_kh·p−Rm^p_ik·∇_khpi

= −Rm^p_ik·∇_khpi, and, by the soliton identities,

div(Rm∗h) = ∇_i(Rm·klihkl) = div(Rm)·klhkl+ Rm·kli∇_ihkl

= Rm·kjl∇_jf h_kl+ Rm·kli∇_ih_kl

= −(Rm∗h)(∇f) + Rm·kli∇_ih_kl. Therefore, [∇_i,∇_k]∇_kh·i=−Rm^p_ik·∇_kh_pi.

Finally, again, by the soliton identities and the first Bianchi identity,

∂t(div_fh) = ∆_f(div_fh)−div_fh

2 +h(div Ric + Ric(∇f)) + div(Rm∗h) + [∇_i,∇_k]∇_kh·i+ Rm∗h(∇f)

= ∆_f(div_fh)−div_fh

2 .

With the help of the Gronwall lemma, one can prove the following :

Theorem 2.3. Let (M, g,∇f) be a gradient Ricci soliton, i.e.

Ric +g

2 =∇²f,

with∈ {−1,0,1} which is strictly stable with constantλ as in definition 2.

Let (h_t)_t∈[0,T], withT >0, be a one-parameter family of2-symmetric tensors satisfying the heat equation∂_th_t= ∆_fh_t+ 2 Rm∗h_t such that, for any t∈[0, T],

ht∈L²_f and divfht∈L²_f.

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Then h_t and the vector field div_fh_t satisfy for anyt∈[0, T], kdivfhtk_L2

f ≤ e⁻²^tkdivfh0k_L2 f, kh_tk_L2

f ≤ e^−λtkh₀k_L2 f. 3. Existence

Fix now once for all a background gradient Ricci soliton (M, g₀,∇^g⁰f₀) satisfying (H).

We begin by investigating theC⁰-evolution of the difference g(t)−g0 :

Lemma 3.1. Let g(·) be a solution to the modified Ricci harmonic map heat flow on (0, T).

Then,

∂

∂tg_ij = g^ab∇^g_a⁰∇^g_b⁰g_ij+∇^g_∇⁰g0f0g_ij (3)

−g^klgipRm(g0)_jklp−g^klgjpRm(g0)_iklp+g_ikRic(g0)_kj +g_jkRic(g0)_ki (4)

+(g⁻¹∗g⁻¹∗ ∇^g⁰g∗ ∇^g⁰g)ij, (5)

where, if A and B are two tensors, A∗B means some linear combination of contractions of A⊗B.

In particular, if g(·) is a solution to the modified Ricci harmonic map heat flow on (0, T) which is β-close to g0 for some positive β sufficiently small, then

∂

∂t|h|² ≤ g⁻¹∗ ∇^g⁰∇^g⁰|h|²+∇^g⁰f₀· |h|²−2(1−β)|∇^g⁰h|² +4 Rm(g₀)(h, h) +Cβ|h|²,

where

h:=g−g₀,

g⁻¹∗ ∇^g⁰∇^g⁰|h|² :=g^ij∇^g_i⁰∇^g_j⁰|h|², Rm(g0)(h, h) := Rm(g0)ikljhklhij,

and C is a positive constant depending on the dimension n andk₀= sup_M|Rm(g₀)|only.

Proof. We do the proof in the case of an expanding gradient Ricci soliton (= 1). According to Shi [Lemma 2.1, [Shi89]],

∂

∂tgij = g^ab∇^g_a⁰∇^g_b⁰gij −g^klgipRm(g0)jklp−g^klgjpRm(g0)iklp

+(g⁻¹∗g⁻¹∗ ∇^g⁰g∗ ∇^g⁰g)ij +L^∇^g⁰^f0(g)ij−gij,

where, ifA and B are two tensors, A∗B means some linear combination of contractions of A⊗B.

