UR Scholarship Repository UR Scholarship Repository The Surface Diffusion and the Willmore Flow for Uniformly The Surface Diffusion and the Willmore Flow for Uniformly Regular Hypersurfaces Regular Hypersurfaces

. We consider the surface diﬀusion and Willmore ﬂows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diﬀusion and Willmore ﬂows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a speciﬁc singular structure. We establish well–posedness of both ﬂows for initial surfaces that are C 1+ α –regular and parameterized over a uniformly regular hypersurface. For the Willmore ﬂow, we also show long–term existence for initial surfaces which are C 1+ α –close to a sphere, and we prove that these solutions become spherical as time goes to inﬁnity.


Introduction
The surface diffusion and Willmore flows are geometric evolution equations that describe the motion of hypersurfaces in Euclidean space (or, more generally, in an ambient Riemannian manifold). The normal velocity of evolving surfaces is determined by purely geometric quantities. For both flows, the mean curvature is involved in the evolution equations, while the Willmore flow additionally depends upon Gauss curvature.
In this paper, we consider uniformly regular hypersurfaces. It should be emphasized that these surfaces may be non-compact. The concept of uniformly regular Riemannian manifolds was introduced by Amann [3,4] and it contains the class of compact Riemannian manifolds as a special case. The study of geometric flows on non-compact manifolds is an active research topic, both from the point of view of PDE theory and in relation to its applications in geometry and topology. To the best of our knowledge, the current literature on the surface diffusion and Willmore flows for non-compact manifolds all concern surfaces defined over an infinite cylinder or entire graphs over R m , or the Willmore flow with small initial energy, cf. [8,16,17,21,22]. Our work generalizes the study of these two flows to a larger class of manifolds.
In our main result we establish well-posedness for initial surfaces that are C 1+αregular and parameterized over a uniformly regular hypersurface. Moreover, we show that solutions instantaneously regularize and become smooth, and even analytic in case Σ is analytic. In order to obtain our results, we show that the pertinent underlying evolution equations can be formulated as parabolic quasilinear equations of fourth order over the reference surface Σ. Our analysis relies on the theory of continuous maximal regularity and the results and techniques developed in [22,33,34].
The results in Theorem 4.3 and Theorem 5.1 are new. However, we note that in case Σ is an infinitely long cylinder embedded in R 3 , an analogous result to Theorem 4.3 was obtained in [22] for the surface diffusion flow.
For the Willmore flow, Theorem 5.1 is also new even if Σ is a compact (smooth, closed) surface. Previous results impose more regularity on the initial surface, for instance C 2+α in [35].
Theorem 5.2, where global existence and convergence to a sphere is shown for surfaces that are C 1+α -close to a sphere, also seems to be new. A corresponding result was obtained in [35] for surfaces close to a sphere in the C 2+α -topology. The authors in [18] showed the existence of a lower bound on the lifespan of a smooth solution, which depends only on how much the curvature of the initial surface is concentrated in space. In [17,19], the authors proved convergence to round spheres under suitable smallness assumptions on the total energy of the surface. Here we note that the energy used in [17,19] involves second-order derivatives, whereas we only need smallness in the C 1+α -topology. In particular, we obtain global existence and convergence for non-convex initial surfaces. The organization of the paper is as follows: In Sections 2.1 and 2.2, we introduce the concept of uniformly regular manifolds and define the function spaces used in this paper. In Sections 2.3 and 2.4, we review continuous maximal regularity theory and its applications to quasilinear parabolic equations with singular nonlinearity. These results form the theoretic basis for the study of the surface diffusion and Willmore flows.
In Section 3, we introduce the concept of uniformly regular hypersurfaces with a uniform tubular neighborhood (called (URT)-hypersurfaces) and work out several examples. We utilize these concepts to parameterize the evolving hypersurfaces driven by surface diffusion and Willmore flows as normal graphs over a (URT)reference hypersurface.
In Section 4, we establish our main results regarding existence, uniqueness, regularity, and semiflow properties for solutions to the surface diffusion flow over (URT)hypersurfaces in R m+1 . In Section 5, we likewise establish well-posedness properties for solutions to the Willmore flow over (URT)-hypersurfaces in R 3 . Additionally, we show stability of Euclidean spheres under perturbations in the C 1+α -topology.
We conclude the paper with an appendix where we state and prove some additional properties of normal graphs over (URT)-hypersurfaces.
Notation: For two Banach spaces X and Y , X . = Y means that they are equal in the sense of equivalent norms. L(X, Y ) denotes the set of all bounded linear maps from X to Y and Lis(X, Y ) is the subset of L(X, Y ) consisting of all bounded linear isomorphisms from X to Y . For x ∈ X, B X (x, r) denotes the (open) ball in X with radius r and center x. We sometimes write B(x, r), in lieu of B X (x, r), in case the setting is clear, and we write B m (x, r) when X = R m . We denote by g m the Euclidean metric in R m . Given an embedded hypersurface M in R m , g m | M means the metric on M induced by g m . Finally, we set N 0 = N ∪ {0}.

