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Willmore flow of surfaces in Riemannian spaces I: Concentration-compactness

In this paper we study the local regularity of closed surfaces immersed in a Riemannian 3-manifold flowing by Willmore flow. We establish a pair of concentration-compactness alternatives for the flow, giving a lower bound on the maximal time of existence of the flow proportional to the concentration of the curvature and area at initial time. The estimate from the first theorem is purely in terms of the concentration of curvature at initial time but applies only to ambient spaces with non-positive sectional curvature. The second requires additional information on the concentration of area at initial time but applies in more general background spaces. Applications of these results shall appear in forthcoming work.


Introduction
Suppose f : Σ → N is a smooth immersion of the surface Σ into the smooth three-dimensional complete Riemannian manifold (N, ·, · ). Let us equip Σ with the metric induced by f , so that (Σ, f * ·, · ) is a Riemannian manifold. We assume that (Σ, f * ·, · ) is closed and complete. Consider the functional where H is the mean curvature and dµ the measure induced via f . Let us use A to denote the second fundamental form of the immersion f , and A o to denote its tracefree part. Surfaces which are critical for W satisfy the Euler-Lagrange equation where Ric denotes the Ricci curvature of (N, ·, · ) and ν is the exterior unit normal vectorfield along f . Note that (1) is invariant under change of orientation and reparametrisation.
We state the following local existence result. The proof follows by first writing the solution as a graph over the initial manifold in the direction of the unit normal ν and then applying the standard theory of higher-order degenerate quasilinear parabolic equations. Details can be found in [1,Chapter 3]. See also [4,Chapter 5], [15] and [12]. The family f is called a Willmore flow with initial data f 0 .
Our main result is the following pair of concentration-compactness alternatives for the flows (2), sometimes termed Lifespan Theorems [13,16].
Remark 1 (Dependence on the metric of N ). Covering arguments are used in the proof of Theorem 2 above and Theorem 3 below. Here the number of balls needed to cover a ball with a given radius in N depends on the metric locally around the largest ball. In addition to bounds on the curvature of N and the injectivity radius of N , this is the only way in which the constants depend on the metric of N . Note for Theorem 2 the hypotheses imply that the injectivity radius of N is unbounded.
In spaces with some positive curvature, the required Sobolev inequality fails for relatively 'large' submanifolds. In order to compensate for this, we supplement the argument used for Theorem 2 with control of the concentration of area along the flow. Our result in this case is the following.
Theorem 3 (Lifespan theorem for Willmore flow in a Riemannian 3-manifold). Let (N, ·, · ) be a smooth Riemannian 3-manifold with positive injectivity radius ρ N . Suppose f : Σ × [0, T ) → N is a one-parameter family of immersed surfaces with smooth initial data evolving by (2). Then there are constants ρ > 0,ε 0 > 0, σ 0 > 0 and c < ∞ depending only on the metric of N and ∇ (k) Ric ∞ for k ∈ {0, 1, 2, 3, 4, 5} such that if ρ is chosen with for any x ∈ N , and then the maximal time T of smooth existence satisfies and we have the estimate Remark 2. The universal constants ε 0 ,ε 0 and σ are, given N , computable and not the result of an abstract existence proof.
In a more global sense, we present Theorems 2 and 3 with a perspective toward further analysis of the flow (2). In particular, as the statement depends on the concentration of the curvature of the initial surface, the result is particularly relevant to the analysis of asymptotic behaviour in the following respect. When considering a blowup of a singularity formed at some time T < ∞ of the constrained Willmore flow, we wish to have that some amount of the curvature concentrates in space. For example, from Theorem 2, if ρ(t) denotes the largest radius such that (4) holds at time t, then ρ(t) ≤ 4 c(T − t) and so at least ε 0 of the curvature concentrates in a ball f −1 (B ρ(T ) (x)). That is, is understood to be the centre of a ball where the integral above is maximised. This will be a fundamental property of blowups considered in an upcoming paper.

