A No Expanding Breather Theorem for Noncompact Ricci Flows

Home , Ricci flow

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Diff(M) stands for the group of self-diffeomorphisms and scalings, then a breather is a periodic orbit and a soliton is a static one. Therefore, the no breather theorem is tantamount to saying that the periodic orbits must also be static. Perelman’s proofs of the no breather theorems require the existence of minimizers for the -functional, the -functional, and the normalized -functional, respectively. When the manifoldW is compact,F such existence results are straightfoF rward applications of variational problems. A natural question to ask is, under what condition can the no breather theorems be established for noncompact manifolds, and how to carry our the proof. One may certainly attempt to find the minimizers of these functionals, and this is possible under certain geometric conditions. For results of this type, the reader may refer to [14] and [15]. We remark here that, in the noncomapct category, this approach is almost impossible for the no steady or expanding breather theorems. The reason is that the -functional or the normalized -functional are generally not finite on noncompact steady orF expanding gradient Ricci solitons,F respectively, when these functionals are evaluated using the potential functions of the corresponding solitons. (One may think of a Bryant soliton for example.) Another approach was initiated by the result of Lu-Zheng [12], where they constructed a Type I ancient solution using a shrinking breather, and proved, under certain geometric conditions, that the backward blow-down limit of this ancient solution must be the breather itself, which must also be a shrinking gradient Ricci soliton by Naber [9]. This method was refined, and the conclusion improved, by the second author [16], where the condition for the no shrinking breather theorem is reduced to bounded curvature alone. The authors [4] recently further reduced the condition for the no shrinking breather theorem to a lower bound of the Ricci curvature alone. In this paper, we continue our study in [4] and apply our method to noncompact expanding breathers. Recall that Feldman-Ilmanen-Ni [6] established some forward monotonicity formulas for the Ricci flow as the dual version of Perelman’s [13], whose equalities are fulfilled on expanding gradient Ricci solitons. We will be implementing Feldman-Ilmanen-Ni’s forward reduced geometry in this paper in the same way as we have applied Perelman’s reduced geometry in [4]. However, since the forward reduced geometry does not behave as nicely as Perelman’s reduced geometry (The reason, intuitively speaking, is this, that on steady or expanding solitons, as it is in the case of a shrinking soliton, the forward reduced volumes should coincide with the -functional or the normalized -functional, respectively, evaluated using the corresponding potentialF functions, whereas the latterF two are generally infinite in the noncompact case), we will need to impose some strong curvature conditions. Theorem 1.3. Let (M, g(t)) be a complete and noncompact expanding breather of the Ricci flow. Assume g(t) has bounded curvature on each time-slice and either one of the following is true. (1) g(t) satisfies the weak PIC 2 condition. (2) (M, g(t)) is Kähler with nonnegative− bisectional curvature. Then (M, g(t)) is the canonical form of an expanding gradient Ricci soliton. Remark 1.4. Our proof depends heavily on Hamilton’s Harnack estimate ([1], [2], and [11]), and this is why we assumed the above curvature conditions. The reader may easily verify that, if Hamilton’s trace Harnack is assumed to be valid, then a nonnegative Ricci curvature assumption is sufficient for the proof. Lott [8] gave an example of a complete but nongradient expanding soliton 3 on noncompact manifold (see page 635 in therein). This soliton has bounded curvature but does not have nonnegative Ricci curvature. Therefore, Theorem 1.3 cannot be proved in general without any curvature positivity condition. Before we conclude our introduction, a word is to be said about our method. Similar to [4], we constructed a Type III immortal solution starting from the given expanding breather and considered the monotonicity of the quantity 1 ˜ ℓ+ θ+(t) = n e− dµt, (4πt) 2 ZM where ℓ+ is the forward reduced distance constructed in [6]. Note that the monotonicity of this quantity relies on Hamilton’s Harnack, as proved in Theorem 4.3 of [10]. However, the forward reduced volume defined in [6]

1 ℓ+ θ+(t) = n e dµt, (4πt) 2 ZM being automatically decreasing along all Ricci flows on compact manifolds, is yet generally infinite on noncompact manifolds. The organization of the paper is as follows. In section 2, we review basic +-geometry L introduced by Feldman-Ilmanen-Ni [6]. In section 3, we estimate the ℓ+-distance on Type III immortal Ricci flows. In section 4, we prove the main theorem.