Now, by the soliton equation,

L^∇^g⁰^f0(g)−g = ∇^g_∇⁰g0f0g+1

2Sym(L^∇^g⁰^f0(g0)⊗g)−g

= ∇^g_∇⁰g0f0g+ Sym(Ric(g₀)⊗g).

Hence the result.

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If the solution is β close tog₀, then :

∂

∂t|h|² = 2 ∂

∂tg, h

= 2hg⁻¹∗ ∇^g⁰∇^g⁰g−Rm(g0)∗g⁻¹∗g, hi

−2hg−L^∇^g⁰f0(g), hi+h∇^g⁰g∗ ∇^g⁰g, hi

≤ g⁻¹∗ ∇^g⁰∇^g⁰|h|²−2(1−β)|∇^g⁰h|²

−2hg−L^∇^g⁰^f0(g), hi −2hRm(g₀)∗g⁻¹∗g, hi,

where (Rm(g0)∗g⁻¹∗g)ij := g^klgipRm(g0)jklp+g^klgjpRm(g0)iklp. Now, using the soliton equation satisfied byg₀,

hg−L^∇^g⁰^f0(g), hi=|h|²− hL^∇^g⁰^f0(h), hi −2hRic(g₀), hi.

Moreover, as

L^∇^g⁰^f0(h) = ∇^g_∇⁰g0f0h+1

2(L^∇^g⁰^f0(g₀)⊗h+h⊗L^∇^g⁰^f0(g₀)),

= ∇^g_∇⁰g0f0h+h+ Ric(g0)⊗h+h⊗Ric(g0), we get,

∂

∂t|h|² ≤ g⁻¹∗ ∇^g⁰∇^g⁰|h|²−2(1−β)|∇^g⁰h|²−2hRm(g₀)∗g⁻¹∗g, hi +4hRic(g₀), hi+∇^g⁰f₀· |h|²+ 2hL^∇^g⁰f0(g₀)⊗h, hi

≤ g⁻¹∗ ∇^g⁰∇^g⁰|h|²+∇^g⁰f0· |h|²−2(1−β)|∇^g⁰h|²

+4hRic(g₀)⊗h, hi+ 4hRic(g₀), hi −2hRm(g₀)∗g⁻¹∗g, hi.

Finally, let us compute the curvature term in coordinates at a fixed point and a fixed time such that (g_0ij) = (δ_ij) and (g_ij) = (λ_iδ_ij) whereλ_i are positive numbers.

hRm(g₀)∗g⁻¹∗g, hi = 2 Rm(g0)_ikkiλ⁻¹_k λi(λi−1)

= 2 Rm(g₀)_ikki(λ⁻¹_k −1)λ_i(λ_i−1) +2 Rm(g₀)_ikkiλ_i(λ_i−1)

= −2 Rm(g₀)_ikki(λ_k−1)(λ_i−1)λ_i λ_k

+2 Rm(g0)_ikki(λi−1)²+ 2 Rm(g0)_ikki(λi−1)

= −2 Rm(g₀)(h, h) + Rm(g0)∗h^∗3 +2hRic(g₀)⊗h, hi+ 2hRic(g₀), hi, where we used λi/λ_k= (λi/λ_k−1) + 1 and the fact that

|λ_i/λ_k−1| ≤ 2 1− |h||h|

≤ C|h|,

ifβ is less than a sufficiently small fixed constant (say 1/2 here).

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Summing it up, we get

∂

∂t|h|² ≤ g⁻¹∗ ∇^g⁰^,2|h|²+∇^g_∇⁰g0f0|h|²

−2(1−β)|∇^g⁰h|²+ 4 Rm(g₀)(h, h) +Cβ|Rm(g₀)∗h^∗2|.

We use this previous differential inequality to solve Dirichlet problems of (MRHF) on geodesic balls of (M, g0).

Corollary 3.2. Let (g(t))_t∈[0,T) with0< T <∞, be a solution of (MRHF) onB_g₀(p, R)with g(t)|_∂B_g

0(p,R)=g0 |_∂B_g

0(p,R) .