Notation and setting.
In this section we collect various general formulae from the differential geometry of submanifolds which we need for later analysis. We use notation similar to that of Hamilton [5] and Huisken [7,8]. We have as our principal object of study a smooth immersion f : Σ → (N, ·, · ) of an orientable surface Σ into a Riemannian 3-manifold (N, ·, · ). The definitions below are understood with respect to a local orthonormal frame {e 0 , e 1 , e 2 } of N , such that restricted to Σ we have e 0 = ν and e i = ∂f ∂xi for i = 1, 2. Further, we take this frame to induce normal coordinates on Σ, so that the Christoffel symbols on Σ and N vanish at a single point. Throughout the paper we perform all calculations with respect to this local orthonormal frame and these coordinates.
The immersion induces a Riemannian metric on Σ with components so that the pair (M, g) is a Riemannian submanifold of (N, ·, · ). We use to denote the inner product between X and Y with respect to the metric g. Here and for the rest of the paper we have adopted the convention that repeated indices are summed from 1 to 2. The Riemannian metric induces an inner product structure on all tensor fields over Σ, not just vector fields. This is realised as the trace over pairs of indices with the metric: In the above formula g ij = (g − 1) ij where g − 1 is the inverse of g. For tensors T defined purely on N , we use the convention that |T | 2 = T , T unless otherwise specified. The mean curvature H is given by Here we use ∇ to refer to the Levi-Civita connection on N . Throughout the paper we use a bar to indicate quantities on N . We note that this is the opposite sign convention of Huisken [7,8], and the same sign convention as Kuwert & Schätzle [10,11]. The Christoffel symbols of the induced connection are determined by the metric, with the covariant derivative on Σ of a vector X and of a covector Y is The second fundamental form is symmetric and satisfies the Codazzi equations: Here R denotes the curvature tensor of N The curvature tensor R on Σ is defined analogously. The second fundamental relation between components of the Riemann curvature tensor R on Σ, the curvature tensor R on N , and the second fundamental form A on Σ, is given by Gauss' equation with contractions where Sc, Sc is the scalar curvature of (Σ, g) and (N, ·, · ) respectively. We will need to interchange covariant derivatives; for vectors X and covectors Y we obtain where ∇ i1...in = ∇ i1 · · · ∇ in . Further for T a tensor field of type (p, q) we define ∇ (r) T to be the tensor field of order (p, q + r) with components ∇ i1...ir T k1...kq j1...jp . We also use for tensors T and S the notation T * S (as in Hamilton [5]) to denote a linear combination of new tensors, each formed by contracting pairs of indices from T and S by the metric g with multiplication by a universal constant. The resultant tensor will have the same type as the other quantities in the equation it appears. Keeping these in mind we also denote polynomials in the iterated covariant derivatives of these terms by where k 1 , . . . , k j ≥ 0 and the constants c k1···kj ∈ R are absolute. We use the convention that P i j (T ) = 0 if i < 0 or j ≤ 0. At times we will need to consider polynomials in multiple tensors or functions, for which we shall use the notation where k 1 , . . . , k j , β 1 , . . . , β α ≥ 0 and the constants c k1···kj β1···βα ∈ R are absolute. As is common for the * -notation, we slightly abuse these constants when certain subterms do not appear in our P -style terms. For example . This will occur throughout the paper without further comment.
The Laplace-Beltrami operator on Σ acting on a tensor T is given by Using the Codazzi equation with the interchange of covariant derivative formula given above, we obtain Simons' identity [8, Lemma 2.1]: In most of our integral estimates, we include a function γ : Σ → R in the integrand. Eventually, this will be specialised to a smooth cutoff function between concentric geodesic balls on Σ. For now however, let us only assume that γ =γ • f , where (γ) 0 ≤γ ≤ 1, and γ C 2 (N ) ≤ cγ < ∞.