2. Preliminaries Let g(t) be a metric evolving by the Ricci flow equation on M [0, T]. We always assume that either M is compact or g(t) has bounded curvature at each× time. Fixing a point x, Feldman-Ilmanen-Ni [6] defined the following dual version of Perelman’s reduced distance, called the forward reduced distance function. t 1 2 ℓ+(y, t) = inf √η R(γ(η), η) + γ′(η) dη, (2.1) γ g(η) 2 √t Z0 | | where (y, t) M (0, T], and the infimum is taken among all piecewise smooth curves γ : [0, t] ∈ × → M satisfying γ(0) = x and γ(t) = y. (x, 0) is called the base point of ℓ+. We remark that since the minimizing curve in (2.1) also satisfies an +-geodesic equation, whose form is very similar to that of an -geodesic equation, one may easilyL modify the arguments in, say, Chapter 7 in [5], L to verify that ℓ+ is locally Lipschitz under our assumption. We then summarize some equations and inequalities satisfied by ℓ+. The reader may note their similarity to the case of Perelman’s -geometry. L Theorem 2.1 (Corollary 2.1 in [6]). The ℓ+ function satisfies the following equalities for almost every (y, t) M (0, T] ∈ × ∂ℓ+ ℓ+ K = R 3 , (2.2) ∂t − t − 2t 2 2 ℓ+ K ℓ+ = R + 3 , (2.3) |∇ | t − t 2 4 LIANG CHENG AND YONGJIA ZHANG

Moreover, ℓ+ satisﬁes the following inequalities in the barrier sense or in the sense of distribution. n K ∆ℓ+ R + 3 , (2.4) ≤ 2t − 2t 2 ∂ℓ+ 2 n + ∆ℓ+ + ℓ+ R 0, (2.5) ∂t |∇ | − − 2t ≤ 2 ℓ+ + n 2∆ℓ+ + ℓ+ R 0, (2.6) |∇ | − − t ≤ where t 3 K = η 2 H(X)dη, (2.7) Z0

X is the velocity of the minimizing +-geodesic connecting (x, 0) and (y, t), and L ∂R R H(X) = + 2 < R, X > +2Rc(X, X) + ∂t ∇ t is Hamilton’s trace Harnack. Furthermore, ℓ+(y, t) = X(t) whenever the minimizing +-geodesic connecting (x, 0) and (y, t) is unique. ∇ L

Theorem 2.2 (Theorem 4.3 in [10]). Let (M, g(t)) be a Ricci ﬂow with bounded curvature at each time slice. Assume Hamilton’s trace Harnack is nonnegative, then the quantity 1 ˜ ℓ+ θ+(t) = n e− dµt (4πt) 2 ZM is monotonically decreasing in t, where µt is the Riemannian measure of g(t) and ℓ+ is the forward reduced distance based at some ﬁxed point on M 0 . ×{ } Sketch of proof. Combing (2.2), (2.3), and (2.4), we have

∂ℓ+ 2 n K ∆ℓ+ + ℓ+ + R + 3 0 (2.8) ∂t − |∇ | 2t ≥ t 2 ≥ in the barrier sense or in the sense of distribution. Then, taking for granted the integration by parts at inﬁnity, we compute

n d ℓ+(x,t) (4πt)− 2 e− dµt dt ZM

n ∂ℓ+ 2 n ℓ+(x,t) = ∆ℓ+ + ℓ+ + R + (4πt)− 2 e− dµt − ZM ∂t − |∇ | 2t ! K n 2 ℓ+(x,t) = 3 (4πt)− e− dµt 0. − ZM t 2 ≤ Note that the above computation is valid in our case according to the estimates in the next section. 5

