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A monotonicity formula for mean curvature flow with surgery
Journal of Functional Analysis 272 (2017) 3647–3668
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Journal of Functional Analysis www.elsevier.com/locate/jfa
A monotonicity formula for mean curvature flow with surgery Simon Brendle 1 Department of Mathematics, Stanford University, Stanford, CA 94305, United States
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 August 2016 Accepted 3 November 2016 Available online 10 November 2016 Communicated by B. Chow
We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken’s monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is, in fact, free of surgeries. This result is used in the analysis of mean curvature flow with surgery in Riemannian three-manifolds (cf. [5]). © 2016 Published by Elsevier Inc.
Keywords: Mean curvature flow Singularities
1. Introduction One of the main tools in the study of mean curvature flow is Huisken’s monotonicity formula (cf. [12]). In the special case of smooth solution of mean curvature flow in R3 , the monotonicity formula implies that the Gaussian integral Mt
1
1 4π(t0 − t)
|x−p|2
e− 4(t0 −t)
E-mail address: [email protected]. The author was supported in part by the National Science Foundation under grant DMS-1201924.
http://dx.doi.org/10.1016/j.jfa.2016.11.004 0022-1236/© 2016 Published by Elsevier Inc.
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S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
is monotone decreasing in t, as long as t0 − t > 0. This monotonicity property plays a crucial role in the analysis of singularities; see e.g. [6,10,16–18]. Our goal in this paper is to adapt the monotonicity formula to surgically modified flows. Motivated by earlier work of Hamilton [8,9] and Perelman [14,15] on the Ricci flow, Huisken and Sinestrari [13] introduced a notion of mean curvature flow with surgery for two-convex hypersurfaces in Rn+1 , where n ≥ 3. In a joint work with Gerhard Huisken [4], we extended this construction to the case n = 2. As a result, we obtained a notion of mean curvature flow with surgery for embedded, mean convex surfaces in R3. One of the key ingredients in the proof is a sharp estimate for the inscribed radius established earlier in [2]. An alternative construction was given by Haslhofer and Kleiner [11]. We note that the surgery construction in [4] can be extended to flows of embedded, mean convex surfaces in three-manifolds; see [3] and [5] for details. Throughout this paper, we will focus on mean curvature flow with surgery for embedded, mean convex surfaces in three-manifolds. We were unable to show that the Gaussian density considered by Huisken is monotone across surgery times. The reason is that, as we replace a neck by a cap, the area element may increase. To get around this technical problem, we modify the Gaussian density by including a term involving the mean curvature. More precisely, if Mt is a mean curvature flow with surgery in R3 , we show that the quantity Mt
1 4π(t0 − t)
|x−p|2
e− 4(t0 −t) − 200 H1 H
is monotone decreasing in t, as long as t0 − t ≥ 59 H1−2 . Here, H1 is a positive constant which represents the so-called surgery scale; in other words, each neck on which we 1 perform surgery has radius between 2H and H11 . A similar monotonicity property holds 1 for mean curvature flow with surgery in a Riemannian three-manifold. In Section 2, we establish a number of auxiliary results. These results will be used in Section 3 to deduce a monotonicity formula for mean curvature flow with surgery in R3 . In Section 4, we extend the monotonicity formula to solutions of mean curvature flow with surgery in Riemannian three-manifolds. 2. Behavior of a Gaussian integral under a single surgery In this section, we analyze how the Gaussian integral changes under a single surgery. The strategy will be to estimate the integral of the Gaussian density over each vertical cross section. To that end, we require some preliminary estimates involving integrals over curves in R2 . In the following, S 1 will denote the unit circle in R2 centered at the origin. Lemma 2.1. There exists a real number β > 0 with the following significance. Suppose that Γ is a curve in R2 which is β-close to the unit circle S 1 in the C 1 -norm. Moreover, suppose that ψ is a real-valued function defined on Γ satisfying supΓ |ψ − 1| < β. Then
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
ψ e−
|x|2 4τ
+q,x
10 11
−
3649
|x|2 + q, x ≥ 0 2τ
Γ
for all τ ≥
5 9
and all q ∈ R2 .
Proof. Let us fix a large constant Q with the property that |q| 2 e ψ ≥ 16 ψ Γ∩{q,x≥ |q| 2 }
Γ∩{q,x≤0}
for all points q ∈ R2 satisfying |q| ≥ Q. This implies 2 |q| − |x| 2 4τ e ψe ≥4 Γ∩{q,x≥ |q| 2 }
for all τ ≥
5 9
ψ e−
|x|2 4τ
Γ∩{q,x≤0}
and all q ∈ R2 satisfying |q| ≥ Q. From this, we deduce that
2
ψe
− |x| 4τ +q,x
|q| |q| q, x ≥ e2 2
Γ∩{q,x≥ |q| 2 }
ψ e−
|x|2 4τ
Γ∩{q,x≥ |q| 2 }
ψ e−
≥ 2 |q|
|x|2 4τ
Γ∩{q,x≤0}
≥−
ψ e−
|x|2 4τ
+q,x
q, x
Γ∩{q,x≤0}
for all τ ≥
5 9
and all q ∈ R2 satisfying |q| ≥ Q. Therefore, we obtain
ψ e−
|x|2 4τ
+q,x
q, x ≥ 0,
Γ
hence
ψ e−
|x|2 4τ
+q,x
10 11
−
|x|2 + q, x ≥ 0 2τ
Γ
for all τ ≥ 59 and all q ∈ R2 satisfying |q| ≥ Q. On the other hand, by choosing β > 0 sufficiently small, we can arrange that
ψ e−
|x|2 4τ
+q,x
10 11
−
|x|2 + q, x ≥ 0 2τ
Γ
for all τ ≥
5 9
and all q ∈ R2 satisfying |q| ≤ Q. This proves the assertion.
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
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Lemma 2.2. Let β > 0 be chosen as in Lemma 2.1. Suppose that Γ is a curve in R2 which is β-close to the unit circle S 1 in the C 1 -norm. Moreover, suppose that ψ is a real-valued function defined on Γ satisfying supΓ |ψ − 1| < β. Then 10
ρ 11
ψ e−
ρ2 |x|2 4τ
+ρ q,x
ψ e−
≤
Γ
|x|2 4τ
+q,x
Γ
for all ρ ∈ (0, 1), all τ ≥ 59 , and all q ∈ R2 . Proof. Let us fix a real number τ ≥ 59 and a point q ∈ R2 . Moreover, let ρ ∈ (0, 1). Applying Lemma 2.1 with τ˜ = ρτ2 ≥ 59 and q˜ = ρ q gives
ψ e−
ρ2 |x|2 4τ
+ρ q,x
10 11
−
ρ2 |x|2 + ρ q, x ≥ 0 2τ
Γ
for all ρ ∈ (0, 1). Therefore, we obtain ρ
d dρ
ψ e−
ρ2 |x|2 4τ
+ρ q,x
=
Γ
ψ e−
Γ
10 ≥− 11
ρ2 |x|2 4τ
+ρ q,x
ψe
−ρ
2 |x|2 4τ
−
ρ2 |x|2 + ρ q, x 2τ
+ρ q,x
Γ
for all ρ ∈ (0, 1). Consequently, the function 10
ρ → ρ 11
ψ e−
ρ2 |x|2 4τ
+ρ q,x
Γ
is monotone increasing for ρ ∈ (0, 1). From this, the assertion follows. ˆ ε, L)-neck N of size 1 (see [4] α, δ, In the remainder of this section, we consider an (ˆ ˆ By definition, we can for the definition). It is understood that ε is much smaller than δ. find a simple closed, convex curve Γ with the property that distC 20 (N, Γ × [−L, L]) ≤ ε. Since distC 20 (N, Γ × [−L, L]) ≤ ε, we can find a collection of curves Γs such that {(γs (t), s) : s ∈ [−(L − 1), L − 1], t ∈ [0, 1]} ⊂ N and ∂ k ∂ l k l (γs (t) − γ(t)) ≤ O(ε). ∂s ∂t
k+l≤20
Here, we have used the notation Γ = {γ(t) : t ∈ [0, 1]} and Γs = {γs (t) : t ∈ [0, 1]}.
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
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As in [4], we may translate the neck N in space so that the center of mass of Γ is at the origin. Using the curve shortening flow, we can construct a homotopy γ˜r (t), (r, t) ∈ [0, 1] × [0, 1], with the following properties: • γ˜r (t) = γ(t) for r ∈ [0, 14 ]. • γ˜r (t) = (cos(2πt), sin(2πt)) for r ∈ [ 12 , 1]. ˜ r is 1 -noncollapsed. • For each r ∈ [0, 1], the curve Γ
1+δˆ 2 ∂ ∂2 sup(r,t)∈[0,1]×[0,1] | ∂r γ˜r (t)| + | ∂r∂ ∂t γ˜r (t)| + | ∂r ˜r (t)| 2γ
• We have as δˆ → 0.
ˆ where ω(δ) ˆ →0 ≤ ω(δ),
Finally, we choose a smooth cutoff function χ : R → R such that χ = 1 on (−∞, 1] and χ = 0 on [2, ∞). We next define a surface F˜Λ : [−L, Λ] × [0, 1] → R3 by ⎧ ⎪ (γ (t), s) ⎪ ⎪ s ⎪ ⎨((1 − e− 4Λ s ) γ (t), s) s F˜Λ (s, t) = 4Λ − ⎪((1 − e s ) (χ(s/Λ 14 ) γs (t) + (1 − χ(s/Λ 14 )) γ(t)), s) ⎪ ⎪ ⎪ ⎩ 4Λ ((1 − e− s ) γ˜s/Λ (t), s)
for s ∈ [−(L − 1), 0] 1
for s ∈ (0, Λ 4 ] 1
1
for s ∈ (Λ 4 , 2 Λ 4 ] 1
for s ∈ (2 Λ 4 , Λ].