When we write "for a function γ : Σ → R as in (γ)" we mean a function γ : Σ → R as above, satisfying all conditions labeled (γ), which additionally achieves the values zero and one in at least two points on Σ. We note that ifγ is a cutoff function on a geodesic ball in N of radius ρ, then we may choose c γ = c ρ where c is a universal constant and we have used that cγ = cγ(ρ). This also explains our choice of scaling for the bound in (γ).

Evolution equations for integrals of curvature.
To begin, we state the following elementary evolution equations, whose proof is standard.
Using F = W = ∆H + H|A o | 2 + HRic(ν, ν), the P -notation introduced in the previous section, and the formula R ijkl = Ric(e i , e l ) e j , e k − Ric(e i , e k ) e j , e l − Ric(e j , e l ) e i , e k + Ric(e j , e k ) e i , e l − 1 2 Sc e i , e l e j , e k − e i , e k e j , e l , we write the evolution of the second fundamental form as ∂ ∂t A ij = − ∇ ij ∆H + P 2 3 + P 0 5 (A) + P 2 1,1 + P 0 3,1 + P 0 1,2 + P 1 2,1 (A, Ric) .
Interchanging covariant derivatives and applying (12) then gives the following lemma.
Let us define K i to be the bound for the i-th ambient derivative of the Ricci curvature of (N, ·, · ): We now establish a basic energy estimate for (2). In estimates such as these, integral quantities are evaluated at each immersion f (·, t) for t ∈ [0, T ). This t-dependence is not typically noted, unless possibly ambiguous or when integrals are evaluated at different times. This situation arises when integrating estimates such as Lemma 6 below.
Proof. (Sketch). We use an induction argument analogous to that found in [11,Lemma 2.4].
Step one of induction for k = 0 can be easily obtained from the result of Lemma 5 and using the conversion from ∇ to ∇ when acting on Ric where the error terms involve the curvature and its derivatives, which we have already computed in the proof of Lemma 6. To prove the induction step from k to k + 1 we use [11,Lemma 2.3] and again the conversion between the two types of derivative and absorb the extra terms arriving from the different more complicated speed of the flow into the summations involving the Ricci curvature.
Proof. Integrating by parts and interchanging covariant derivatives, we obtain A standard argument with integration by parts and Young's inequality yields the interpolation inequalities The estimates (15), (16) above allow us to interpolate away several terms on the right hand side of (14). For the remaining terms we use In the last estimate above and for the rest of the proof we shall no longer write the dependence of c on c γ explicitly. Combining (14), (15), (16), and (17) we obtain the estimate where c depends only on K 0 , s, c γ and δ. The estimate (18) is the leading order term in our main estimate below for the evolution of the concentration of curvature in L 2 .
We compute where (18) was used in the last step. The rightmost triple of integrals are estimated with where we again applied (17). Combining (20) with (19) and using (15), (16), we have This finishes the proof.
For later application we split out the cases k = 1 and k = 2 from Lemma 8. (2). There exists a constant c ∈ (0, ∞) depending only on K i for i = 0, 1, 2, 3, s and c γ such that the evolution of the concentration of ∇A in L 2 is estimated by The constant on the right hand side depends on K i for i = 0, 1, 2. From this point onward we also allow the constant c to depend upon K 3 . We have used the notation |P j i | to mean that the norm of each term in that sum of contractions is taken. Using Young's inequality we estimate the right hand side by Let us now estimate each of the terms on the right hand side of (22) in turn. We begin with An analogous computation yields Combining (23) and (24) we find Estimate (25) deals with the third term on the right hand side of (22), while the fourth term on the right hand side of (22) is estimated by first applying (24) and then using (25). Observe that the fifth term is already no problem: Choosing δ sufficiently small and absorbing finishes the proof.