Let δ be a positive number. Then there exists β₁=β₁(n, T, δ, k₀)>0 such that kg(0)−g₀k_L∞(Bg0(p,R))≤β₁

implies

kg(t)−g₀k_L∞(Bg0(p,R)×[0,T))≤δ.

Proof. Define β₁² := δ²e^C(n,k⁰^)T where the choice of C(n, k₀) will be made clearer below.

Assume without loss of generality thatδ is smaller thanβ0 whereβ0 is as in lemma 3.1.

Then lemma 3.1 and the maximum principle applied to h(t) :=g(t)−g0 imply that kg(t)−g0k²_L∞(Bg0(p,R)) ≤ kg(0)−g0k²_L∞(Bg0(p,R))e^C(n,k⁰^)t

≤ β₁²e^C(n,k⁰^)T ≤δ², fort∈[0, T) as long askg(t)−g0k_L^∞_(B_g

0(p,R))≤β0 is verified. Now,kg(t)−g0k²_L∞(Bg0(p,R))≤ δ² implies that the condition kg(t)−g0k_L^∞_(B_g

0(p,R)) ≤β0 remains true for t∈[0, T). Hence the result.

Remark 3.3. Corollary 3.2 also holds for global solutions, i.e. defined on M as soon as the maximum principle is applicable, e.g. if, a priori the difference |g(t)−g₀|is bounded for any time t∈[0, T).

Now, if we manage to prove a priori that our solution is sufficiently close to the background metric g0, this solution will exist for all time. This is the content of the following theorem which is the analogue of theorem 2.1 of [SSS11].

Theorem 3.4. There exists a positive constantδ˜= ˜δ(n, k₀)such that the following holds. Let 0< β < δ ≤δ. Then every metric˜ g(0)β-close to g₀ has aδ-maximal solution g(t)_t∈[0,T_(g(0))) with T(g(0)) positive and kg(t)−g₀k_L∞(M) < δ for all t ∈ [0, T(g(0))). The solution is δ- maximal in the following sense. Either T(g(0)) = +∞ and kg(t)−g₀k_L∞(M) < δ for any nonnegative time t or we can extend(g(t))_t to a solution on Mⁿ×[0, T(g(0)) +τ), for some positive τ =τ(n, k0) and kg(T(g(0)))−g0k_L^∞_(M)=δ.

Proof. The proof follows exactly the proof of theorem 2.1 of [SSS11] : their proof is based on a previous work by Simon [Sim02] dealing with the usual modified Ricci harmonic map heat flow∂tg=−2 Ric(g) +LV(g)(g). In our case, the equation differs only by the presence of the covariant derivative ∇^g_∇⁰g0f0gas seen in the proof of lemma 3.1.

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In [SSS11], the authors only use the following ingredients.

Interior estimates from [Section 2-4, [Sim02]] on the covariant derivatives of a solution to the usual (MRHF) that holds here since the gradient ∇^g⁰f₀ is bounded on compact subsets of M. Note however that we are not claiming for the moment that global bounds on the covariant derivatives of a solution to (MRHF) hold. Indeed, the unboundedness of ∇^g⁰f₀ in the expanding case could cause some trouble, it turns out that it does not : lemma 4.3.

Then, a diagonal argument combined with Arzela-Ascoli’s theorem shows the existence of a global solution (g(t))_t∈[0,τ_]withτ =τ(n, k₀) such thatg(t)→g(0) ast→0 in the C_loc⁰ -norm.

Then, with the help of the maximum principle applied to lemma 3.1, one can extend the solution in order to get aδ-maximal solution.

With the help of the a priori estimates of corollary 3.2, a useful corollary for the sequel consists in the following :

Corollary 3.5. With the same assumptions as in theorem 3.4, if T > 0 is given, and if β(n, T, δ, k0) is small enough, then the solution (g(t))_t∈[0,T_(g(0))+τ) satisfiesT(g(0))≥T.