3. Estimates for ℓ+ function on Type III Ricci flows In this section, we use similar methods as in [9] to derive some estimates for the forward reduced distance on a Type III immortal Ricci ﬂow. Since Theorem 3.1(2)—(4) are already well established in [9], we shall be relatively brief in their proofs. Recall that an immortal solution (M, g(t))t [0, ) ∈ ∞ is called Type III, if there exists a constant C0 > 0, such that

C Rm 0 everywhere on M (0, ). (3.1) | |≤ t × ∞

n Theorem 3.1. Let (M , g(t))t [0, ) be an n-dimensional Type III immortal Ricci ﬂow with ∈ ∞ nonnegative Ricci curvature everywhere. Let ℓ+ be the forward reduced distance function based a ﬁxed point (x, 0). Furthermore, assume that there exists a sequence of space-time points (x j, t j) such that t j and { }∞j=1 ր∞

ℓ+(x j, t j) A < for all j 1. (3.2) ≤ ∞ ≥ Then, there exists a positive constant Q depending only on A, α (0, 1), n, and the Type III constant ∈ n 1 C0 in (3.1), such that the following inequalities hold for all (y, t) M [α− ,α] and for all j 1, understood in the barrier sense if any diﬀerentiation is involved. ∈ × ≥ 2 2 dg (t)(x j, y) dg (t)(x j, y) (1) K j (y, t) Q √t 1 + j , K j (y, t) Q 1 + j , | | ≤ √t ! |∇ | ≤ √t ! 2 ∂ Q dg (t)(x j, y) K j (y, t) 1 + j , ∂t ≤ ! √t √t d (x , y) 2 d (x , y) 2 1 g j(t) j j g j(t) j (2) 1 + Q ℓ+(y, t) Q 1 + , Q √t ! − ≤ ≤ √t ! j Q dg j(t)(x j, y) (3) ℓ+ (y, t) 1 + , |∇ | ≤ √t √t ! j 2 ∂ℓ Q dg (t)(x j, y) (4) + (y, t) 1 + j , ∂t ≤ t ! √t

1 j where g j(t) := t−j g(tt j) is the Ricci flow sequence obtained by Type III scaling, K is as defined in j (2.7) for the scaled Ricci flow g j, and ℓ ( , t) := ℓ+( , tt j) is the forward reduced distance based at + · · (x, 0) and with respect to the Ricci flow g j(t).

Proof. In this proof, we will use the capital letter C to denote a general estimation constant, which depends on α, n, A, and C0 as indicated in the statement of this theorem, and could vary from line to line. Since the Ricci ﬂow is Type III, by Shi’s gradient estimates we have

C C ∂R C ∂ C R (y, t) , Rm (y, t) , (y, t) , R , 3 2 5 | | ≤ t |∇ | ≤ t 2 ∂t ≤ t ∇∂t ≤ t 2

6 LIANG CHENG AND YONGJIA ZHANG for all (y, t) M (0, ). Let (y, t) M (0, ) be such that the minimizing +-geodesic γ ∈ × ∞ ∈ × ∞ L connecting (x, 0) and (y, t) is unique. We denote X := γ′ and calculate that t t 3 ∂R R 3 C C C 2 K = η 2 + + 2 R, X + 2Rc(X, X) dη η 2 + X + X dη | | ∂η η h∇ i ! ≤ η2 3 | | η | |  Z0 Z0  η 2    t C t t   2 √ 2 √ 1 dη + √η X dη 2C t + C √η( X + R)dη = 2C t(1 + ℓ+(y, t)). (3.3) ≤ Z0 η 2 Z0 | | ≤ Z0 | | Since the Type III scaling process does not alter the Type III constant in (3.1), we obtain from (2.2), (2.3), and (3.3) that j ∂ℓ+ C j (y, t) (1 + ℓ+(y, t)), (3.4) ∂t ≤ t

