Proposition 1

(Local umbilic estimate) Every mean curvature flow in R n + 1 , which is properly defined and α -pinched in B 2 L × [ 0 , T ) , satisfies

where A ˚ is the trace-free part of A , Θ ≑ sup B 2 L × { 0 } ∪ B 2 L ⧹ B L × ( 0 , T ) H , and C ε = C ( n , α , ε ) .

Proof

The claim follows from the m = 0 case of [13, Theorem 1] since the integral hypothesis is in this case superfluous. Indeed, by the pinching hypothesis and the area formula,

for each t ∈ [ 0 , T ) , where K is the Gauss curvature.□

Lemma 2

There exists C = C ( n ) with the following property. Let { ∂ Ω t } t ∈ [ 0 , T ) be a convex solution to mean curvature flow in R n + 1 . If B r ( p ) ⊂ Ω t and sup B 2 L r H ( ⋅ , 0 ) ≤ Θ r − 1 , where Θ ≥ 1 and L > 1 , then

Proposition 3

(Existence) Let Ω ⊂ R n + 1 be an (unbounded) convex body with smooth boundary ∂ Ω satisfying sup ∂ Ω ∣ A ∣ < ∞ . There exist δ > 0 and a family of (unbounded) convex bodies whose boundaries { ∂ Ω t } t ∈ ( 0 , δ ] are smooth and evolve by mean curvature, converge locally uniformly to Ω 0 ≑ Ω as t → 0 , and satisfy sup t ∈ [ 0 , δ ] sup ∂ Ω t ∣ A ∣ < ∞ .

Proof

We may assume that Ω contains no lines – else it splits as a product of an affine subspace with a lower dimensional convex body that contains no lines; the solution we seek is then obtained as a product of an affine subspace with the lower dimensional solution we shall construct. In that case, Ω is either bounded or, up to a rotation, the graph of a function u : D → R over some convex domain D ⊂ R n × { 0 } with compact sublevel sets { u ≤ h } . When Ω is unbounded, we consider for each height h > 0 , the bounded convex body Ω h defined by intersecting Ω with its reflection about the plane { x n + 1 = h } (if Ω is bounded, just take Ω h = Ω in what follows). Let B h be the largest ball contained in Ω h and denote by r h : ∂ B n → R the radial graph height of ∂ Ω h . We now mollify Ω h to obtain, for any sufficiently small ε > 0 , a smooth convex body Ω h , ε with a corresponding smooth radial graph function r h , ε : ∂ B n → R (e.g., we could mollify the radial graph functions r h using, say, the heat kernel on S n ).

For ε sufficiently small, we can evolve r h , ε smoothly by radial graphical mean curvature flow using parabolic existence theory. Since, by the avoidance principle, the time of existence of the approximating solutions is bounded from below by the square of their inradius (which is bounded uniformly from below as ε → 0 and h → ∞ ), Lemma 2 and the higher-order estimates of Ecker and Huisken [9] yield the claims upon taking ε → 0 and then h → ∞ .□

Proposition 4

(Pinching is preserved) Let { M t } t ∈ [ 0 , δ ] be a family of convex, locally uniformly convex hypersurfaces evolving by mean curvature flow with sup t ∈ [ 0 , δ ] sup M t ∣ A ∣ < ∞ . If κ 1 ≥ α H on M 0 , then

Proof

Since M 0 does not contain any lines, we can find p ∈ R n + 1 and e ∈ S n so that β ≑ inf X ∈ M 0 X − p ∣ X − p ∣ , e > 0 . If we define

where C ≑ sup t ∈ [ 0 , δ ] sup M t ∣ A ∣ 2 , then

Fix ε > 0 and set S ≑ A − α H g . We claim that the tensor

remains non-negative definite on [ 0 , δ ] . Suppose that this is not the case. Then, since S is positive definite on M 0 and ψ → ∞ as ∣ X ∣ → ∞ , there must exist t 0 ∈ ( 0 , δ ] , X 0 ∈ M t 0 n , and V 0 ∈ T X 0 M t 0 n such that S ( X , t ) ε > 0 for each X ∈ M t n , t ∈ [ 0 , t 0 ) , but S ( X 0 , t 0 ) ε ( V 0 , V 0 ) = 0 . Extend V 0 locally in space by solving

along radial g t 0 -geodesics γ emanating from X 0 , and then extend the resulting local vector field locally in the time direction by solving

where ∇ t is the time-dependent connection of Andrews and Baker [2]. We find that ∇ V ( X 0 , t 0 ) = 0 , ∇ t V ( X 0 , t 0 ) = 0 , and Δ V ( X 0 , t 0 ) = 0 .