4. Convergence

Again, consider a background gradient Ricci soliton (M, g₀,∇^g⁰f₀) satisfying (H).

If (g(t))t∈[0,T) is close enough to g0, we consider the truncated L²_f₀ norm of the difference h(t) :=g(t)−g0 as a possible Lyapunov function as in [SSS11].

Proposition 4.1. Assume (M, g0,∇^g⁰f0) is strictly stable. Let (g(t))_t∈[0,T₎ be a solution to (MRHF) onB_g₀(p, R)withg(t) =g₀on∂B_g₀(p, R)×[0, T). Assume thatkg−g₀k_L∞(Bg0(p,R)×[0,T)) ≤ β≤β(λ, k₀, n). Then we have

Z

Bg0(p,R)

|g(t)−g₀|²dµ_f₀ ≤e⁻^˜^λt Z

Bg0(p,R)

|g(0)−g₀|²dµ_f₀, where ˜λ= ˜λ(β)<2λ such thatlimβ→0λ(β) = 2λ.˜

Proof. These estimates follow by integrating the pointwise estimates of lemma 3.1 and by using integration by parts :

d dt

Z

Bg0(p,R)

|h|²dµ_f₀ ≤ Z

Bg0(p,R)

g⁻¹∗ ∇^g⁰^,2|h|²+∇^g_∇⁰g0f0|h|²dµ_f₀ +

Z

Bg0(p,R)

−2(1−β)|∇^g⁰h|²+ 4 Rm(g₀)(h, h) +C(n, k₀)β|h|²dµ_f₀

≤ Z

Bg0(p,R)

∆_f₀|h|²+ g⁻¹−g₀⁻¹

∗ ∇^g⁰^,2|h|²dµ_f₀ +

Z

Bg0(p,R)

−2(1−β)|∇^g⁰h|²+ 4 Rm(g0)(h, h) +C(n, k0)β|h|²dµ_f₀

≤ −2(1−β) Z

Bg0(p,R)

|∇^g⁰h|²−2 Rm(g₀)(h, h)dµ_f₀ +C(n, k₀)β

Z

Bg0(p,R)

|h|²dµ_f₀ − Z

Bg0(p,R)

hdiv_f₀(g⁻¹),∇^g⁰|h|²idµ_f₀.

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To get rid of the divergence term, as the vector field ∇^g⁰f₀ is not necessarily bounded, we cannot simply bound this term by the norm of the covariant derivative of h. Therefore, we need the following identity inspired by the work of Koiso [Koi82] :

2<∇^g⁰T,∇^g⁰T >_L²

f0

= <Cod(T),Cod(T)>_L²

f0

+2<div_f₀(T),div_f₀(T)>_L²

f0

−2<Ric_f₀(T), T >_L²

f0

+2<Rm(g0)∗T, T >_L²

f0

= kCod(T)k²_L2 f0

+ 2kdiv_f₀(T)k²_L2 f0

+kTk²_L2 f0

+ 2<Rm(g₀)∗T, T >_L2 f0

,

for any T ∈ C₀^∞(M, S²T^∗M) where Cod(T)_ijk := ∇^g_i⁰T_jk − ∇^g_j⁰T_ik is the Codazzi tensor associated to T and where Ricf0 := Ric(g0)− ∇^g⁰^,2f0 is the Bakry- ´Emery tensor. Here 2 Ricf0 = −g₀ in the case of expanding gradient Ricci solitons. Hence the L²_f₀-norm of the divergence of T is bounded by the L²_f

0-norm of T and ∇^g⁰T. Therefore, by applying the Young inequality,

d dt

Z

Bg0(p,R)

|h|²dµf0 ≤ −(2−C(n, k0)β) Z

Bg0(p,R)

|∇^g⁰h|²−2 Rm(g0)(h, h)dµf0

+C(n, k₀)β Z

Bg0(p,R)

|h|²dµ_f₀.