C ℓ j 2 (y, t) (1 + ℓ j (y, t)). (3.5) |∇ +| ≤ t + In view of the fact j ℓ (x j, 1) = ℓ+(x j, t j) A, + ≤ we may integrate (3.4) to obtain j 1 ℓ (x j, t) C for all t [α− ,α]. (3.6) + ≤ ∈ Integrating (3.5) in space and applying (3.6), we have 2 j dg j(t)(x j, y) 1 ℓ+(y, t) Q 1 + for all (y, t) M [α,α− ]. (3.7) ≤ √t ! ∈ × Conclusion (3), (4), the second inequality of conclusion (2), and the first inequality of conclusion (1) now follow from (3.3), (3.4), (3.5), and (3.7). 1 To obtain the first inequality of conclusion (2), we fix (y, t) M [α,α− ] and let γ1(s) and γ2(s) j ∈ × be minimizing +-geodesics from (x, 0) to (x , t) and to (y, t), respectively, all with respect to g j(t). L We denote f (s) := dg j(s)(γ1(s), γ2(s)). Then we have d d − f (s) = d , γ (s) + d , γ (s) + − d (γ (s), γ (s)) g j(s) 1′ g j(s) 2′ g j(τ) 1 2 ds h∇ i h∇ i dτ ! τ=s

j j d− = dg (s), ℓ (γ1(s), s) + dg (s), ℓ (γ2(s), s) + dg (τ )(γ1(s), γ2(s)) h∇ j ∇ + i h∇ j ∇ + i dτ j ! τ=s d ℓ j (γ (s), s) + ℓ j (γ (s), s) + − g (τ)(σ ,σ ) , + 1 + 2 j ′ ′ ≤ |∇ | |∇ | dτ Zσ q ! τ=s where σ is a unit speed minimizing geodesic from γ1(s) to γ2(s) with respect to metric g j(s), and d f f (s+h) f (s) − = lim inf − is the lower forward Dini derivative. By Lemma 18.1 in [5], we have ds h 0+ h → d− g j(τ)(σ′,σ′) = min Rcg j(s)(η′, η′) 0, dτ Z ! − η (s) Z ≤ σ q τ=s ∈Z η

7 where (s) denotes the set of all unit speed minimizing geodesics from γ (s) to γ (s) with respect Z 1 2 to metric g j(s). Then, by (3.5), we get d C C − f (s) 2 (1 + ℓ j (γ (s), s)) + 2 (1 + ℓ j (γ (s), s)). (3.8) ds ≤ r s + 1 r s + 2 On the other hand, since for all s (0, t], we have ∈ s t j 1 2 1 2 √t j ℓ+(γ2(s), s) = √η(R + X )dη √η(R + X )dη = ℓ+(y, t), 2 √s Z0 | | ≤ 2 √s Z0 | | √s

j √t j √t ℓ+(γ1(s), s) ℓ+(x j, t) C , ≤ √s ≤ √s where in the latter formula we have applied (3.6). Then, (3.8) becomes 1 d− t 4 j f (s) C 3 1 + 1 + l+(y, t) for all s (0, t]. ds ≤ s 4 q ! ∈ Integrating the above inequality from 0 to t, we obtain the ﬁrst inequality of conclusion (2). Finally, to obtain the last two inequalities of conclusion (1), we let γ be a minimizing +- L geodesic with respect to g, which connects (x, 0) and (y, t), and Y an +-Jacobi ﬁeld along γ, satisfying [X, Y] = 0 and L

1 2 η 2 η C XY = Ric(Y) + Y, Y(η) = Y(t) = , XY 1 1 . (3.9) ∇ 2η | | t | | t |∇ |≤ t 2 η 2 Then, we may compute t 3 ∂ 1 2 δY K = η 2 R, Y + R, Y + 2 R, X Y + 2 R, XY (3.10) | | Z ∇∂η ηh∇ i h∇ ⊗ i h∇ ∇ i 0 D E t 3 1 1 1 2 + 2 Ric(Y, X, X) + 2Ric(X, Y) dη C η 2 + X + X dη X  1 1 3 1  ∇ ∇ ≤ Z0 t 2 η2 t 2 η 2 | | t 2 η| |   t t   C 1 C 2   1 dη + √η( X + R)dη C(1 + ℓ+(y, t)). ≤ √t Z0 η 2 √t Z0 | | ≤ Since the Type III constant in (3.1) is not aﬀected by the Type III scaling, we obtain the second inequality of conclusion (1) by (3.10). Next, we observe that