As in Lemma 9 above we use Young's inequality to estimate As earlier we estimate Estimate (28) deals with the third term on the right hand side of (27), while the fourth term on the right hand side of (27) is estimated by first applying (25) and then using (28). The fifth term is estimated by applying (24), then (25) and finally (28). Observe that the sixth term is of the form Choosing δ sufficiently small and absorbing in (26) we find For the last term we integrate by parts and estimate using Young's inequality repeatedly to obtain Absorbing yields Combining (30) with (29) and absorbing with estimates (25) and then (28) finishes the proof.
To deal with the sixth term on the right hand side of (22) we have to apply the Hoffman-Spruck Sobolev inequality, which takes various forms depending on the geometry of (N, ·, · ). This is done in the next section.

4.
Integral estimates with small concentration of curvature.
We will primarily use the Hoffman-Spruck Sobolev inequality [6], which is the famous Michael-Simon Sobolev inequality [14] adapted to submanifolds of Riemannian spaces. We state here a version of [6, Theorem 2.1] which is tailored to our situation: Theorem 11 (Hoffman-Spruck Sobolev inequality for solutions of (2)). Let f : Σ × [0, T ) → N be a family of immersed surfaces and u ∈ C 1 c (Σ × [0, T )). Assume that , and (32) where ρ N is the injectivity radius of N . Then we have Remark 3. If N is simply connected, complete, and with non-positive sectional curvature, the condition (31) is automatically satisfied regardless of the value of |Σ| along the flow. Furthermore, by the Cartan-Hadamard theorem the injectivity radius ρ N = ∞ on such a manifold, and so (32) is also satisfied. This means that the assumption (3) implies that we may apply Theorem 11 along the flow for any differentiable test function u with compact support at any time.
A straightforward consequence of Theorem 11 is the following multiplicative Sobolev inequality. With Theorem 11 in hand, the proof follows the same argument as in [11,Lemma 4.2], and so we omit it.
Lemma 12. Let f : Σ × [0, T ) → N be a family of immersions and let γ be as in (γ) satisfying the assumptions (31), (32) of Theorem 11 with u = γ. If s ≥ 4 then we have where c is a constant depending only on s.
For solutions in simply-connected 3-manifolds with non-positive Ricci curvature, we immediately obtain our desired control of the concentration of curvature along the flow.
Proof. Using Lemma 12 we estimate the right hand side of the energy estimate Lemma 6 as follows.
Choosing δ 0 = 1 2 , clearly for ε smaller than 1 2c0 we may absorb the first term on the left and obtain d dt Integrating the above differential inequality yields the desired estimate (34).
We use Lemma 9 and Lemma 10 to improve this to pointwise control. (2) and let γ be as in (γ) such that the assumptions (31) and (32) of Theorem 11 with u = γ are satisfied. There is an ε 2 > 0 and s 1 depending only on c γ and where c 2 is a constant depending only on c γ , K i for i ∈ {0, 1, 2, 3}, A 2 2,[γ>0] t=0 , and ∇A 2 2,[γ>0] t=0 . Proof. Throughout the proof we apply several of the previous results. This requires that we choose an s sufficiently large to allow application of each of them. Of course, since there are only finitely many choices, s is an absolute constant. We begin by using Lemma 12, the smallness assumption, and estimate (25) to improve the statement of Lemma 9 to d dt Σ We now need to deal with the P -style terms above. Let us begin by estimating To control the second term on the right hand side we use new multiplicative Sobolev inequalities Modifying the proof of (30) we see that the |∇A| 4 component can be estimated with Since we already have good control over A 6 6,γ s , our control over the right hand side is as good as our control over A∇ (2) A 2 2,γ s . The last component is also controlled by the first. Recall that we assume (31) and (32) are satisfied for u = γ, and clearly the support of ϕγ is contained in the support of γ for any function ϕ : Σ → R. We may thus apply the Hoffman-Spruck inequality to find The critical term to control is thus A∇ (2) A 2 2,γ s . Using integration by parts we first compute that We continue by absorbing, using (37), (38) and estimating to obtain Absorbing once again by choosingδ sufficiently small and using Lemma 12 followed by (25) we finally have Now combining each of the estimates (36), (37), (38), (39) with (35) and choosing δ sufficiently small we have |∇ (4) Ric| 2 dµ .