Now, as (M, g0,∇^g⁰f0) is strictly stable, and ifβ is small enough, we get : d

dt Z

Bg0(p,R)

|h|²dµ_f₀ ≤ −((2−C(n, k₀)β)λ−C(n, k₀)β) Z

Bg0(p,R)

|h|²dµ_f₀, with (2−C(n, k0)β)λ−C(n, k0)β >0.

Since we are building a solution with the help of Dirichlet exhaustions, the previous estimate extends to M×[0, T) :

Corollary 4.2. Assume (M, g0,∇^g⁰f0) is strictly stable. Let T > 0. Assume g(0)−g0 ∈ L²_f

0(M, S²T^∗M).

Then there exists β₀ =β₀(n, T, k₀, λ) such that if g(0) is β₀-close to g₀ then a solution to (MRHF) (g(t))_t∈[0,T) onM exists, is β-close with β as in proposition 4.1 and satisfies

kg(t)−g₀k²_L2

f0(M,S²T^∗M)≤e⁻^λt^˜ kg(0)−g₀k²_L2

f0(M,S²T^∗M), for all t∈[0, T) withλ˜ defined in proposition 4.1.

Such an exponential decay of the L²_f

0-norm of g(t)−g0 implies an exponential decay of the supremum norm ofg(t)−g0. Before stating this result, we need to uniformly control the covariant derivatives ofg(t) in time :

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Lemma 4.3. Let (g(t))_t∈[0,T₎ be a solution of (MRHF) on M such that g(t) isβ-close to g₀ for any t∈[0, T) withβ ≤β(n, k0).Then,

sup

M

|∇^g⁰^,jg(t)| ≤ c(n, j, k0) t^j , for all t∈(0,min{1, T}].

Proof. The proof is almost the same as the proof of theorem 4.3 in [Sim02]. The only thing that could cause trouble is the presence of the vector field ∇^g⁰f0 that is unbounded in the expanding case. But it turns out that it does not by considering adequate cut-off functions built with the help of the potential function. To convince the reader, we reproduce the proof for the first covariant derivative.

As (Mⁿ, g₀,∇^g⁰f₀) is a gradient Ricci soliton with bounded curvature, the covariant derivatives of the curvature operator are bounded in the following way by Shi’s estimates [Chap. 6, [CLN06]] :

sup

M

|∇^kRm(g₀)| ≤C(n, k, k₀), for any nonnegative integerk and wherek0 = sup_M|Rm(g0)|.

• First of all, since (g(t))_t∈[0,T₎satisfies (3), by using commutation formula and identities on gradient Ricci solitons, one has onM×[0, T], similarly to the proof of [lemma 4.1, [Shi89]] :

∂t|∇^g⁰g|² ≤ g⁻¹∗ ∇^g⁰^,2|∇^g⁰g|²+∇^g_∇⁰g0f0|∇^g⁰g|²−1

2|∇^g⁰^,2g|² (6) +c(n, k0)|∇^g⁰g|²+c(n)|∇^g⁰g|⁴+c(n, k0). (7) Indeed, formally speaking, the only term we have to take care of is the term involving the gradient of the potential function :

∇^g_i⁰∇^g_∇⁰g0f0g = ∇^g_i⁰(∇^g_k⁰f0∇^g_k⁰g)

= ∇^g_ik⁰^,2f₀∇^g_k⁰g+∇^g_k⁰f₀∇^g_k⁰∇^g_i⁰g+∇^g_k⁰f₀[∇^g_i⁰,∇^g_k⁰]g, which implies,

∇^g⁰∇^g_∇⁰g0f0g=∇^g_∇⁰g0f0∇^g⁰g+∇^g⁰^,2f0∗ ∇^g⁰g+ (Rm(g0)∗ ∇^g⁰f0)∗ ∇^g⁰g,

where Rm(g₀)∗ ∇^g⁰f₀(·,·,·) := Rm(g₀)(·,∇^g⁰f₀,·,·). Now, by the soliton equation,

∇^g⁰^,2f0is bounded onM by a constantc(n, k0), and so is Rm(g0)∗ ∇^g⁰f0 by lemmata A.1, A.2. Hence the differential inequality given by (6), (7).