It is clear that F˜Λ is smooth. Moreover, F˜Λ is axially symmetric for s ≥ Λ2 . As in [4], we may extend the immersion F˜Λ by gluing in an axially symmetric cap. To do that, we fix 1 a smooth, convex, even function Φ : R → R such that Φ(z) = |z| for |z| ≥ 100 . We then define a = 1 − e−4 +
1 1 (1 − e−4 )2 Λ− 4 3
and
vΛ (s) = 1 − e − Λ− 4 1
1
Λ + 2 Λ4 − s 1 a + Λ + 2 Λ4 − s
1 1 4Λ Λ + 2 Λ4 − s − s 4 Φ Λ 1−e −a 1 a + Λ + 2 Λ4 − s
− 4Λ s
+a
1
for s ∈ [Λ, Λ + Λ 4 ]. Moreover, we put
vΛ (s) = 2a 1
1
1
Λ + 2 Λ4 − s 1 a + Λ + 2 Λ4 − s
for s ∈ [Λ + Λ 4 , Λ + 2Λ 4 ]. The axially symmetric cap is chosen so that its cross section at 1 height s ∈ [Λ, Λ + 2Λ 4 ] is a circle of radius 21 vΛ (s). The resulting surface will be denoted ˜. by N
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
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1
We first consider the region s ∈ (0, Λ 4 ]. Lemma 2.3. There exist numerical constants δ∗ > 0 and Λ∗ > 0 with the following significance. If δˆ < δ∗ and Λ > Λ∗ , then we have
e−
2 x2 1 +x2 4τ
|∇N˜ x
˜ ∩{x3 =s} N
e−
≤
1
˜ +q1 x1 +q2 x2 −r0 H
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
3|
1 |∇N x3 |
1
1 for all s ∈ (0, Λ 4 ], all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . 1 ˜ ∩ {x3 = s} is obtained by dilating Proof. For each s ∈ (0, Λ 4 ], the cross section N 4Λ the cross section N ∩ {x3 = s} by the factor ρ(s) := 1 − e− s . Moreover, we have the pointwise inequalities
4Λ Λ |∇N x3 | ≤ 1 + C 2 e− x3 ˜ x3 |∇N x3 |
and 2
˜ − H ≥ c Λ e− x3 H x43
4Λ
1
for x3 ∈ (0, Λ 4 ]. Hence, if we choose Λ sufficiently large, then we have ˜
e−r0 H
1 1 ≤ e−r0 H N |∇ x3 | |∇N˜ x3 |
1
for x3 ∈ (0, Λ 4 ]. From this, we deduce that
e−
˜ ∩{x3 =s} N
2 x2 1 +x2 4τ
≤ ρ(s)
1
˜ +q1 x1 +q2 x2 −r0 H
e−
|∇N˜ x 2 ρ(s)2 (x2 1 +x2 ) +ρ(s) 4τ
3|
(q1 x1 +q2 x2 )−r0 H
N ∩{x3 =s}
1 |∇N x3 |
1
for s ∈ (0, Λ 4 ]. On the other hand, applying Lemma 2.2 with ψ = er0 (1−H) 10
ρ(s) 11 N ∩{x3 =s}
e−
2 ρ(s)2 (x2 1 +x2 ) +ρ(s) 4τ
(q1 x1 +q2 x2 )−r0 H
1 |∇N x
3|
1 |∇N x3 |
gives
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
e−
≤
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
3653
1 |∇N x3 |
1
for s ∈ (0, Λ 4 ]. Putting these facts together, we conclude that
e−
˜ ∩{x3 =s} N
2 x2 1 +x2 4τ
1
1 |∇N˜ x3 |
˜ +q1 x1 +q2 x2 −r0 H
e−
≤ ρ(s) 11
2 x2 1 +x2 4τ
1 |∇N x3 |
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s} 1
for s ∈ (0, Λ 4 ]. This proves the assertion. 1
We now consider the intermediate region s ∈ (Λ 4 , Λ4 ]. Lemma 2.4. We can find numerical constants δ∗ > 0 and Λ∗ > 0, and a positive function E∗ (Λ) with the following property. If δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ), then we have
e−
2 x2 1 +x2 4τ
|∇N˜ x
˜ ∩{x3 =s} N
e−
≤
1
˜ +q1 x1 +q2 x2 −r0 H
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
3|
1 |∇N x3 |
1
1 for all s ∈ (Λ 4 , Λ4 ], all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 .
Proof. We first observe that the claim is true if q12 + q22 is sufficiently large. More precisely, we can find positive real numbers δ∗ and Λ1 , and positive functions E1 (Λ) and Q(Λ) with the following significance. If δˆ < δ∗ , Λ > Λ1 , and ε < E1 (Λ), then we have
e−
2 x2 1 +x2 4τ
˜ ∩{x3 =s} N
≤ N ∩{x3 =s}
e−
1
˜ +q1 x1 +q2 x2 −r0 H
2 x2 1 +x2 4τ
|∇N˜ x +q1 x1 +q2 x2 −r0 H
3|
1 |∇N x
3|
1 1 whenever s ∈ (Λ 4 , Λ4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and q12 + q22 > Q(Λ). 1 Therefore, it remains to consider the case q12 + q22 ≤ Q(Λ). For each s ∈ (Λ 4 , Λ4 ], ˜ ∩ {x3 = s} is obtained by dilating the cross section N ∩ {x3 = s} by the cross section N 4Λ the factor ρ(s) := 1 − e− s , up to errors of order O(ε). Moreover, we have the pointwise inequalities
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
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2 Λ 4Λ 4Λ |∇N x3 | e− x3 + C ε ≤ 1 + C Λ−2 e− x3 + C ε ≤1+C ˜ 2 N x |∇ x3 | 3 and ˜ −H ≥0 H 1
for x3 ∈ (Λ 4 , Λ4 ]. Hence, we obtain
e−
2 x2 1 +x2 4τ
|∇N˜ x
˜ ∩{x3 =s} N
≤ (1 + C Λ−2 e− ·
1
˜ +q1 x1 +q2 x2 −r0 H
4Λ s
3|
+ C(Λ) ε) ρ(s) e−
2 ρ(s)2 (x2 1 +x2 ) +ρ(s) 4τ
(q1 x1 +q2 x2 )−r0 H
N ∩{x3 =s} 1
1 whenever s ∈ (Λ 4 , Λ4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and 1 r0 (1−H) using Lemma 2.2 with ψ = e gives |∇N x3 |
ρ(s)
10 11
e−
2 ρ(s)2 (x2 1 +x2 ) +ρ(s) 4τ
q12 + q22 ≤ Q(Λ). On the other hand,
(q1 x1 +q2 x2 )−r0 H
N ∩{x3 =s}
e−
≤
2 x2 1 +x2 4τ
1 |∇N x3 |
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x3 |
1 |∇N x
3|
1
1 whenever s ∈ (Λ 4 , Λ4 ], r0 ∈ [ 1000 , 1], and τ ≥ 59 . Putting these facts together, we obtain
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
˜ ∩{x3 =s} N
≤ (1 + C Λ−2 e− ·
4Λ s
1 |∇N˜ x3 | 1
+ C(Λ) ε) ρ(s) 11 e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x
3|
1 1 whenever s ∈ (Λ 4 , Λ4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and q12 + q22 ≤ Q(Λ). Clearly, we can find a positive constant Λ2 and a positive function E2 (Λ) such that (1 + C Λ−2 e−
4Λ s
1
+ C(Λ) ε) ρ(s) 11 ≤ 1
if Λ > Λ2 and ε < E2 (Λ). Hence, if we put Λ∗ = max{Λ1 , Λ2 } and E∗ (Λ) = min{E1 (Λ), E2 (Λ)}, then the assertion follows.
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1
In the next step, we consider the region s ∈ ( Λ4 , Λ + Λ 4 ]. Lemma 2.5. We can find numerical constants δ∗ > 0 and Λ∗ > 0 with the following property. If δˆ < δ∗ and Λ > Λ∗ , then we have 2 x2 1 1 +x2 ˜ e− 4τ +q1 x1 +q2 x2 −r0 H ˜ N |∇ x3 | ˜ ∩{x3 =s} N
e−
≤
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x3 |
1
1 for all s ∈ ( Λ4 , Λ + Λ 4 ], all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 .
Proof. Again, the claim is true if q12 + q22 is sufficiently large. More precisely, we can find positive real numbers δ1 and Λ1 , and a positive constant Q with the following significance. If δˆ < δ1 and Λ > Λ1 , then we have 2 x2 1 1 +x2 ˜ e− 4τ +q1 x1 +q2 x2 −r0 H ˜ N |∇ x3 | ˜ ∩{x3 =s} N
e−
≤
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x3 |
1 1 whenever s ∈ ( Λ4 , Λ + Λ 4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and q12 + q22 > Q. 1 Hence, it remains to consider the case q12 + q22 ≤ Q. For each s ∈ ( Λ4 , Λ + Λ 4 ], the ˜ ∩ {x3 = s} is close cross section N ∩ {x3 = s} is close to a circle of radius 1, whereas N ˜ − 4Λ N N s to a circle of radius ρ(s) := 1 − e . Moreover, |∇ x3 | and |∇ x3 | are close to 1. By Lemma 2.2, we have 2 x2 1 +x2 e− 4τ +q1 x1 +q2 x2 {x21 +x22 =ρ(s)2 }
≤ ρ(s)
1 11
e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2
{x21 +x22 =1}
for all (q1 , q2 ) ∈ R2 . Hence, we can find positive real numbers δ2 and Λ2 with the following property: if δˆ < δ2 and Λ > Λ2 , then 2 x2 1 1 +x2 ˜ e− 4τ +q1 x1 +q2 x2 −r0 H |∇N˜ x3 | ˜ ∩{x3 =s} N
≤ N ∩{x3 =s}
e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
1 |∇N x
3|
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S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
1 1 whenever s ∈ ( Λ4 , Λ + Λ 4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and q12 + q22 ≤ Q. Therefore, if we put δ∗ = min{δ1 , δ2 } and Λ = max{Λ1 , Λ2 }, then the assertion follows. 1
1
Finally, we consider the region s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ). Lemma 2.6. There exist numerical constants δ∗ > 0 and Λ∗ > 0 with the following property. If δˆ < δ∗ and Λ > Λ∗ , then we have
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
˜ ∩{x3 =s} N
e−
≤
2 x2 1 +x2 4τ
1 |∇N˜ x3 |
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s} 1
1 |∇N x3 |
1
1 for all s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . 1 1 ˜ at height s is a circle of Proof. For each s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ), the cross section of N radius
ρ(s) = a where a = 1 − e−4 +
1 3
b−s , a+b−s
(1 − e−4 )2 Λ− 4 < 1 and b = Λ + 2 Λ 4 . Note that 1
1 |∇N˜ x
3|
1
=
1 + ρ (s)2
and ˜ ≥ 1 H a ˜ ∩ {x3 = s}. Therefore, on the set N
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
˜ ∩{x3 =s} N
≤
e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −
r0 a
˜ ∩{x3 =s} N
= ρ(s)
1 |∇N˜ x3 | 1 |∇N˜ x3 |
r0 ρ(s)2 1 + ρ (s)2 e− 4τ − a eρ(s) (q1 x1 +q2 x2 ) S1
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
a2 a a4 + e 4τ 4 a + b − s 4 (a + b − s)
1−
=a
·
e
a
b−s a+b−s
a a2 a+b−s − 4τ
−
3657
r0 a
(q1 x1 +q2 x2 )
S1
for all τ ≥
5 9
1
1
and all s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ). It is elementary to check that 1−z+
z4 ≤ e−z , 4
hence 1−z+
z4 z e2 ≤ 1 4
for all z ∈ [0, 1]. Consequently, we have
1− for all τ ≥
5 9
a2 a4 a + e 4τ 4 a + b − s 4 (a + b − s)
1
a a+b−s
≤1
1
and s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ). From this, we deduce that
e−
2 x2 1 +x2 4τ
˜ ∩{x3 =s} N a2
r0
≤ a e− 4τ − a
˜ +q1 x1 +q2 x2 −r0 H
e
a
b−s a+b−s
1 |∇N˜ x3 |
(q1 x1 +q2 x2 )
S1 1
1
for all τ ≥ 59 and s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ). On the other hand, if the neck N is sufficiently close to a cylinder of radius 1, then we have
e S1
a
b−s a+b−s
(q1 x1 +q2 x2 )
≤
e
2 1−x2 1 −x2 4τ
1 +q1 x1 +q2 x2 +r0 ( a −H)
N ∩{x3 =s} 1
1
1 |∇N x
3|
1 for all s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . Putting these facts together, we conclude that
˜ ∩{x3 =s} N
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
1 |∇N˜ x
3|
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S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
≤ ae
1−a2 4τ
e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s} 1
1 |∇N x3 |
1
1 for all s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . Since a
is close to 1 − e−4 for Λ large, we have a e follows.