Absorbing again for ε 2 sufficiently small and integrating finishes the proof. (2) and let γ be as in (γ) such that the assumptions (31) and (32) of Theorem 11 with u = γ are satisfied. There is an ε 3 > 0 and s 2 depending only on c γ and K i for i ∈ {0, 1, 2, 3, 4} such that if where c 3 is a constant depending only on c γ , K i for i ∈ {0, 1, 2, 3, 4}, T , and Proof. We begin by using Lemma 12, the smallness assumption, and estimates (23), (28) to improve the statement of Lemma 10 to We need to deal with the P -style terms above. Let us use integration by parts to estimate In order to deal with these terms we use the following interpolation inequality. The proof is by induction on an easy modification of the argument used to prove (25). For each δ > 0 there exist constants c ∈ (0, ∞) and s ∈ (4, ∞) depending only on δ, n, p, r, c γ and k such that We will also need the estimate (44) Σ |A| 6 γ s−p dµ ≤ cε which holds for s ≥ max{s 1 + p, 2p + 4}, and for c depending additionally on T . This dependence is affine, so that T < ∞ gives c < ∞. The proof of this is similar to that of Lemma 12, except we use the additional information given to us by Proposition 14. For the reader's convenience we reproduce the details here. Applying Theorem 11 we have whereupon absorbing finishes the proof of (44). We applied Proposition 14 in the last estimate, which requires s 2 ≥ s 1 . Note that the constant depends on T and the initial data, but is finite so long as T < ∞ and f 0 ∈ W 3,2 .
We now continue with the main proof. Applying Theorem 11 we estimate the last term on the right hand side of (42) by using (44) in the last two steps and requiring s ≥ max{2s 1 + 2p, 4p + 8}. We continue by using Theorem 11, integration by parts, and standard estimates to control the second last term: so that (44) (with s ≥ 2s 1 + 4), Proposition 14, and absorbing give Squaring (47) and estimating with (43) we arrive at Combining (48) with (46) and absorbing we have The first term on the right hand side of (49) can be estimated exactly as in the proof of Proposition 14 and the second term can be estimated via (43). In light of the work completed in the proof of Proposition 14 we have thus improved (42) to The second integral on the right hand side of (50) can be absorbed for ε 1 sufficiently small. The first integrand of the third integral is controlled by the second two, thanks to the following estimate. Using the Hoffman-Spruck Sobolev inequality and (39), we have Combining (51) with (50) and absorbing (for ε 1 sufficiently small) we find The second integrand in the second integral on the right hand side of (52) can be estimated by which upon absorption and using Proposition 14 yields Note that s is already assumed larger than required for the last step. To estimate this pair of integrals we apply (43) twice to obtain Plugging the above estimate (54) into (52) we find Estimating the remaining term on the right hand side of (55) is similar to the estimates we have already obtained: Using again the estimate (43) and absorbing we finally have which we combine with (55) to obtain Combining the estimate (56) with (41), choosing δ, ε 3 sufficiently small and absorbing yields the result.
These last two estimates combine to give a pointwise curvature bound which is crucial.
Then for any t ∈ [0, T ) we have where c is a constant depending only on c γ , K i for i ∈ {0, 1, 2, 3, 4}, K ρ,i for i ∈ {3, 4, 5}, and Proof. Let x ∈ N be arbitrary and set γ to be as in (γ), satisfying in addition Lemma 4.3 from [11] holds whenever Theorem 11 holds, which is for any u, due to the hypotheses on the ambient space. Thus, applying Proposition 15, For solutions of (2) in 3-manifolds which are not simply-connected or have somewhere positive Ricci curvature, it is not clear that the Hoffman-Spruck Sobolev inequality is applicable along the flow. The surface area may grow to violate one or both of (31), (32). This is somewhat tricky: it is absolutely critical that the Sobolev constant in Theorem 11 does not depend on the geometry of f (Σ). Our strategy is to show that if the initial concentration of area is small enough, and we have good control on certain curvature integrals in L 1 ([0, T )), then the concentration of area for a distinct time interval remains small.