• Secondly, define φ:=a(n) +Pn

k=1λ^m_k(t) where λ_k(t) are the eigenvalues of g(t) and m=m(n) is an integer. Again, a straightforward adaptation of the computations of the proof of [lemma 4.1, [Shi89]], gives on M×[0, T],

∂_tφ≤g⁻¹∗ ∇^g⁰^,2φ+∇^g_∇⁰g0f0φ− m²

8 |∇^g⁰g|²+c(n, k₀), with suitablea,m andβ depending onn.

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• Hence, ifψ(x, t) :=φ(t, x)|∇^g⁰g|²(t, x), we get (proof of [lemma 4.1, [Shi89]]) :

∂tψ≤g⁻¹∗ ∇^g⁰^,2ψ+∇^g_∇⁰g0f0ψ−ψ²

16 +c(n, k0), (8)

on M×(0, T].

• Now, we adapt some arguments from the proof of [lemma 4.1, [Sim02]].

Define the function F(x, t) := tψ(x, t) on M ×[0, T]. By (8), we get on M × (0,min{1, T}],

∂_tF ≤g⁻¹∗ ∇^g⁰^,2F+∇^g_∇⁰g0f0F− 1

16tF²+c(n, k₀) +F t.

As the potential function of a gradient Ricci soliton is defined up to an additive constant and because of its growth at infinity given by lemma A.4, we assume that min_Mf₀ = 1. Let η :M → [0,1] be a smooth compactly supported function on M defined by : η(x) := ˜η(p

f0(x)/r) with r > 1 and ˜η : [0,+∞[→ [0,1] is a smooth compactly supported function such that

η|˜_[0,1/2]≡1, η|˜_[1,+∞[≡0, η˜⁰ ≤0, η˜⁰²

˜

η ≤c, η˜⁰⁰≥ −c, wherec is a universal constant. Now,

∇^g⁰η= η˜⁰ r∇^g⁰p

f₀, ∇^g⁰^,2η = η˜⁰⁰ r²∇^g⁰p

f₀⊗ ∇^g⁰p f₀+η˜⁰

r ∇^g⁰^,2p

f₀. (9)

Now, let us compute the evolution equation ofηF :

∂t(ηF) ≤ g⁻¹∗ ∇^g⁰^,2(ηF)−2g^ij∇^g_i⁰η∇^g_j⁰F −(g⁻¹∗ ∇^g⁰^,2η)F +∇^g_∇⁰g0f0(ηF)−(∇^g_∇⁰g0f0η)F − η

16tF²+c(n, k₀) +ηF t ,

Therefore, by considering a point (x₀, t₀) where the function ηF attains its maximum onM×[0, T], using the relations

∇^g⁰(ηF)(x0, t0) = 0, ∂t(ηF)(x0, t0)≥0, g⁻¹∗ ∇^g⁰^,2(ηF)(x0, t0)≤0,

with the help of the previous differential inequality, one gets the following, after mul- tiplying byη implicitly evaluated at (x0, t0) :

(ηF)² 16t₀ ≤

2

ηg^ij∇^g_i⁰η∇^g_j⁰η−g⁻¹∗ ∇^g⁰^,2η+ 1

t₀ − ∇^g_∇⁰g0f0η

(ηF) +c(n, k0).

Now, 2

ηg^ij∇^g_i⁰η∇^g_j⁰η≤c(n, β(n, k₀))|∇^g⁰p f₀|²η˜⁰²

rη˜

−g⁻¹∗ ∇^g⁰^,2η ≤c(β(n, k0))(−˜η⁰⁰) r² |∇^g⁰p

f0|²+(−˜η⁰) r ∆g0

pf0

−∇^g_∇⁰g0f0η≤ (−˜η⁰) 2√

f0r|∇^g⁰f0|².