1−a2 4τ
≤ ae
9(1−a2 ) 20
≤ 1. From this, the assertion
Combining the previous results, we can draw the following conclusion: Proposition 2.7. There exist numerical constants δ∗ > 0 and Λ∗ > 0, and a function E∗ (Λ) with the following property. If δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ), then we have
2
e
˜ − |x−p| −r0 H 4τ
1
e−
≤
|x−p|2 4τ
−r0 H
1
˜ ∩{0≤x3 ≤Λ+2Λ 4 } N
N ∩{0≤x3 ≤Λ+2Λ 4 }
1 for all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all p ∈ R3 .
Proof. In view of Lemma 2.3, Lemma 2.4, Lemma 2.5, and Lemma 2.6, we can find positive real numbers δ∗ , and Λ∗ , and a positive function E∗ (Λ) with the following property. If δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ), then we have
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
˜ ∩{x3 =s} N
e−
≤
2 x2 1 +x2 4τ
1 |∇N˜ x3 |
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x3 |
1
1 for all s ∈ (0, Λ + 2Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . This implies
e−
|x−p|2 4τ
˜ ∩{x3 =s} N
≤ N ∩{x3 =s}
e−
1
˜ −r0 H
|x−p|2 4τ
|∇N˜ x −r0 H
3|
1 |∇N x3 |
1
1 for all s ∈ (0, Λ + 2Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all p ∈ R3 . If we integrate over s and apply the co-area formula, the assertion follows.
3. A monotonicity formula for mean curvature flow with surgery in R3 Proposition 3.1. Let Mt be a family of mean convex surfaces in R3 which evolve under smooth mean curvature flow. For each r0 > 0, the function
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
Mt
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|x−p|2
1 4π(t0 − t)
e− 4(t0 −t) −r0 H
is monotone decreasing for t0 − t > 0. Proof. We compute ∂ − Δ e−r0 H = −r0 e−r0 H − Δ H − r02 e−r0 H |∇H|2 ∂t ∂t = −r0 e−r0 H |A|2 H − r02 e−r0 H |∇H|2 ≤ 0.
∂
Hence, the assertion follows from Ecker’s weighted monotonicity formula (see Theorem 4.13 in [7]). We next consider a mean curvature flow with surgery in R3 . We assume that ˆ ε, L)-neck of size each surgery procedure involves performing Λ-surgery on an (ˆ α, δ, 1 1 r ∈ [ 2H1 , H1 ] (see [4] for definitions). Theorem 3.2. Let δ∗ , Λ∗ , and E∗ (Λ) be defined as in Proposition 2.7. Moreover, suppose that Mt is a mean curvature flow with surgery, and that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ). Then the function Mt
|x−p|2
1 4π(t0 − t)
is monotone decreasing for t0 − t ≥
5 9
e− 4(t0 −t) − 200 H1 H
H1−2 .
Proof. Proposition 3.1 guarantees that the monotonicity formula holds in between surgery times. Moreover, it follows from Proposition 2.7 that the monotonicity property holds across surgery times. Theorem 3.2 allows us to draw the following conclusion: Corollary 3.3. We can find positive real numbers δ∗ , Λ∗ , L, and a positive function E∗ (Λ) with the following property. Suppose that Mt is a mean curvature flow with surgery, and that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ). Finally, suppose ˆ ε, L)-neck of size r ∈ [ 1 , 1 ], and p is a point in ambient that Mt0 − contains an (ˆ α, δ, 2H1 H1 space which lies on the axis of that neck. Then Mt
for all τ ≥
5 9
H1−2 .
−2 5 0 + 9 H1 −τ
1 − |x−p|2 e 4τ ≥ 1.01 4πτ
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Proof. It is elementary to check that inf 1
r∈[ 2H , H1 1 1
]
H 1 r e−
2 r2 9 H1 20
− 200 1H
1 r
=
1 − 49 e 400 . 2
Hence, on an exact cylinder S 1 (r) × R, we have
9 H12 − 9 H1220|x|2 − 200HH 1 = e 20π
S 1 (r)×R
2 r2 9 H1 1 9π H1 r e− 20 − 200 H1 r 5
9π − 49 e 400 20 ≥ 1.02 ≥
1 for all r ∈ [ 2H , 1 ]. By assumption, the surface Mt0 − contains a neck of size r ∈ 1 H1 1 1 [ 2H1 , H1 ], and p lies on the axis of that neck. Hence, if L is sufficiently large, then we obtain
|x−p|2 9 H12 − 9 H12 20 − 200HH 1 ≥ 1.01. e 20π
M t0 −
Using Theorem 3.2, we obtain Mt
1 − |x−p|2 e 4τ 4πτ
−2 5 0 + 9 H1 −τ
≥ Mt
2 H 1 − |x−p| e 4τ − 200 H1 4πτ
−2 5 0 + 9 H1 −τ
≥
|x−p|2 9 H12 − 9 H12 20 − 200HH 1 e 20π
M t0 −
≥ 1.01 provided that τ ≥
5 9
H1−2 and L is sufficiently large.
¯ Theorem 3.4. Fix an open interval I and a compact interval J ⊂ I. Suppose that M is an embedded smooth solution of mean curvature flow which is defined for t ∈ I. Moreover, suppose that M(j) is a sequence of mean curvature flows with surgery, each of which is defined for t ∈ I. We assume that each surgery of the flow M(j) involves ˆ ε, L)-neck of size r ∈ [ 1(j) , 1(j) ], where H (j) → ∞. performing Λ-surgery on an (ˆ α, δ, 1 2H1
H1
We assume further that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ).
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¯ in the sense of geometric measure Finally, we assume that the flows M(j) converge to M (j) theory. Then, if j is sufficiently large, the flow M is free of surgeries for all t ∈ J. ¯ as j → ∞. Furthermore, the flows M(j) converge smoothly to M Proof. We first show that, for j sufficiently large, the flow M(j) is free of surgeries for all t ∈ J. Suppose that each flow M(j) has at least one surgery time tj ∈ J. For each j, ˆ ε, L)-neck in M (j) of size rj ∈ [ 1(j) , 1(j) ]. Let pj be a point in we can find an (ˆ α, δ, tj − 2H1
H1
ambient space which lies on the axis of that neck. Using Corollary 3.3, we obtain M
1 − |x−pj |2 4τ e ≥ 1.01 4πτ
(j) (j) tj + 5 (H1 )−2 −τ 9
(j) for all τ ≥ 59 (H1 )−2 . We now pass to the limit as j → ∞. If we define t¯ = limj→∞ tj ∈ J and p¯ = limj→∞ pj , then we obtain
¯2 1 − |x−p| e 4τ ≥ 1.01 4πτ
¯ t¯−τ M
¯ is smooth, we have for each τ ∈ (0, inf J − inf I). On the other hand, since M
¯2 1 − |x−p| e 4τ → 1 4πτ
¯ t¯−τ M
as τ → 0. This is a contradiction. Therefore, the flow M(j) is free of surgeries for t ∈ J. Using standard local regularity theorems for mean curvature flow (cf. [1,18]), we conclude ¯ as j → ∞. that the flows M(j) converge smoothly to M 4. Adaptation to the Riemannian setting Let Mt be a mean convex solution of mean curvature flow in a compact Riemannian three-manifold X. Let us fix a time t0 and point p in ambient space. Let ϕ be a smooth cutoff on X such that ϕ(x) = 1 for d(p, x) ≤ 14 inj(X) and ϕ(x) = 0 for d(p, x) ≥ 12 inj(X). Moreover, we put Φ(x, t) =
1 4π(t0 − t)
d(p,x)2
e− 4(t0 −t) ϕ(x)2
for t0 − t > 0. Proposition 4.1. Let K be a positive constant with the property that the ambient threemanifold X has Ricci curvature is at least −K. Then we have
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d dt
1 Φ exp (t0 − t) 2 − r0 e−K (t0 −t) H ≤ C |Mt |
Mt
whenever t0 − t ∈ (0, 1]. The constant C depends only on the ambient three-manifold X. Proof. In the following, we assume that t0 − t ∈ (0, 1]. In the region d(p, x) ≤ we have
1 4
inj(X),
¯ ν Φ 2 d(p, x)2 D + Δ − H2 Φ ≤ − H + Φ. Φ+C ∂t Φ t0 − t
∂
This implies ∂
d(p, x)2 + Δ − H2 Φ ≤ C Φ, ∂t t0 − t
hence ∂ 1 + Δ − H 2 Φ exp (t0 − t) 2 ∂t d(p, x)2 1 1 ≤ C − Φ exp (t0 − t) 2 ≤ C 1 t0 − t 2 2 (t0 − t) for d(p, x) ≤
1 4
inj(X). Moreover, in the region
1 4
inj(X) ≤ d(p, x) ≤
1 2
inj(X), we have
¯ ν Φ 2 D + Δ − H2 Φ ≤ − H + Φ + C. ∂t Φ
∂
This gives ∂ 1 + Δ − H 2 Φ exp (t0 − t) 2 ≤ C ∂t for
1 4
inj(X) ≤ d(p, x) ≤
1 2
inj(X). To summarize, we have shown that
∂ 1 + Δ − H 2 Φ exp (t0 − t) 2 ≤ C ∂t at each point in Mt . Since the Ricci curvature of X is bounded from below by −K, we have ∂ ∂t hence
− Δ (e−K(t0 −t) H) ≥ e−K(t0 −t) |A|2 H ≥ 0,
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∂ − Δ exp − r0 e−K(t0 −t) H ∂t ∂ = − exp − r0 e−K(t0 −t) H − Δ (r0 e−K(t0 −t) H) ∂t − exp − r0 e−K(t0 −t) H |∇(r0 e−K(t0 −t) H)|2 ≤ 0. Putting these facts together, we obtain d dt
1 2
Φ exp (t0 − t) − r0 e
−K(t0 −t)
H
Mt
≤C
exp − r0 e−K(t0 −t) H ≤ C |Mt |,
Mt
provided that t0 − t ∈ (0, 1]. This completes the proof. We now consider mean curvature flow with surgery in a Riemannian manifold. We begin with a definition. Definition 4.2. Let M be a closed surface in a Riemannian three-manifold, and let N be ˆ ε, L)-neck of size r if there exists a point o ∈ N a region in M . We say that N is an (ˆ α, δ, −1 ˆ ε, L)-neck of size r in Euclidean α, δ, with the property that the surface expo (N ) is an (ˆ space (see [4] for the definition). We next explain how to do on a neck in Riemannian three-manifold. Namely, if o lies ˆ ε, L)-neck N in X, then exp−1 (N ) is an (ˆ ˆ ε, L)-neck in Euat the center of an (ˆ α, δ, α, δ, o clidean space. Hence, we can perform the surgery procedure described in [4] on exp−1 o (N ). We then paste the surgically modified surface back into X using the exponential map expo . In the next step, we analyze how the quantity in Proposition 4.1 changes under a single surgery. To that end, it will be convenient to work in geodesic normal coordinates around o; that is, we will identify a point in To N = R3 with its image under the expoˆ ε, L)-neck in R3 . α, δ, nential map expo . With this identification, we can view N as an (ˆ Without loss of generality, we may assume that the axis of the neck N is parallel to the x3 -axis. Note that the origin lies on N , so the axis of the neck does not pass through the ˜ the surface obtained from N by performing a Λ-surgery origin. Finally, we denote by N on N . ˆ Λ, and ε satisfy δˆ < δ∗ , Λ > Λ∗ , Proposition 4.3. Suppose that the surgery parameters δ, and ε < E∗ (Λ), where δ∗ , Λ∗ , and E∗ are defined as in Proposition 2.7. Moreover, let N ˆ ε, L)-neck N of size r in X, and let N ˜ denote the surgically modified surface. α, δ, be an (ˆ Then
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e−
d(p,x)2 4τ
˜ −r0 e−Kτ H
dμ
1
˜ ∩{0≤x3 ≤Λ+2Λ 4 } N
e−
≤
d(p,x)2 4τ
−r0 e−Kτ H
dμ + C(Λ) τ r2
1
N ∩{0≤x3 ≤Λ+2Λ 4 }
provided that d(o, p) ≤
1 8
inj(X),
1 K
≥τ ≥
5 9
1 r2 and r0 ∈ [ 300 r, r].