∂ d 3 ∂R R K (y, t) = K K, X (y, t) t 2 + + 2 R, X + 2Rc(X, X) + K X ∂t dη η=t − h∇ i ≤ ∂t t h∇ i |∇ || |

C 2 C 2 + C X + C √t X + C(1 + ℓ+(y, t)) X (1 + ℓ+(y, t)) + C √t ℓ+ (y, t) ≤ √t | | | | | |≤ √t |∇ | C (1 + ℓ+(y, t)). ≤ √t Here we have used X(t) = ℓ+(y, t) and formula (3.5). Again, by the scaling invariance of the Type III constant, we obtain the∇ third inequality of conclusion (1). 8 LIANG CHENG AND YONGJIA ZHANG

4. the proof of the main theorem In this section, we prove the main theorem by implementing the method of, [4], [12], and [16]. Since our techniques and arguments are very similar to that of [4], we are not including obvious modiﬁcations, and the readers are referred to this paper for more detailed treatment. Let (M, g0(t)) be an expanding breather as described in the statement of Theorem 1.3. Note that the assumptions therein guarantees the validity of Hamilton’s Harnack estimate ([1], [2] and [11]). After rescaling and translating in time, we may assume that there exists α (0, 1) and a diﬀeomorphism φ : M M, such that ∈ →

αg0(1) = φ∗g0(0). (4.1)

Similar to [4], for each j 0 we deﬁne ≥ j k t j = α− , t0 = 1, Xk=0 j j j g j(t) = α− (φ )∗g0(α (t t j 1)), t [t j 1, t j]. − − ∈ −

Obviously, t j . We may then define the spliced immortal solution ր∞ g (t), t [0, 1], g(t) = 0 ∈ ( g j(t), t [t j 1, t j]. ∈ − = C It is straightforward to check that g j(t j 1) g j 1(t j 1) and Rmg j(t) t for all j 1, where the constant C depends only on α and the curvature− − bound− of the| original| ≤ breather. Then≥ the immortal solution g(t) is of Type III and is smooth by the uniqueness of the Ricci flow (c.f. [3] and [7]). ( j+1) Next, we fix a point p M and define x j = φ (p ) for j 0. Let ℓ+ be the reduced distance 0 ∈ − 0 ≥ from (p0, 0), and we shall proceed to show

lim sup ℓ+(x j, t j) < . (4.2) j ∞ →∞ Let σ : [0, 1] Mn be a smooth curve satisfying σ(0) = p and σ(1) = x = φ 1(p ). We deﬁne → 0 0 − 0 σ j : [t j, t j+ ] M and γ j : [0, t j+ ] M as 1 → 1 → ( j+1) j+1 σ j(t) = φ− σ(α (t t j)), t [t j, t j+ ], (4.3) ◦ − ∈ 1 σ(t), t [0, 1], γ j(t) = ∈ (4.4) ( σi(t), t [ti, ti+1], 0 i j. ∈ ≤ ≤ Since

( j+1) j j j σ j(t j) = φ− σ(0) = φ− σ(1) = φ− σ(α (t j t j 1)) = σ j 1(t j), ◦ ◦ ◦ − − − 9

0 we have that γ j(t) deﬁned as (4.4) is a piecewise smooth C curve satisfying γ j(0) = p0, γ j(t j) = x j. We then compute