Now, set s ε ( X , t ) ≑ S ( X , t ) ε ( V ( X , t ) , V ( X , t ) ) for ( X , t ) near ( X 0 , t 0 ) . We now find at ( X 0 , t 0 ) that

which is absurd. Hence, S ε indeed remains positive definite in [ 0 , δ ] . Now take ε → 0 .□

Proposition 5

(Curvature bound at infinity) Let { ∂ Ω t } t ∈ − 1 2 n R 2 , 0 be a family of convex boundaries evolving by mean curvature. Suppose that 0 ∈ Ω 0 . Given ε > 0 and δ > 0 , there exists L < ∞ such that, given any X ∈ M 0 ⧹ B L R ( 0 ) ,

Proof

It suffices to prove the claim when R = 1 . So, contrary to the claim, defining P r ( X , t ) ≑ B r ( X ) × ( t − 1 2 n r 2 , t ] assumes that we can find ε > 0 , δ > 0 , and a sequence of points X j ∈ M 0 such that

where P r ( X , t ) ≑ B r ( X ) × t − 1 2 n r 2 , t .

Point selection yields a sequence of points ( Y j , s j ) with the following properties:

1. ( Y j , s j ) ∈ P δ 2 H ( X j , 0 ) ( X j , 0 ) (and hence κ 1 H ( Y j , s j ) > ε ).

2. H ( Y j , s j ) ≥ H ( X j , 0 ) .

3. H ≤ 2 H ( Y j , s j ) in P δ 4 H ( Y j , s j ) ( Y j , s j ) .

So, ( X j 1 , t j 1 ) satisfies (1) and (2). If ( X j 1 , t j 1 ) also satisfies (3), choose ( Y j , s j ) = ( X j 1 , t j 1 ) ; if not, choose ( X j 2 , t j 2 ) ∈ P δ 4 H ( X j 1 , t j 1 ) ( X j 1 , t j 1 ) such that

Since P δ 4 H ( X j 1 , t j 1 ) ( X j 1 , t j 1 ) ⊂ P δ 4 H ( X j , 0 ) + δ 8 H ( X j , 0 ) ( X j , 0 ) , this point satisfies properties (1) and (2). Since ∑ k = 1 ∞ 2 − k = 1 and H is finite on the set P δ H ( X j , 0 ) ( X j , 0 ) , continuing in this way we find, after some finite number of steps k , a point ( Y j , s j ) ≑ ( X j k , t j k ) satisfying (1), (2), and (3).

Now, translate ( Y j , s j ) to the spacetime origin and rescale by λ j ≑ H ( Y j , s j ) to obtain a sequence of rescaled flows { M t j } t ∈ ( − λ j 2 ( s j + 1 2 n ) , 0 ] defined by M t j ≑ λ j ( M λ j − 2 t + s j − Y j ) . Passing to a subsequence, the final timeslices M 0 j converge locally uniformly in the Hausdorff topology to a convex hypersurface M 0 ∞ = ∂ Ω 0 ∞ .

We claim that M 0 ∞ splits off a line. To see this, consider the segments ℓ i joining Y j ∈ ∂ Ω s j to 0 ∈ Ω 0 ⊂ Ω s j . Passing to a subsequence, these segments converge to a ray, ℓ , emanating from 0. Observe that ℓ ⊂ Ω 0 for all j . Indeed, if this were not the case, then we could find a point p ∈ ℓ ∩ ∂ Ω 0 . But then, since ℓ ∩ ∂ Ω − 1 ∕ 2 = ∅ , the tangent hyperplane to ∂ Ω t parallel to T p ∂ Ω 0 must travel an infinite distance in finite time, contradicting the uniform curvature bound. Now, consider the segment ℓ i ′ obtained from ℓ i by reflection across the hyperplane orthogonal to ℓ and through Y j . Since Ω s j is convex, the triangle bounded by ℓ i , ℓ i ′ , and ℓ lies in Ω ¯ s j for all j . The claim follows, since the angle between ℓ i and ℓ i ′ goes to π as i → ∞ and λ i ≥ δ > 0 .

Since (after passing to a subsequence) the convergence is smooth in P δ 8 , we find that H = 1 , κ 1 = 0 , and κ 1 H ≥ ε at the spacetime origin, which is absurd.□

Proposition 6

(No pinched solitons) There exist no locally uniformly convex pinched mean curvature flow translators or expanders.

Proof

We proceed as in [5,19]. First, let M n ⊂ R n + 1 , n ≥ 2 , be a locally uniformly convex, pinched mean curvature flow translator. Then, M satisfies

for some e ∈ R n + 1 ⧹ { 0 } . Observe that the vector field V ≑ e ⊤ satisfies

and

where L denotes the Weingarten map (see, e.g., Lemma 13.32 in the book of Chow et al. [3]).