Proof. It follows from Proposition 2.7 that
e−
|x−p|2 4τ
˜ eucl −r0 e−Kτ H
dμeucl
1
˜ ∩{0≤x3 ≤Λ+2Λ 4 } N
e−
≤ 1 N ∩{0≤x3 ≤Λ+2Λ 4
|x−p|2 4τ
−r0 e−Kτ Heucl
dμeucl
}
1 provided that τ ≥ 59 r2 and r0 ∈ [ 300 r, r]. Here, |x − p| denotes the Euclidean distance ˜ eucl denote the mean of x and p in geodesic normal coordinates around o; Heucl and H ˜ curvatures of N and N with respect to the Euclidean metric; and dμeucl denotes the area form with respect to the Euclidean metric on R3 . In the next step, we compare the Riemannian distance d(p, x) to the Euclidean distance |x − p|. To that end, we perform a Taylor expansion of the function x → 12 d(p, x)2 around the origin o. The value of this function at o is given by 12 |p|2 . Its gradient at o equals −p. Moreover, the Hessian of the function x → 12 d(p, x)2 at o equals g + O(|p|2 ). Finally, the third derivatives of the function x → 12 d(p, x)2 at o are bounded by O(|p|). Putting these facts together, we obtain
1 1 1 d(p, x)2 = |p|2 − p, x + |x|2 + O(|p|2 |x|2 + |p| |x|3 + |x|4 ). 2 2 2 In other words, 1 1 d(p, x)2 = |x − p|2 + O((|p|2 + |x|2 ) |x|2 ). 2 2 ˜ ) ∩ {0 ≤ x3 ≤ Λ + 2Λ 41 }, then we have |x|2 ≤ C(Λ) r2 , hence |x|2 ≤ C(Λ) τ . If x ∈ (N ∪ N This implies d(p, x)2 |x − p|2 − ≤ C(Λ) (|p|2 + τ ) 4τ 4τ ˜ ) ∩ {0 ≤ x3 ≤ Λ + 2Λ 14 }. Consequently, we have for all points x ∈ (N ∪ N e−
|x−p|2 4τ
≤ (1 + C(Λ) (|p|2 + τ )) e−
d(p,x)2 4τ
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1
for all points x ∈ N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }. Since |H − Heucl | ≤ C(Λ) r for all points 1 x ∈ N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }, it follows that e−
|x−p|2 4τ
−r0 e−Kτ Heucl
≤ (1 + C(Λ) (|p|2 + τ )) e−
d(p,x)2 4τ
−r0 e−Kτ H
1
for all points x ∈ N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }. Thus,
e−
1 N ∩{0≤x3 ≤Λ+2Λ 4
|x−p|2 4τ
−r0 e−Kτ Heucl
}
dμeucl
e−
≤ (1 + C(Λ) (|p|2 + τ ))
d(p,x)2 4τ
−r0 e−Kτ H
dμ.
1
N ∩{0≤x3 ≤Λ+2Λ 4 }
Similarly, we have e−
d(p,x)2 4τ
≤ (1 + C(Λ) (|p|2 + τ )) e−
|x−p|2 4τ
˜ ∩ {0 ≤ x3 ≤ Λ + 2Λ 14 }. Since |H ˜ −H ˜ eucl | ≤ C(Λ) r for all points for all points x ∈ N 1 ˜ x ∈ N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }, it follows that e−
d(p,x)2 4τ
˜ −r0 e−Kτ H
≤ (1 + C(Λ) (|p|2 + τ )) e−
|x−p|2 4τ
˜ eucl −r0 e−Kτ H
˜ ∩ {0 ≤ x3 ≤ Λ + 2Λ 14 }. This gives for all points x ∈ N
e−
d(p,x)2 4τ
˜ −r0 e−Kτ H
dμ
1
˜ ∩{0≤x3 ≤Λ+2Λ 4 } N
e−
≤ (1 + C(Λ) (|p| + τ )) 2
1 ˜ ∩{0≤x3 ≤Λ+2Λ 4 N
|x−p|2 4τ
˜ eucl −r0 e−Kτ H
dμeucl .
}
Putting these facts together, we conclude that 1 ˜ ∩{0≤x3 ≤Λ+2Λ 4 N
e−
d(p,x)2 4τ
}
˜ −r0 e−Kτ H
dμ
e−
≤ (1 + C(Λ) (|p|2 + τ )) 1 N ∩{0≤x3 ≤Λ+2Λ 4
d(p,x)2 4τ
−r0 e−Kτ H
dμ.
}
Finally, we have the pointwise estimate (|p|2 + τ ) e− 1 N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }. Therefore, we obtain
d(p,x)2 4τ
≤ C(Λ) τ at each point on
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3666
e−
(|p|2 + τ )
d(p,x)2 4τ
−r0 e−Kτ H
dμ ≤ C(Λ) τ r2 ,
1
N ∩{0≤x3 ≤Λ+2Λ 4 }
hence
e−
1 ˜ ∩{0≤x3 ≤Λ+2Λ 4 N
d(p,x)2 4τ
˜ −r0 e−Kτ H
dμ
}
e−
≤
d(p,x)2 4τ
−r0 e−Kτ H
dμ + C(Λ) τ r2 .
1
N ∩{0≤x3 ≤Λ+2Λ 4 }
This proves the assertion. Combining Proposition 4.1 and Proposition 4.3, we arrive at the following conclusion: Theorem 4.4. Let Mt be a mean curvature flow with surgery in a Riemannian threemanifold X. Suppose that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ), where δ∗ , Λ∗ , and E∗ (Λ) are defined as in Proposition 2.7. Then
1 e−K(t0 −tˆ) H Φ exp (t0 − tˆ) 2 − dμ 200 H1
Mtˆ
≤
1 e−K(t0 −t˜) H Φ exp (t0 − t˜) 2 − dμ 200 H1
Mt˜
+ C |Mt˜| (t0 − t˜) + C(Λ) L−1 |Mt˜|, provided that
1 K
≥ t0 − t˜ ≥ t0 − tˆ ≥
5 9
H1−2 .
Note that the error term C(Λ) L−1 |Mt˜| can be made arbitrarily small by choosing L very large (depending on Λ). Proof. By Proposition 4.1, the quantity
1 e−K(t0 −t) H Φ exp (t0 − t) 2 − dμ 200 H1
Mt
increases at a rate of at most C |Mt | in between surgery times. Moreover, Proposition 4.3 implies that, during each surgery, the quantity Mt
1 e−K(t0 −t) H Φ exp (t0 − t) 2 − dμ 200 H1
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1 increases by at most C(Λ) H1−2 , provided that K ≥ t0 − t ≥ 59 H1−2 . On the other hand, −2 1 the surface area decreases by at least 10 L H1 during each surgery. Hence, there are at −1 2 most C L H1 |Mt˜| surgeries after time t˜. Consequently, the quantity considered above increases by at most C |Mt˜| (t0 − t˜) + C(Λ) L−1 |Mt˜| between time t˜ and time tˆ.
As a consequence of Theorem 4.4, we obtain an analogue of Theorem 3.4 in the Riemannian setting. Theorem 4.5. Fix an open interval I and a compact interval J ⊂ I. Moreover, suppose ¯ is an embedded smooth solution of mean curvature flow in a Riemannian threethat M manifold which is defined for t ∈ I. Moreover, suppose that M(j) is a sequence of mean curvature flows with surgery in the same Riemannian three-manifold, each of which is defined for t ∈ I. We assume that each surgery of the flow M(j) involves performing ˆ ε, L)-neck of size r ∈ [ 1(j) , 1(j) ], where H (j) → ∞. We assume Λ-surgery on an (ˆ α, δ, 1 2H1
H1
further that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , ε < E∗ (Λ), and that L is chosen sufficiently large depending on Λ. Finally, we assume that the flows M(j) con¯ in the sense of geometric measure theory. Then, if j is sufficiently large, verge to M the flow M(j) is free of surgeries for all t ∈ J. Furthermore, the flows M(j) converge ¯ as j → ∞. smoothly to M By combining Theorem 4.5 with results of Brian White [16,17], we are able to characterize the longtime behavior of mean curvature flow with surgery in Riemannian three-manifolds. This is discussed in [5]. References [1] K. Brakke, The Motion of a Surface by Its Mean Curvature, Princeton University Press, 1978. [2] S. Brendle, A sharp bound for the inscribed radius under mean curvature flow, Invent. Math. 202 (2015) 217–237. [3] S. Brendle, An inscribed radius estimate for mean curvature flow in Riemannian manifolds, Ann. Sc. Norm. Super. Pisa (2016), in press. [4] S. Brendle, G. Huisken, Mean curvature flow with surgery of mean convex surfaces in R3 , Invent. Math. 203 (2016) 615–654. [5] S. Brendle, G. Huisken, Mean curvature flow with surgery of mean convex surfaces in threemanifolds, J. Eur. Math. Soc. (2016), in press. [6] T. Colding, W. Minicozzi, Generic mean curvature flow I: generic singularities, Ann. of Math. 175 (2012) 755–833. [7] K. Ecker, Regularity Theory for Mean Curvature Flow, Birkhäuser, Boston, 2004. [8] R. Hamilton, The formation of singularities in the Ricci flow, in: Surveys in Differential Geometry, vol. II, International Press, Somerville, MA, 1995, pp. 7–136. [9] R. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997) 1–92. [10] R. Haslhofer, B. Kleiner, Mean curvature flow of mean convex hypersurfaces, arXiv:1304.0926. [11] R. Haslhofer, B. Kleiner, Mean curvature flow with surgery, arXiv:1404.2332. [12] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990) 285–299. [13] G. Huisken, C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math. 175 (2009) 137–221. [14] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:0211159.
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[15] G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:0303109. [16] B. White, The size of the singular set in mean curvature flow of mean convex sets, J. Amer. Math. Soc. 13 (2000) 665–695. [17] B. White, The nature of singularities in mean curvature flow of mean convex sets, J. Amer. Math. Soc. 16 (2003) 123–138. [18] B. White, A local regularity theorem for mean curvature flow, Ann. of Math. 161 (2005) 1487–1519.
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Journal of Functional Analysis www.elsevier.com/locate/jfa
A monotonicity formula for mean curvature flow with surgery Simon Brendle 1 Department of Mathematics, Stanford University, Stanford, CA 94305, United States
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 August 2016 Accepted 3 November 2016 Available online 10 November 2016 Communicated by B. Chow
We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken’s monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is, in fact, free of surgeries. This result is used in the analysis of mean curvature flow with surgery in Riemannian three-manifolds (cf. [5]). © 2016 Published by Elsevier Inc.