j ti+1 2 2 t j+ ℓ+(x j+ , t j+ ) +(σ) + √t R(σi(t), t) + σ′(t) dt 1 1 1 ≤ L | i |g(t) Xi=1 Z p ti j ti+1 j C i+1 i+1 D + √t + Aα dt D + C α− 2 ≤ t ≤ Xi=1 Z Xi=1 ti j+1 D + Cα− 2 , ≤ for all j 0, where A := max σ′(t) g (t), C, and D are all constants independent of j. Since ≥ t [0,1] | | 0 ( j+1) ∈ t j+ α , we obtain (4.2) from the above computation. Theorem 3.1 is now applicable to 1 ≥ − (M, g(t))t [0, ) along the space-time sequence (x j, t j) ∞= . ∈ ∞ { } j 1 From the construction of g(t), we easily observe that 1 t M, t−j g(t jt), x j j+1 M, g (t), p0 1 , t 1, t → ∞ t [1,α− ] ∈ j ∈ where g and g0 diﬀer only by a scaling and a time-shifting. Furthermore, by Theorem 3.1, we have that∞ there exists a function ℓ : M [1,α 1] R, such that +∞ × − → j ℓ ℓ∞ + → + 0,α 1,2 j in the C sense and the weak W sense, where ℓ ( , t) = ℓ+( , tt j). Our next goal is to show that loc ∗ loc + · · ℓ+∞ gives rise to an expander structure on (M, g (t)). Arguing as in section 4 of [9] or section 6 of∞ [4] and applying the monotonicity in theorem 2.2 in 1 the same way as one have applied Perelman’s reduced geometry, we have that ℓ+∞ : M [1,α− ] R is a smooth function, satisfying × →

∂ℓ+∞ 2 n ∆ℓ+∞ + ℓ+∞ + R + = 0. (4.5) ∂t − |∇ | ∞ 2t 1 Furthermore, by Theorem 3.1(1), we may ﬁnd a function K∞ : M [1,α− ] R, such that j 0,α 1,2 × → 1 K K∞ in the Cloc sense and the weak Wloc sense. Fixing arbitrary 1 < s1 < s2 < α− and an arbitrary→ nonnegative and smooth time-independent∗ cut-oﬀ function ϕ compactly supported on M, we obtain the following by (2.8)

s2 1 n j j j 2 ℓ+ 0 3 K ϕ(4πt)− e− dµt dt ≤ Zs1 ZM t 2 s2 ∂ j j n n j j 2 ℓ+ ϕ ℓ+ ϕ, ℓ+ + R jϕ + ϕ (4πt)− e− dµt dt ≤ Zs1 ZM ∂t − h∇ ∇ i 2t ! s 2 ∂ n n ℓ+∞ ϕ ℓ+∞ ϕ, ℓ+∞ + R ϕ + ϕ (4πt)− 2 e− dµt∞dt → Zs1 ZM ∂t − h∇ ∇ i ∞ 2t ! = 0. 10 LIANG CHENG AND YONGJIA ZHANG

This shows that 1 K∞ 0 everywhere on M [1,α− ]. (4.6) ≡ × Since equations (2.2) and (2.3) are both carried to the limit in the sense of distribution, and since ℓ is smooth, the following hold on M [1,α 1] +∞ × − ∂ ℓ+∞ 2 ℓ+∞ ℓ+∞ = R , ℓ+∞ = R . ∂t ∞ − t |∇ | t − ∞ In combination with (4.5), we then have

∂ℓ+∞ 2 n + ∆ℓ+∞ + ℓ+∞ R = 0, ∂t |∇ | − ∞ − 2t 2 ℓ+∞ + n 2∆ℓ+∞ + ℓ+∞ R = 0. |∇ | − ∞ − t Then, by Theorem 1.2 in [6],

∂ n 2 ℓ+∞ + ∆ R t(2∆ℓ+∞ + ℓ+∞ R ) ℓ+∞ n (4πt)− 2 e ∂t ∞ ∞ − ! |∇ | − − − 2 g n ℓ+∞ = 2t Rc ℓ+∞ + ∞ (4πt)− 2 e = 0. − ∞ − ∇∇ 2t

Hence ℓ+∞ satisﬁes the following gradient expanding soliton equation 1 Rc ℓ+∞ = g . ∞ − ∇∇ −2t ∞ n ℓ Note that Theorem 3.1(2) guarantees that (4πt)− 2 e +∞ > 0 everywhere. This ﬁnishes the proof.

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School of Mathematics and Statistics &Hubei Key Laboratory of Mathematical Sciences,Central China Normal University,Wuhan, 430079, P.R.China Email address: [email protected]

School of Mathematics,University of Minnesota,Twin Cities, MN, 55414, USA Email address: [email protected]