Proposition 17 (Control on the concentration of area). Let f : Σ × [0, T ) → N be a solution to (2). Set There is a tuple (ε 0 ,σ) depending only on the metric of N and c γ such that the following statement holds. Assume ρ > 0 is such that Proof. Using Lemma 4 we compute Estimating, we have Since γ has compact support, we can cover its image under f with finitely many balls of radius ρ, with ρ as in (57). Therefore for a parameter c 4 to be later specified. We will show that this leads to a contradiction. Note that forσ 0 andε 0 small the right hand side of (60) is large. The conditions (31) and (32) for the Hoffman-Spruck Sobolev inequality are satisfied on [0, t 0 ). We may therefore apply Propositions 13, 14 and 15. Using these with a cutoff function η in these propositions satisfying [η > 0] ⊂ [γ = 1], we have for t ∈ [0, t 0 ). Now consider a cutoff function ϕ satisfying the conditions of (γ) and Combining (62) with (61) yields Integrating, squaring then using Hölder's inequality gives Applying a version of Gronwall's inequality [3, Theorem 5] and supposingσ 0 < 1 we find where c 4 is a finite, absolute constant, depending only on c from (64) and K 0 , K 4 , K 5 . The maximality of t 0 implies that contradicting the maximality of t 0 . Therefore as required. To see that this proves the applicability of Theorem 11 on [0, t 0 ), let u : Σ → R be any function with compact support satisfying in addition Clearly then |Σ| [u>0] ≤ |Σ| [γ=1] ≤ |Σ| γ 4 ≤ c supp , so that both (31) and (32) are satisfied.
Remark 4. The above estimate will also be applicable to the case where f is an entire solution to (2). This is because we have kept all estimates local and not used any bound on |Σ|. If |Σ| is initialy bounded, a global estimate is easy to obtain via a simplified version of the above argument.
Remark 5. For submanifolds of a Riemannian space, it makes sense to consider rescalings which scale both the ambient space and the submanifold simultaneously by the same factor. The concentration of area is not scale invariant: under a scaling with magnitude ρ, it scales like ρ 2 . The Ricci curvature scales like ρ −2 , and so the right hand side of (58) also like ρ 2 . This implies that despite the concentration of area not being scale invariant, the condition (58) is.
By combining Propositions 13 and 17 we conclude the following concentration of curvature estimate for solutions in a space with some positive curvature.
where c is a constant depending only on the metric of N , c γ , K i for i ∈ {0, 1, 2, 3, 4, 5}, and ∇ (i) A 2 2 t=0 for i ∈ {0, 1, 2}. Remark 6. If T < ∞, then we may always take ρ small enough in (57) to force T * = T . We may therefore assume, up to imposing ρ small enough, that Corollary 18 above and Corollary 19 below hold on the entire time interval [0, T ), so long as T < ∞. If T = ∞ then for any compact subinterval I there is a ρ 0 > 0 small enough such that the pointwise estimates hold on I. In the case of a sequence of compact subintervals I j where sup{t ∈ I j } → ∞, ρ 0 → 0 and the estimates do not hold in the limit.
The Hoffman-Spruck inequality is the only critical ingredient needed for Propositions 14 and 15. We therefore obtain a pointwise curvature estimate also in this case.

Proof of the main theorems
Let us assume first the hypotheses of Theorem 2. We shall describe the modifications required to prove Theorem 3 after we finish the proof of Theorem 2.