Keywords: Mean curvature flow Singularities
1. Introduction One of the main tools in the study of mean curvature flow is Huisken’s monotonicity formula (cf. [12]). In the special case of smooth solution of mean curvature flow in R3 , the monotonicity formula implies that the Gaussian integral Mt
1
1 4π(t0 − t)
|x−p|2
e− 4(t0 −t)
E-mail address: [email protected]. The author was supported in part by the National Science Foundation under grant DMS-1201924.
http://dx.doi.org/10.1016/j.jfa.2016.11.004 0022-1236/© 2016 Published by Elsevier Inc.
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is monotone decreasing in t, as long as t0 − t > 0. This monotonicity property plays a crucial role in the analysis of singularities; see e.g. [6,10,16–18]. Our goal in this paper is to adapt the monotonicity formula to surgically modified flows. Motivated by earlier work of Hamilton [8,9] and Perelman [14,15] on the Ricci flow, Huisken and Sinestrari [13] introduced a notion of mean curvature flow with surgery for two-convex hypersurfaces in Rn+1 , where n ≥ 3. In a joint work with Gerhard Huisken [4], we extended this construction to the case n = 2. As a result, we obtained a notion of mean curvature flow with surgery for embedded, mean convex surfaces in R3. One of the key ingredients in the proof is a sharp estimate for the inscribed radius established earlier in [2]. An alternative construction was given by Haslhofer and Kleiner [11]. We note that the surgery construction in [4] can be extended to flows of embedded, mean convex surfaces in three-manifolds; see [3] and [5] for details. Throughout this paper, we will focus on mean curvature flow with surgery for embedded, mean convex surfaces in three-manifolds. We were unable to show that the Gaussian density considered by Huisken is monotone across surgery times. The reason is that, as we replace a neck by a cap, the area element may increase. To get around this technical problem, we modify the Gaussian density by including a term involving the mean curvature. More precisely, if Mt is a mean curvature flow with surgery in R3 , we show that the quantity Mt
1 4π(t0 − t)
|x−p|2
e− 4(t0 −t) − 200 H1 H
is monotone decreasing in t, as long as t0 − t ≥ 59 H1−2 . Here, H1 is a positive constant which represents the so-called surgery scale; in other words, each neck on which we 1 perform surgery has radius between 2H and H11 . A similar monotonicity property holds 1 for mean curvature flow with surgery in a Riemannian three-manifold. In Section 2, we establish a number of auxiliary results. These results will be used in Section 3 to deduce a monotonicity formula for mean curvature flow with surgery in R3 . In Section 4, we extend the monotonicity formula to solutions of mean curvature flow with surgery in Riemannian three-manifolds. 2. Behavior of a Gaussian integral under a single surgery In this section, we analyze how the Gaussian integral changes under a single surgery. The strategy will be to estimate the integral of the Gaussian density over each vertical cross section. To that end, we require some preliminary estimates involving integrals over curves in R2 . In the following, S 1 will denote the unit circle in R2 centered at the origin. Lemma 2.1. There exists a real number β > 0 with the following significance. Suppose that Γ is a curve in R2 which is β-close to the unit circle S 1 in the C 1 -norm. Moreover, suppose that ψ is a real-valued function defined on Γ satisfying supΓ |ψ − 1| < β. Then
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
ψ e−
|x|2 4τ
+q,x
10 11
−
3649
|x|2 + q, x ≥ 0 2τ
Γ
for all τ ≥
5 9
and all q ∈ R2 .
Proof. Let us fix a large constant Q with the property that |q| 2 e ψ ≥ 16 ψ Γ∩{q,x≥ |q| 2 }
Γ∩{q,x≤0}
for all points q ∈ R2 satisfying |q| ≥ Q. This implies 2 |q| − |x| 2 4τ e ψe ≥4 Γ∩{q,x≥ |q| 2 }
for all τ ≥
5 9
ψ e−
|x|2 4τ
Γ∩{q,x≤0}
and all q ∈ R2 satisfying |q| ≥ Q. From this, we deduce that
2
ψe
− |x| 4τ +q,x
|q| |q| q, x ≥ e2 2
Γ∩{q,x≥ |q| 2 }
ψ e−
|x|2 4τ
Γ∩{q,x≥ |q| 2 }
ψ e−
≥ 2 |q|
|x|2 4τ
Γ∩{q,x≤0}
≥−
ψ e−
|x|2 4τ
+q,x
q, x
Γ∩{q,x≤0}
for all τ ≥
5 9
and all q ∈ R2 satisfying |q| ≥ Q. Therefore, we obtain
ψ e−
|x|2 4τ
+q,x
q, x ≥ 0,
Γ
hence
ψ e−
|x|2 4τ
+q,x
10 11
−
|x|2 + q, x ≥ 0 2τ
Γ
for all τ ≥ 59 and all q ∈ R2 satisfying |q| ≥ Q. On the other hand, by choosing β > 0 sufficiently small, we can arrange that
ψ e−
|x|2 4τ
+q,x
10 11
−
|x|2 + q, x ≥ 0 2τ
Γ
for all τ ≥
5 9
and all q ∈ R2 satisfying |q| ≤ Q. This proves the assertion.
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
3650
Lemma 2.2. Let β > 0 be chosen as in Lemma 2.1. Suppose that Γ is a curve in R2 which is β-close to the unit circle S 1 in the C 1 -norm. Moreover, suppose that ψ is a real-valued function defined on Γ satisfying supΓ |ψ − 1| < β. Then 10
ρ 11
ψ e−
ρ2 |x|2 4τ
+ρ q,x
ψ e−
≤
Γ
|x|2 4τ
+q,x
Γ
for all ρ ∈ (0, 1), all τ ≥ 59 , and all q ∈ R2 . Proof. Let us fix a real number τ ≥ 59 and a point q ∈ R2 . Moreover, let ρ ∈ (0, 1). Applying Lemma 2.1 with τ˜ = ρτ2 ≥ 59 and q˜ = ρ q gives
ψ e−
ρ2 |x|2 4τ
+ρ q,x
10 11
−
ρ2 |x|2 + ρ q, x ≥ 0 2τ
Γ
for all ρ ∈ (0, 1). Therefore, we obtain ρ
d dρ
ψ e−
ρ2 |x|2 4τ
+ρ q,x
=
Γ
ψ e−
Γ
10 ≥− 11
ρ2 |x|2 4τ
+ρ q,x
ψe
−ρ
2 |x|2 4τ
−
ρ2 |x|2 + ρ q, x 2τ
+ρ q,x
Γ
for all ρ ∈ (0, 1). Consequently, the function 10
ρ → ρ 11
ψ e−
ρ2 |x|2 4τ
+ρ q,x
Γ
is monotone increasing for ρ ∈ (0, 1). From this, the assertion follows. ˆ ε, L)-neck N of size 1 (see [4] α, δ, In the remainder of this section, we consider an (ˆ ˆ By definition, we can for the definition). It is understood that ε is much smaller than δ. find a simple closed, convex curve Γ with the property that distC 20 (N, Γ × [−L, L]) ≤ ε. Since distC 20 (N, Γ × [−L, L]) ≤ ε, we can find a collection of curves Γs such that {(γs (t), s) : s ∈ [−(L − 1), L − 1], t ∈ [0, 1]} ⊂ N and ∂ k ∂ l k l (γs (t) − γ(t)) ≤ O(ε). ∂s ∂t
k+l≤20
Here, we have used the notation Γ = {γ(t) : t ∈ [0, 1]} and Γs = {γs (t) : t ∈ [0, 1]}.
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
3651
As in [4], we may translate the neck N in space so that the center of mass of Γ is at the origin. Using the curve shortening flow, we can construct a homotopy γ˜r (t), (r, t) ∈ [0, 1] × [0, 1], with the following properties: • γ˜r (t) = γ(t) for r ∈ [0, 14 ]. • γ˜r (t) = (cos(2πt), sin(2πt)) for r ∈ [ 12 , 1]. ˜ r is 1 -noncollapsed. • For each r ∈ [0, 1], the curve Γ
1+δˆ 2 ∂ ∂2 sup(r,t)∈[0,1]×[0,1] | ∂r γ˜r (t)| + | ∂r∂ ∂t γ˜r (t)| + | ∂r ˜r (t)| 2γ
• We have as δˆ → 0.
ˆ where ω(δ) ˆ →0 ≤ ω(δ),
Finally, we choose a smooth cutoff function χ : R → R such that χ = 1 on (−∞, 1] and χ = 0 on [2, ∞). We next define a surface F˜Λ : [−L, Λ] × [0, 1] → R3 by ⎧ ⎪ (γ (t), s) ⎪ ⎪ s ⎪ ⎨((1 − e− 4Λ s ) γ (t), s) s F˜Λ (s, t) = 4Λ − ⎪((1 − e s ) (χ(s/Λ 14 ) γs (t) + (1 − χ(s/Λ 14 )) γ(t)), s) ⎪ ⎪ ⎪ ⎩ 4Λ ((1 − e− s ) γ˜s/Λ (t), s)
for s ∈ [−(L − 1), 0] 1
for s ∈ (0, Λ 4 ] 1
1
for s ∈ (Λ 4 , 2 Λ 4 ] 1
for s ∈ (2 Λ 4 , Λ].