By Proposition 5, H attains its maximum at some point o ∈ M . Since A > 0 , (2) implies that o is a zero of V . We claim that V vanishes nowhere else. To see this, fix X ∈ M ⧹ { o } and let γ : [ 0 , d ( X ) ] → M be any minimizing unit speed geodesic joining o to X , where d denotes the intrinsic distance to o . Observe that

This is positive (and hence V X ≠ 0 ) since, by (3), ∇ V is positive definite. In fact, since ∣ V ∣ 2 = 1 − H 2 and A ≥ α H g for some α > 0 , we obtain (we will use this in a moment)

It follows that M ⧹ { o } is foliated by integral curves σ : ( 0 , ∞ ) → M of V with σ ( s ) → o as s → 0 . Let σ be such a curve. By (2) and the pinching hypothesis,

Combining (5) and (6) yields

Since M is convex and non-flat, it follows from the Ecker–Huisken interior estimates that λ M converges as λ → 0 locally uniformly in C ∞ ( R n + 1 ⧹ { 0 } ) to a non-planar convex cone. But this violates pinching.

Now, let M n ⊂ R n + 1 , n ≥ 2 , be a locally uniformly convex, pinched mean curvature flow expander. Then, M satisfies

Observe that the vector field V ≑ 1 2 X ⊤ satisfies (2) and

(see, e.g., [3, Lemma 10.14]).

By Proposition 5, H attains its maximum at some point o ∈ M . Since A > 0 , (2) implies that o is a zero of V , arguing as in (4), we find that V vanishes nowhere else. In fact, we obtain

for some α > 0 . On the other hand,

Since (2) holds, (6) also holds (with V = 1 2 X ⊤ ) on an expander. Combining (6), (8), and (9), we conclude that

The claim now follows as before.□

Theorem 7

(Hamilton [10]) Every pinched, convex hypersurface with bounded curvature in R n + 1 , n ≥ 2 , is either a hyperplane or compact.

Proof

So let M = ∂ Ω ⊂ R n + 1 , n ≥ 2 , be a convex hypersurface with bounded curvature, which is α -pinched for some α > 0 .

By Proposition 3, we obtain a family { M t } t ∈ [ 0 , T ) of convex boundaries M t = ∂ Ω t with M 0 = M , which evolve by mean curvature and satisfy sup t ∈ [ 0 , σ ] sup M t ∣ A ∣ < ∞ for all σ ∈ [ 0 , T ) and either T = ∞ or

By Proposition 4, M t is α -pinched for each t ∈ ( 0 , T ) . By applying the strong maximum principle to the evolution equation for H (see, e.g., [3, Equation (6.18)]), we may assume that H > 0 on M t for all t ∈ ( 0 , T ) .

Case 1: T < ∞ . Since Proposition 5 implies that

the local umbilic estimate (Proposition 1) yields

for every ε > 0 , where Θ ≑ sup M 0 H < ∞ .

Since ∣ A ( X , t ) ∣ → 0 as ∣ X ∣ → ∞ , we can find a sequence of times t j → T and points X j ∈ M t j such that

Translating ( X j , t j ) to the spacetime origin and rescaling by λ j yield a sequence of mean curvature flows { M t j } t ∈ ( − λ j 2 t j , 0 ] defined by M t j ≑ λ j ( M λ j − 2 t + t j − X j ) . Since max M t j H ≤ 1 = H ( 0 , 0 ) and λ j → ∞ as j → ∞ , a subsequence of the rescaled flows converges locally uniformly in C ∞ ( R n + 1 × ( − ∞ , 0 ] ) to an ancient flow { M t ∞ } t ∈ ( − ∞ , 0 ] . By (11), this limit is umbilic, and hence a shrinking sphere. So M is compact.

Case 2: T = ∞ . Suppose first that the flow is type-III, i.e., Λ ≑ sup t ∈ ( 0 , ∞ ) t max M t H < ∞ . Given a sequence of times t j > 0 with t j → ∞ , consider the rescaled flow { M t j } t ∈ ( − λ j 2 t j , ∞ ) defined by M t j ≑ λ j M λ j − 2 t + t j , where λ j ≑ 1 t j . Since λ j 2 t j ≡ 1 and

we obtain a subsequential limit defined for t ∈ ( − 1 , ∞ ) . Furthermore,

By the differential Harnack inequality and the type-III hypothesis, the limit on the right exists (and is positive) independently of t . Thus, by the rigidity case of the differential Harnack inequality, the limit is a nontrivial expander, which violates Proposition 6.

So, suppose that the flow is type-IIb, i.e., sup t ∈ ( 0 , ∞ ) t max M t H = ∞ . For each j , choose ( x j , t j ) such that

and set λ j ≑ H ( x j , t j ) . The corresponding rescaled flows satisfy

for all t ∈ ( α j , ω j ) , where α j ≑ − λ j 2 t j and ω j ≑ λ j 2 ( j − t j ) . Since

we find that α j → − ∞ and ω j → ∞ , and hence obtain an eternal limit flow { M t } t ∈ ( − ∞ , ∞ ) . Since max H is attained at the spacetime origin (where it is positive), the differential Harnack inequality implies that { M t } t ∈ ( − ∞ , ∞ ) evolves by translation. This violates Proposition 6.□