We make the definition  where λ is a parameter to be specified later. We assume that ε 0 ≤ min{ε 1 , ε 2 , ε 3 }. The proof continues in three steps. First, we show that it must be the case that t 0 = min(T, λ) for a finite parameter λ to be chosen. Second, we show that if t 0 = λ, then we can conclude the lifespan theorem. Finally, we prove by contradiction that if T = ∞, then t 0 = T . Note that of course if T were infinite, then λ < T and by step 2 we conclude the lifespan theorem. We label these steps as t 0 = min(T, λ), We now give the proof of (69). From the hypothesis of Theorem 2, η(0) ≤ ε 0 ≤ ε 3 < 3c η ε 3 , and therefore by the definition (68) and smoothness of the flow we have t 0 > 0. Assume for the sake of contradiction that t 0 < min(T, λ). Then (68) and the continuity of η gives (72) η(t 0 ) = 3c η ε 0 .
Now set γ to be a cutoff function as in (γ) such that for some x ∈ N . By varying ρ in (4) we may choose ε 0 small enough that 3c η ε 0 = ε 3 . Definition (68) then implies that the smallness condition (33) is satisfied on [0, t 0 ). Therefore we may apply Proposition 13 to obtain for all t ∈ [0, t 0 ). We combine this with (67) to conclude where 0 ≤ t < t 0 . Since η is continuous, we can let t → t 0 . This is in contradiction with (72). Therefore, with the choice of λ in equation (73), the assumption that t 0 < min(T, λ) is incorrect. We have thus proven (69). This argument also establishes (70). Making explicit the dependence of c 1 on ρ we have λ = 1 3c 1 c η ρ 4 wherec 1 is now a constant depending only on K i for i = 0, 1, 2. Now if t 0 = λ then which is (5). Also, (74) implies (6). That is, we have proved if t 0 = λ, then the lifespan theorem holds, which is the second step (70).
It only remains to prove equation (71). We assume t 0 = T = ∞ , and will derive a contradiction with the finite maximality of T . The recent existence result due to Lamm and Koch [9] applied for the Willmore flow of C 0 graphs in R n with small Lipschitz constant, and gives global existence. It is a straightforward adaptation of their argument to obtain local existence for C 0,1 graphs with any Lipschitz constant. By considering graphs over the initial datum instead of graphs over R 2 , this additionally holds for the setting we consider here: closed immersed submanifolds of a smooth Riemannian space. The maximality of T in light of this local existence theorem implies that as t → T either the Lipschitz constant of f is becoming unbounded, or the surface is no longer a surface. We now claim that any sequence of immersions f (·, t j ) where t j → T has a convergent subsequence in the C 1 topology, with limit a C 1 immersed surface. This is an immediate consequence of the pointwise bound for the second fundamental form from Corollary 16 combined with an application of the main compactness theorem in [2]. This implies that the Lipschitz constant of f is not blowing up and that the geometry of the surface is not devolving; i.e., the evolving metrics are uniformly bounded. We thus have the desired contradiction to the finite maximality of T , establishing (71) and proving the theorem.
The modifications required to prove Theorem 3 are the following. As the ambient space is no longer simply-connected and non-positively curved, the assumptions (31), (32) are no longer automatically satisfied along the flow. We have to use Proposition 17 to first control the concentration of the area, and then use this to allow us to apply Theorem 11 and conclude our estimates.
Observe that the constant λ = 1 3c1cη is universal, depending only on the metric of N and K i for i = 0, 1, 2. Additionally observe that the argument for step 3 above need only be performed in the case where T < λ. We may without loss of generality assume that ρ < ρ 0 (an allowable choice for ρ 0 would be the extrinsic diameter of Σ) and then forceσ 0 andε 0 small enough (by taking ρ smaller in the hypotheses of Theorem 3) so that Proposition 17 implies the conditions (31), (32) are satisfied on [0, λρ 4 0 ), which is a big enough interval to perform the entire argument above. In place of Proposition 13 we use Corollary 18, and Corollary 19 holds on the interval [0, λρ 4 0 ) ⊃ [0, λρ), which is all that is needed. The argument goes through exactly as above.