It is clear that F˜Λ is smooth. Moreover, F˜Λ is axially symmetric for s ≥ Λ2 . As in [4], we may extend the immersion F˜Λ by gluing in an axially symmetric cap. To do that, we fix 1 a smooth, convex, even function Φ : R → R such that Φ(z) = |z| for |z| ≥ 100 . We then define a = 1 − e−4 +
1 1 (1 − e−4 )2 Λ− 4 3
and
vΛ (s) = 1 − e − Λ− 4 1
1
Λ + 2 Λ4 − s 1 a + Λ + 2 Λ4 − s
1 1 4Λ Λ + 2 Λ4 − s − s 4 Φ Λ 1−e −a 1 a + Λ + 2 Λ4 − s
− 4Λ s
+a
1
for s ∈ [Λ, Λ + Λ 4 ]. Moreover, we put
vΛ (s) = 2a 1
1
1
Λ + 2 Λ4 − s 1 a + Λ + 2 Λ4 − s
for s ∈ [Λ + Λ 4 , Λ + 2Λ 4 ]. The axially symmetric cap is chosen so that its cross section at 1 height s ∈ [Λ, Λ + 2Λ 4 ] is a circle of radius 21 vΛ (s). The resulting surface will be denoted ˜. by N
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
3652
1
We first consider the region s ∈ (0, Λ 4 ]. Lemma 2.3. There exist numerical constants δ∗ > 0 and Λ∗ > 0 with the following significance. If δˆ < δ∗ and Λ > Λ∗ , then we have
e−
2 x2 1 +x2 4τ
|∇N˜ x
˜ ∩{x3 =s} N
e−
≤
1
˜ +q1 x1 +q2 x2 −r0 H
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
3|
1 |∇N x3 |
1
1 for all s ∈ (0, Λ 4 ], all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . 1 ˜ ∩ {x3 = s} is obtained by dilating Proof. For each s ∈ (0, Λ 4 ], the cross section N 4Λ the cross section N ∩ {x3 = s} by the factor ρ(s) := 1 − e− s . Moreover, we have the pointwise inequalities
4Λ Λ |∇N x3 | ≤ 1 + C 2 e− x3 ˜ x3 |∇N x3 |
and 2
˜ − H ≥ c Λ e− x3 H x43
4Λ
1
for x3 ∈ (0, Λ 4 ]. Hence, if we choose Λ sufficiently large, then we have ˜
e−r0 H
1 1 ≤ e−r0 H N |∇ x3 | |∇N˜ x3 |
1
for x3 ∈ (0, Λ 4 ]. From this, we deduce that
e−
˜ ∩{x3 =s} N
2 x2 1 +x2 4τ
≤ ρ(s)
1
˜ +q1 x1 +q2 x2 −r0 H
e−
|∇N˜ x 2 ρ(s)2 (x2 1 +x2 ) +ρ(s) 4τ
3|
(q1 x1 +q2 x2 )−r0 H
N ∩{x3 =s}
1 |∇N x3 |
1
for s ∈ (0, Λ 4 ]. On the other hand, applying Lemma 2.2 with ψ = er0 (1−H) 10
ρ(s) 11 N ∩{x3 =s}
e−
2 ρ(s)2 (x2 1 +x2 ) +ρ(s) 4τ
(q1 x1 +q2 x2 )−r0 H
1 |∇N x
3|
1 |∇N x3 |
gives
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
e−
≤
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
3653
1 |∇N x3 |
1
for s ∈ (0, Λ 4 ]. Putting these facts together, we conclude that
e−
˜ ∩{x3 =s} N
2 x2 1 +x2 4τ
1
1 |∇N˜ x3 |
˜ +q1 x1 +q2 x2 −r0 H
e−
≤ ρ(s) 11
2 x2 1 +x2 4τ
1 |∇N x3 |
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s} 1
for s ∈ (0, Λ 4 ]. This proves the assertion. 1
We now consider the intermediate region s ∈ (Λ 4 , Λ4 ]. Lemma 2.4. We can find numerical constants δ∗ > 0 and Λ∗ > 0, and a positive function E∗ (Λ) with the following property. If δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ), then we have
e−
2 x2 1 +x2 4τ
|∇N˜ x
˜ ∩{x3 =s} N
e−
≤
1
˜ +q1 x1 +q2 x2 −r0 H
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
3|
1 |∇N x3 |
1
1 for all s ∈ (Λ 4 , Λ4 ], all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 .
Proof. We first observe that the claim is true if q12 + q22 is sufficiently large. More precisely, we can find positive real numbers δ∗ and Λ1 , and positive functions E1 (Λ) and Q(Λ) with the following significance. If δˆ < δ∗ , Λ > Λ1 , and ε < E1 (Λ), then we have
e−
2 x2 1 +x2 4τ
˜ ∩{x3 =s} N
≤ N ∩{x3 =s}
e−
1
˜ +q1 x1 +q2 x2 −r0 H
2 x2 1 +x2 4τ
|∇N˜ x +q1 x1 +q2 x2 −r0 H
3|
1 |∇N x
3|
1 1 whenever s ∈ (Λ 4 , Λ4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and q12 + q22 > Q(Λ). 1 Therefore, it remains to consider the case q12 + q22 ≤ Q(Λ). For each s ∈ (Λ 4 , Λ4 ], ˜ ∩ {x3 = s} is obtained by dilating the cross section N ∩ {x3 = s} by the cross section N 4Λ the factor ρ(s) := 1 − e− s , up to errors of order O(ε). Moreover, we have the pointwise inequalities
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
3654
2 Λ 4Λ 4Λ |∇N x3 | e− x3 + C ε ≤ 1 + C Λ−2 e− x3 + C ε ≤1+C ˜ 2 N x |∇ x3 | 3 and ˜ −H ≥0 H 1
for x3 ∈ (Λ 4 , Λ4 ]. Hence, we obtain
e−
2 x2 1 +x2 4τ
|∇N˜ x
˜ ∩{x3 =s} N
≤ (1 + C Λ−2 e− ·
1
˜ +q1 x1 +q2 x2 −r0 H
4Λ s
3|
+ C(Λ) ε) ρ(s) e−
2 ρ(s)2 (x2 1 +x2 ) +ρ(s) 4τ
(q1 x1 +q2 x2 )−r0 H
N ∩{x3 =s} 1
1 whenever s ∈ (Λ 4 , Λ4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and 1 r0 (1−H) using Lemma 2.2 with ψ = e gives |∇N x3 |
ρ(s)
10 11
e−
2 ρ(s)2 (x2 1 +x2 ) +ρ(s) 4τ
q12 + q22 ≤ Q(Λ). On the other hand,
(q1 x1 +q2 x2 )−r0 H
N ∩{x3 =s}
e−
≤
2 x2 1 +x2 4τ
1 |∇N x3 |
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x3 |
1 |∇N x
3|
1
1 whenever s ∈ (Λ 4 , Λ4 ], r0 ∈ [ 1000 , 1], and τ ≥ 59 . Putting these facts together, we obtain
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
˜ ∩{x3 =s} N
≤ (1 + C Λ−2 e− ·
4Λ s
1 |∇N˜ x3 | 1
+ C(Λ) ε) ρ(s) 11 e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x
3|
1 1 whenever s ∈ (Λ 4 , Λ4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and q12 + q22 ≤ Q(Λ). Clearly, we can find a positive constant Λ2 and a positive function E2 (Λ) such that (1 + C Λ−2 e−
4Λ s
1
+ C(Λ) ε) ρ(s) 11 ≤ 1
if Λ > Λ2 and ε < E2 (Λ). Hence, if we put Λ∗ = max{Λ1 , Λ2 } and E∗ (Λ) = min{E1 (Λ), E2 (Λ)}, then the assertion follows.
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
3655
1
In the next step, we consider the region s ∈ ( Λ4 , Λ + Λ 4 ]. Lemma 2.5. We can find numerical constants δ∗ > 0 and Λ∗ > 0 with the following property. If δˆ < δ∗ and Λ > Λ∗ , then we have 2 x2 1 1 +x2 ˜ e− 4τ +q1 x1 +q2 x2 −r0 H ˜ N |∇ x3 | ˜ ∩{x3 =s} N
e−
≤
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x3 |
1
1 for all s ∈ ( Λ4 , Λ + Λ 4 ], all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 .
Proof. Again, the claim is true if q12 + q22 is sufficiently large. More precisely, we can find positive real numbers δ1 and Λ1 , and a positive constant Q with the following significance. If δˆ < δ1 and Λ > Λ1 , then we have 2 x2 1 1 +x2 ˜ e− 4τ +q1 x1 +q2 x2 −r0 H ˜ N |∇ x3 | ˜ ∩{x3 =s} N
e−
≤
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x3 |
1 1 whenever s ∈ ( Λ4 , Λ + Λ 4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and q12 + q22 > Q. 1 Hence, it remains to consider the case q12 + q22 ≤ Q. For each s ∈ ( Λ4 , Λ + Λ 4 ], the ˜ ∩ {x3 = s} is close cross section N ∩ {x3 = s} is close to a circle of radius 1, whereas N ˜ − 4Λ N N s to a circle of radius ρ(s) := 1 − e . Moreover, |∇ x3 | and |∇ x3 | are close to 1. By Lemma 2.2, we have 2 x2 1 +x2 e− 4τ +q1 x1 +q2 x2 {x21 +x22 =ρ(s)2 }
≤ ρ(s)
1 11
e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2
{x21 +x22 =1}
for all (q1 , q2 ) ∈ R2 . Hence, we can find positive real numbers δ2 and Λ2 with the following property: if δˆ < δ2 and Λ > Λ2 , then 2 x2 1 1 +x2 ˜ e− 4τ +q1 x1 +q2 x2 −r0 H |∇N˜ x3 | ˜ ∩{x3 =s} N
≤ N ∩{x3 =s}
e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
1 |∇N x
3|
3656
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
1 1 whenever s ∈ ( Λ4 , Λ + Λ 4 ], r0 ∈ [ 1000 , 1], τ ≥ 59 , and q12 + q22 ≤ Q. Therefore, if we put δ∗ = min{δ1 , δ2 } and Λ = max{Λ1 , Λ2 }, then the assertion follows. 1
1
Finally, we consider the region s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ). Lemma 2.6. There exist numerical constants δ∗ > 0 and Λ∗ > 0 with the following property. If δˆ < δ∗ and Λ > Λ∗ , then we have
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
˜ ∩{x3 =s} N
e−
≤
2 x2 1 +x2 4τ
1 |∇N˜ x3 |
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s} 1
1 |∇N x3 |
1
1 for all s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . 1 1 ˜ at height s is a circle of Proof. For each s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ), the cross section of N radius
ρ(s) = a where a = 1 − e−4 +
1 3
b−s , a+b−s
(1 − e−4 )2 Λ− 4 < 1 and b = Λ + 2 Λ 4 . Note that 1
1 |∇N˜ x
3|
1
=
1 + ρ (s)2
and ˜ ≥ 1 H a ˜ ∩ {x3 = s}. Therefore, on the set N
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
˜ ∩{x3 =s} N
≤
e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −
r0 a
˜ ∩{x3 =s} N
= ρ(s)
1 |∇N˜ x3 | 1 |∇N˜ x3 |
r0 ρ(s)2 1 + ρ (s)2 e− 4τ − a eρ(s) (q1 x1 +q2 x2 ) S1
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
a2 a a4 + e 4τ 4 a + b − s 4 (a + b − s)
1−
=a
·
e
a
b−s a+b−s
a a2 a+b−s − 4τ
−
3657
r0 a
(q1 x1 +q2 x2 )
S1
for all τ ≥
5 9
1
1
and all s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ). It is elementary to check that 1−z+
z4 ≤ e−z , 4
hence 1−z+
z4 z e2 ≤ 1 4
for all z ∈ [0, 1]. Consequently, we have
1− for all τ ≥
5 9
a2 a4 a + e 4τ 4 a + b − s 4 (a + b − s)
1
a a+b−s
≤1
1
and s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ). From this, we deduce that
e−
2 x2 1 +x2 4τ
˜ ∩{x3 =s} N a2
r0
≤ a e− 4τ − a
˜ +q1 x1 +q2 x2 −r0 H
e
a
b−s a+b−s
1 |∇N˜ x3 |
(q1 x1 +q2 x2 )
S1 1
1
for all τ ≥ 59 and s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ). On the other hand, if the neck N is sufficiently close to a cylinder of radius 1, then we have
e S1
a
b−s a+b−s
(q1 x1 +q2 x2 )
≤
e
2 1−x2 1 −x2 4τ
1 +q1 x1 +q2 x2 +r0 ( a −H)
N ∩{x3 =s} 1
1
1 |∇N x
3|
1 for all s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . Putting these facts together, we conclude that
˜ ∩{x3 =s} N
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
1 |∇N˜ x
3|
3658
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
≤ ae
1−a2 4τ
e−
2 x2 1 +x2 4τ
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s} 1
1 |∇N x3 |
1
1 for all s ∈ (Λ + Λ 4 , Λ + 2 Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . Since a
is close to 1 − e−4 for Λ large, we have a e follows.
1−a2 4τ
≤ ae
9(1−a2 ) 20
≤ 1. From this, the assertion
Combining the previous results, we can draw the following conclusion: Proposition 2.7. There exist numerical constants δ∗ > 0 and Λ∗ > 0, and a function E∗ (Λ) with the following property. If δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ), then we have
2
e
˜ − |x−p| −r0 H 4τ
1
e−
≤
|x−p|2 4τ
−r0 H
1
˜ ∩{0≤x3 ≤Λ+2Λ 4 } N
N ∩{0≤x3 ≤Λ+2Λ 4 }
1 for all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all p ∈ R3 .
Proof. In view of Lemma 2.3, Lemma 2.4, Lemma 2.5, and Lemma 2.6, we can find positive real numbers δ∗ , and Λ∗ , and a positive function E∗ (Λ) with the following property. If δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ), then we have
e−
2 x2 1 +x2 4τ
˜ +q1 x1 +q2 x2 −r0 H
˜ ∩{x3 =s} N
e−
≤
2 x2 1 +x2 4τ
1 |∇N˜ x3 |
+q1 x1 +q2 x2 −r0 H
N ∩{x3 =s}
1 |∇N x3 |
1
1 for all s ∈ (0, Λ + 2Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all (q1 , q2 ) ∈ R2 . This implies
e−
|x−p|2 4τ
˜ ∩{x3 =s} N
≤ N ∩{x3 =s}
e−
1
˜ −r0 H
|x−p|2 4τ
|∇N˜ x −r0 H
3|
1 |∇N x3 |
1
1 for all s ∈ (0, Λ + 2Λ 4 ), all r0 ∈ [ 1000 , 1], all τ ≥ 59 , and all p ∈ R3 . If we integrate over s and apply the co-area formula, the assertion follows.
3. A monotonicity formula for mean curvature flow with surgery in R3 Proposition 3.1. Let Mt be a family of mean convex surfaces in R3 which evolve under smooth mean curvature flow. For each r0 > 0, the function
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
Mt
3659
|x−p|2
1 4π(t0 − t)
e− 4(t0 −t) −r0 H
is monotone decreasing for t0 − t > 0. Proof. We compute ∂ − Δ e−r0 H = −r0 e−r0 H − Δ H − r02 e−r0 H |∇H|2 ∂t ∂t = −r0 e−r0 H |A|2 H − r02 e−r0 H |∇H|2 ≤ 0.
∂
Hence, the assertion follows from Ecker’s weighted monotonicity formula (see Theorem 4.13 in [7]). We next consider a mean curvature flow with surgery in R3 . We assume that ˆ ε, L)-neck of size each surgery procedure involves performing Λ-surgery on an (ˆ α, δ, 1 1 r ∈ [ 2H1 , H1 ] (see [4] for definitions). Theorem 3.2. Let δ∗ , Λ∗ , and E∗ (Λ) be defined as in Proposition 2.7. Moreover, suppose that Mt is a mean curvature flow with surgery, and that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ). Then the function Mt
|x−p|2
1 4π(t0 − t)
is monotone decreasing for t0 − t ≥
5 9
e− 4(t0 −t) − 200 H1 H
H1−2 .
Proof. Proposition 3.1 guarantees that the monotonicity formula holds in between surgery times. Moreover, it follows from Proposition 2.7 that the monotonicity property holds across surgery times. Theorem 3.2 allows us to draw the following conclusion: Corollary 3.3. We can find positive real numbers δ∗ , Λ∗ , L, and a positive function E∗ (Λ) with the following property. Suppose that Mt is a mean curvature flow with surgery, and that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ). Finally, suppose ˆ ε, L)-neck of size r ∈ [ 1 , 1 ], and p is a point in ambient that Mt0 − contains an (ˆ α, δ, 2H1 H1 space which lies on the axis of that neck. Then Mt
for all τ ≥
5 9
H1−2 .
−2 5 0 + 9 H1 −τ
1 − |x−p|2 e 4τ ≥ 1.01 4πτ
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Proof. It is elementary to check that inf 1
r∈[ 2H , H1 1 1
]
H 1 r e−
2 r2 9 H1 20
− 200 1H
1 r
=
1 − 49 e 400 . 2
Hence, on an exact cylinder S 1 (r) × R, we have
9 H12 − 9 H1220|x|2 − 200HH 1 = e 20π
S 1 (r)×R
2 r2 9 H1 1 9π H1 r e− 20 − 200 H1 r 5
9π − 49 e 400 20 ≥ 1.02 ≥
1 for all r ∈ [ 2H , 1 ]. By assumption, the surface Mt0 − contains a neck of size r ∈ 1 H1 1 1 [ 2H1 , H1 ], and p lies on the axis of that neck. Hence, if L is sufficiently large, then we obtain
|x−p|2 9 H12 − 9 H12 20 − 200HH 1 ≥ 1.01. e 20π
M t0 −
Using Theorem 3.2, we obtain Mt
1 − |x−p|2 e 4τ 4πτ
−2 5 0 + 9 H1 −τ
≥ Mt
2 H 1 − |x−p| e 4τ − 200 H1 4πτ
−2 5 0 + 9 H1 −τ
≥
|x−p|2 9 H12 − 9 H12 20 − 200HH 1 e 20π
M t0 −
≥ 1.01 provided that τ ≥
5 9
H1−2 and L is sufficiently large.
¯ Theorem 3.4. Fix an open interval I and a compact interval J ⊂ I. Suppose that M is an embedded smooth solution of mean curvature flow which is defined for t ∈ I. Moreover, suppose that M(j) is a sequence of mean curvature flows with surgery, each of which is defined for t ∈ I. We assume that each surgery of the flow M(j) involves ˆ ε, L)-neck of size r ∈ [ 1(j) , 1(j) ], where H (j) → ∞. performing Λ-surgery on an (ˆ α, δ, 1 2H1
H1
We assume further that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ).
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¯ in the sense of geometric measure Finally, we assume that the flows M(j) converge to M (j) theory. Then, if j is sufficiently large, the flow M is free of surgeries for all t ∈ J. ¯ as j → ∞. Furthermore, the flows M(j) converge smoothly to M Proof. We first show that, for j sufficiently large, the flow M(j) is free of surgeries for all t ∈ J. Suppose that each flow M(j) has at least one surgery time tj ∈ J. For each j, ˆ ε, L)-neck in M (j) of size rj ∈ [ 1(j) , 1(j) ]. Let pj be a point in we can find an (ˆ α, δ, tj − 2H1
H1
ambient space which lies on the axis of that neck. Using Corollary 3.3, we obtain M
1 − |x−pj |2 4τ e ≥ 1.01 4πτ
(j) (j) tj + 5 (H1 )−2 −τ 9
(j) for all τ ≥ 59 (H1 )−2 . We now pass to the limit as j → ∞. If we define t¯ = limj→∞ tj ∈ J and p¯ = limj→∞ pj , then we obtain
¯2 1 − |x−p| e 4τ ≥ 1.01 4πτ
¯ t¯−τ M
¯ is smooth, we have for each τ ∈ (0, inf J − inf I). On the other hand, since M
¯2 1 − |x−p| e 4τ → 1 4πτ
¯ t¯−τ M
as τ → 0. This is a contradiction. Therefore, the flow M(j) is free of surgeries for t ∈ J. Using standard local regularity theorems for mean curvature flow (cf. [1,18]), we conclude ¯ as j → ∞. that the flows M(j) converge smoothly to M 4. Adaptation to the Riemannian setting Let Mt be a mean convex solution of mean curvature flow in a compact Riemannian three-manifold X. Let us fix a time t0 and point p in ambient space. Let ϕ be a smooth cutoff on X such that ϕ(x) = 1 for d(p, x) ≤ 14 inj(X) and ϕ(x) = 0 for d(p, x) ≥ 12 inj(X). Moreover, we put Φ(x, t) =
1 4π(t0 − t)
d(p,x)2
e− 4(t0 −t) ϕ(x)2
for t0 − t > 0. Proposition 4.1. Let K be a positive constant with the property that the ambient threemanifold X has Ricci curvature is at least −K. Then we have
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
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d dt
1 Φ exp (t0 − t) 2 − r0 e−K (t0 −t) H ≤ C |Mt |
Mt
whenever t0 − t ∈ (0, 1]. The constant C depends only on the ambient three-manifold X. Proof. In the following, we assume that t0 − t ∈ (0, 1]. In the region d(p, x) ≤ we have
1 4
inj(X),
¯ ν Φ 2 d(p, x)2 D + Δ − H2 Φ ≤ − H + Φ. Φ+C ∂t Φ t0 − t
∂
This implies ∂
d(p, x)2 + Δ − H2 Φ ≤ C Φ, ∂t t0 − t
hence ∂ 1 + Δ − H 2 Φ exp (t0 − t) 2 ∂t d(p, x)2 1 1 ≤ C − Φ exp (t0 − t) 2 ≤ C 1 t0 − t 2 2 (t0 − t) for d(p, x) ≤
1 4
inj(X). Moreover, in the region
1 4
inj(X) ≤ d(p, x) ≤
1 2
inj(X), we have
¯ ν Φ 2 D + Δ − H2 Φ ≤ − H + Φ + C. ∂t Φ
∂
This gives ∂ 1 + Δ − H 2 Φ exp (t0 − t) 2 ≤ C ∂t for
1 4
inj(X) ≤ d(p, x) ≤
1 2
inj(X). To summarize, we have shown that
∂ 1 + Δ − H 2 Φ exp (t0 − t) 2 ≤ C ∂t at each point in Mt . Since the Ricci curvature of X is bounded from below by −K, we have ∂ ∂t hence
− Δ (e−K(t0 −t) H) ≥ e−K(t0 −t) |A|2 H ≥ 0,
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∂ − Δ exp − r0 e−K(t0 −t) H ∂t ∂ = − exp − r0 e−K(t0 −t) H − Δ (r0 e−K(t0 −t) H) ∂t − exp − r0 e−K(t0 −t) H |∇(r0 e−K(t0 −t) H)|2 ≤ 0. Putting these facts together, we obtain d dt
1 2
Φ exp (t0 − t) − r0 e
−K(t0 −t)
H
Mt
≤C
exp − r0 e−K(t0 −t) H ≤ C |Mt |,
Mt
provided that t0 − t ∈ (0, 1]. This completes the proof. We now consider mean curvature flow with surgery in a Riemannian manifold. We begin with a definition. Definition 4.2. Let M be a closed surface in a Riemannian three-manifold, and let N be ˆ ε, L)-neck of size r if there exists a point o ∈ N a region in M . We say that N is an (ˆ α, δ, −1 ˆ ε, L)-neck of size r in Euclidean α, δ, with the property that the surface expo (N ) is an (ˆ space (see [4] for the definition). We next explain how to do on a neck in Riemannian three-manifold. Namely, if o lies ˆ ε, L)-neck N in X, then exp−1 (N ) is an (ˆ ˆ ε, L)-neck in Euat the center of an (ˆ α, δ, α, δ, o clidean space. Hence, we can perform the surgery procedure described in [4] on exp−1 o (N ). We then paste the surgically modified surface back into X using the exponential map expo . In the next step, we analyze how the quantity in Proposition 4.1 changes under a single surgery. To that end, it will be convenient to work in geodesic normal coordinates around o; that is, we will identify a point in To N = R3 with its image under the expoˆ ε, L)-neck in R3 . α, δ, nential map expo . With this identification, we can view N as an (ˆ Without loss of generality, we may assume that the axis of the neck N is parallel to the x3 -axis. Note that the origin lies on N , so the axis of the neck does not pass through the ˜ the surface obtained from N by performing a Λ-surgery origin. Finally, we denote by N on N . ˆ Λ, and ε satisfy δˆ < δ∗ , Λ > Λ∗ , Proposition 4.3. Suppose that the surgery parameters δ, and ε < E∗ (Λ), where δ∗ , Λ∗ , and E∗ are defined as in Proposition 2.7. Moreover, let N ˆ ε, L)-neck N of size r in X, and let N ˜ denote the surgically modified surface. α, δ, be an (ˆ Then
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S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
e−
d(p,x)2 4τ
˜ −r0 e−Kτ H
dμ
1
˜ ∩{0≤x3 ≤Λ+2Λ 4 } N
e−
≤
d(p,x)2 4τ
−r0 e−Kτ H
dμ + C(Λ) τ r2
1
N ∩{0≤x3 ≤Λ+2Λ 4 }
provided that d(o, p) ≤
1 8
inj(X),
1 K
≥τ ≥
5 9
1 r2 and r0 ∈ [ 300 r, r].
Proof. It follows from Proposition 2.7 that
e−
|x−p|2 4τ
˜ eucl −r0 e−Kτ H
dμeucl
1
˜ ∩{0≤x3 ≤Λ+2Λ 4 } N
e−
≤ 1 N ∩{0≤x3 ≤Λ+2Λ 4
|x−p|2 4τ
−r0 e−Kτ Heucl
dμeucl
}
1 provided that τ ≥ 59 r2 and r0 ∈ [ 300 r, r]. Here, |x − p| denotes the Euclidean distance ˜ eucl denote the mean of x and p in geodesic normal coordinates around o; Heucl and H ˜ curvatures of N and N with respect to the Euclidean metric; and dμeucl denotes the area form with respect to the Euclidean metric on R3 . In the next step, we compare the Riemannian distance d(p, x) to the Euclidean distance |x − p|. To that end, we perform a Taylor expansion of the function x → 12 d(p, x)2 around the origin o. The value of this function at o is given by 12 |p|2 . Its gradient at o equals −p. Moreover, the Hessian of the function x → 12 d(p, x)2 at o equals g + O(|p|2 ). Finally, the third derivatives of the function x → 12 d(p, x)2 at o are bounded by O(|p|). Putting these facts together, we obtain
1 1 1 d(p, x)2 = |p|2 − p, x + |x|2 + O(|p|2 |x|2 + |p| |x|3 + |x|4 ). 2 2 2 In other words, 1 1 d(p, x)2 = |x − p|2 + O((|p|2 + |x|2 ) |x|2 ). 2 2 ˜ ) ∩ {0 ≤ x3 ≤ Λ + 2Λ 41 }, then we have |x|2 ≤ C(Λ) r2 , hence |x|2 ≤ C(Λ) τ . If x ∈ (N ∪ N This implies d(p, x)2 |x − p|2 − ≤ C(Λ) (|p|2 + τ ) 4τ 4τ ˜ ) ∩ {0 ≤ x3 ≤ Λ + 2Λ 14 }. Consequently, we have for all points x ∈ (N ∪ N e−
|x−p|2 4τ
≤ (1 + C(Λ) (|p|2 + τ )) e−
d(p,x)2 4τ
S. Brendle / Journal of Functional Analysis 272 (2017) 3647–3668
3665
1
for all points x ∈ N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }. Since |H − Heucl | ≤ C(Λ) r for all points 1 x ∈ N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }, it follows that e−
|x−p|2 4τ
−r0 e−Kτ Heucl
≤ (1 + C(Λ) (|p|2 + τ )) e−
d(p,x)2 4τ
−r0 e−Kτ H
1
for all points x ∈ N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }. Thus,
e−
1 N ∩{0≤x3 ≤Λ+2Λ 4
|x−p|2 4τ
−r0 e−Kτ Heucl
}
dμeucl
e−
≤ (1 + C(Λ) (|p|2 + τ ))
d(p,x)2 4τ
−r0 e−Kτ H
dμ.
1
N ∩{0≤x3 ≤Λ+2Λ 4 }
Similarly, we have e−
d(p,x)2 4τ
≤ (1 + C(Λ) (|p|2 + τ )) e−
|x−p|2 4τ
˜ ∩ {0 ≤ x3 ≤ Λ + 2Λ 14 }. Since |H ˜ −H ˜ eucl | ≤ C(Λ) r for all points for all points x ∈ N 1 ˜ x ∈ N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }, it follows that e−
d(p,x)2 4τ
˜ −r0 e−Kτ H
≤ (1 + C(Λ) (|p|2 + τ )) e−
|x−p|2 4τ
˜ eucl −r0 e−Kτ H
˜ ∩ {0 ≤ x3 ≤ Λ + 2Λ 14 }. This gives for all points x ∈ N
e−
d(p,x)2 4τ
˜ −r0 e−Kτ H
dμ
1
˜ ∩{0≤x3 ≤Λ+2Λ 4 } N
e−
≤ (1 + C(Λ) (|p| + τ )) 2
1 ˜ ∩{0≤x3 ≤Λ+2Λ 4 N
|x−p|2 4τ
˜ eucl −r0 e−Kτ H
dμeucl .
}
Putting these facts together, we conclude that 1 ˜ ∩{0≤x3 ≤Λ+2Λ 4 N
e−
d(p,x)2 4τ
}
˜ −r0 e−Kτ H
dμ
e−
≤ (1 + C(Λ) (|p|2 + τ )) 1 N ∩{0≤x3 ≤Λ+2Λ 4
d(p,x)2 4τ
−r0 e−Kτ H
dμ.
}
Finally, we have the pointwise estimate (|p|2 + τ ) e− 1 N ∩ {0 ≤ x3 ≤ Λ + 2Λ 4 }. Therefore, we obtain
d(p,x)2 4τ
≤ C(Λ) τ at each point on
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3666
e−
(|p|2 + τ )
d(p,x)2 4τ
−r0 e−Kτ H
dμ ≤ C(Λ) τ r2 ,
1
N ∩{0≤x3 ≤Λ+2Λ 4 }
hence
e−
1 ˜ ∩{0≤x3 ≤Λ+2Λ 4 N
d(p,x)2 4τ
˜ −r0 e−Kτ H
dμ
}
e−
≤
d(p,x)2 4τ
−r0 e−Kτ H
dμ + C(Λ) τ r2 .
1
N ∩{0≤x3 ≤Λ+2Λ 4 }
This proves the assertion. Combining Proposition 4.1 and Proposition 4.3, we arrive at the following conclusion: Theorem 4.4. Let Mt be a mean curvature flow with surgery in a Riemannian threemanifold X. Suppose that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , and ε < E∗ (Λ), where δ∗ , Λ∗ , and E∗ (Λ) are defined as in Proposition 2.7. Then
1 e−K(t0 −tˆ) H Φ exp (t0 − tˆ) 2 − dμ 200 H1
Mtˆ
≤
1 e−K(t0 −t˜) H Φ exp (t0 − t˜) 2 − dμ 200 H1
Mt˜
+ C |Mt˜| (t0 − t˜) + C(Λ) L−1 |Mt˜|, provided that
1 K
≥ t0 − t˜ ≥ t0 − tˆ ≥
5 9
H1−2 .
Note that the error term C(Λ) L−1 |Mt˜| can be made arbitrarily small by choosing L very large (depending on Λ). Proof. By Proposition 4.1, the quantity
1 e−K(t0 −t) H Φ exp (t0 − t) 2 − dμ 200 H1
Mt
increases at a rate of at most C |Mt | in between surgery times. Moreover, Proposition 4.3 implies that, during each surgery, the quantity Mt
1 e−K(t0 −t) H Φ exp (t0 − t) 2 − dμ 200 H1
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1 increases by at most C(Λ) H1−2 , provided that K ≥ t0 − t ≥ 59 H1−2 . On the other hand, −2 1 the surface area decreases by at least 10 L H1 during each surgery. Hence, there are at −1 2 most C L H1 |Mt˜| surgeries after time t˜. Consequently, the quantity considered above increases by at most C |Mt˜| (t0 − t˜) + C(Λ) L−1 |Mt˜| between time t˜ and time tˆ.
As a consequence of Theorem 4.4, we obtain an analogue of Theorem 3.4 in the Riemannian setting. Theorem 4.5. Fix an open interval I and a compact interval J ⊂ I. Moreover, suppose ¯ is an embedded smooth solution of mean curvature flow in a Riemannian threethat M manifold which is defined for t ∈ I. Moreover, suppose that M(j) is a sequence of mean curvature flows with surgery in the same Riemannian three-manifold, each of which is defined for t ∈ I. We assume that each surgery of the flow M(j) involves performing ˆ ε, L)-neck of size r ∈ [ 1(j) , 1(j) ], where H (j) → ∞. We assume Λ-surgery on an (ˆ α, δ, 1 2H1
H1
further that the surgery parameters satisfy δˆ < δ∗ , Λ > Λ∗ , ε < E∗ (Λ), and that L is chosen sufficiently large depending on Λ. Finally, we assume that the flows M(j) con¯ in the sense of geometric measure theory. Then, if j is sufficiently large, verge to M the flow M(j) is free of surgeries for all t ∈ J. Furthermore, the flows M(j) converge ¯ as j → ∞. smoothly to M By combining Theorem 4.5 with results of Brian White [16,17], we are able to characterize the longtime behavior of mean curvature flow with surgery in Riemannian three-manifolds. This is discussed in [5]. References [1] K. Brakke, The Motion of a Surface by Its Mean Curvature, Princeton University Press, 1978. [2] S. Brendle, A sharp bound for the inscribed radius under mean curvature flow, Invent. Math. 202 (2015) 217–237. [3] S. Brendle, An inscribed radius estimate for mean curvature flow in Riemannian manifolds, Ann. Sc. Norm. Super. Pisa (2016), in press. [4] S. Brendle, G. Huisken, Mean curvature flow with surgery of mean convex surfaces in R3 , Invent. Math. 203 (2016) 615–654. [5] S. Brendle, G. Huisken, Mean curvature flow with surgery of mean convex surfaces in threemanifolds, J. Eur. Math. Soc. (2016), in press. [6] T. Colding, W. Minicozzi, Generic mean curvature flow I: generic singularities, Ann. of Math. 175 (2012) 755–833. [7] K. Ecker, Regularity Theory for Mean Curvature Flow, Birkhäuser, Boston, 2004. [8] R. Hamilton, The formation of singularities in the Ricci flow, in: Surveys in Differential Geometry, vol. II, International Press, Somerville, MA, 1995, pp. 7–136. [9] R. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997) 1–92. [10] R. Haslhofer, B. Kleiner, Mean curvature flow of mean convex hypersurfaces, arXiv:1304.0926. [11] R. Haslhofer, B. Kleiner, Mean curvature flow with surgery, arXiv:1404.2332. [12] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990) 285–299. [13] G. Huisken, C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math. 175 (2009) 137–221. [14] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:0211159.
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[15] G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:0303109. [16] B. White, The size of the singular set in mean curvature flow of mean convex sets, J. Amer. Math. Soc. 13 (2000) 665–695. [17] B. White, The nature of singularities in mean curvature flow of mean convex sets, J. Amer. Math. Soc. 16 (2003) 123–138. [18] B. White, A local regularity theorem for mean curvature flow, Ann. of Math. 161 (2005) 1487–1519.