Preliminaries
2.1. Uniformly regular manifolds. The concept of uniformly regular (Riemannian) manifolds was introduced by H. Amann in [3] and [4]. Loosely speaking, an m-dimensional Riemannian manifold (M, g) is uniformly regular if its differentiable structure is induced by an atlas such that all its local patches are of approximately the same size, all derivatives of the transition maps are bounded, and the pull-back metric of g in every local coordinate is comparable to the Euclidean metric g m .
We will now state some structural properties of uniformly regular manifolds which will be used in the analysis of the the surface diffusion flow and and the Willmore flow in subsequent sections.
An oriented C ∞ -manifold (M, g) of dimension m and without boundary is uniformly regular if it admits an orientation-preserving atlas A := {(O κ , ϕ κ ) : κ ∈ K}, with a countable index set K, satisfying the following conditions.
(R1) There exists K ∈ N such that any intersection of more than K coordinate patches is empty.
where B m is the unit Euclidean ball centered at the origin in R m . Moreover, A is uniformly shrinkable; by which we mean that there exists some r ∈ (0, 1) such that {ψ κ (rB m ) : κ ∈ K} forms a cover for M, Here g m is the Euclidean metric in R m and ψ * κ g denotes the pull-back metric of g by ψ κ . Here (R5) means that there exists some number c ≥ 1 such that Given an open subset U ⊂ R m , a Banach space X, and a mapping u : U → X, u k,∞ := max |α|≤k ∂ α u ∞ is the norm of the space BC k (U, X), which consists of all functions u ∈ C k (U, X) such that u k,∞ < ∞.
Remark 2.1. In [12], the authors showed that a C ∞ -manifold without boundary is uniformly regular iff it is of bounded geometry, i.e. it is geodesically complete, of positive injectivity radius and all covariant derivatives of the curvature tensor are bounded. In particular, every compact manifold without boundary is uniformly regular and the manifolds considered in [20,21] are all uniformly regular.
Given σ, τ ∈ N 0 , we define the (σ, τ )-tensor bundle of M as Throughout the rest of this paper, we will adopt the following convention.
• p always denotes a point on a uniformly regular manifold.
Here, and in the following, it is understood that a partially defined and compactly supported tensor field is automatically extended over the whole base manifold by identifying it to be zero outside its original domain. We further introduce two maps:

2.2.
Hölder and little Hölder spaces on uniformly regular manifolds. In this subsection we follow Amann [4,3], see also [34]. We define ∈ N 0 , we can give an alternative characterization of these spaces on R m . For 0 < s < 1 and 0 < δ ≤ ∞, we define a seminorm by For k < s < k + 1, the space BC s (R m , E) can be equivalently defined as where u s,∞ := u k,∞ + max |α|=k [∂ α u] s−k,∞ ; and . We denote by l ∞ (F s ) the linear subspace of F s consisting of all x = (x κ ) κ∈K such that We define l ∞,unif (bc k ) as the linear subspace of l ∞ (bc k ) consisting of all u = (u κ ) κ∈K such that (∂ α u κ ) κ∈K is uniformly continuous on R m κ for |α| ≤ k, uniformly with respect to κ ∈ K. For k < s < k + 1, we define l ∞,unif (bc s ) as the linear subspace of l ∞,unif (bc k ) of all u = (u κ ) κ∈K such that The following properties of little Hölder spaces were first established in [3,4]. We also refer to [ 2.3. Continuous maximal regularity. For a fixed interval I = [0, T ], µ ∈ (0, 1), and a given Banach space X, we define We equip these two spaces with the natural Fréchet topology induced by the topol- which are themselves Banach spaces when equipped with the norms v E0,µ(I) := sup where γ 0 is the evaluation map at 0, i.e., γ 0 (u) = u(0), and E µ := (E 0 , E 1 ) 0 µ,∞ . In this case, we use the notation

2.4.
Quasilinear equations with singular nonlinearity. Consider the following abstract quasilinear parabolic evolution equation We assume that V µ ⊂ E µ is an open subset of the continuous interpolation space E µ := (E 0 , E 1 ) 0 µ,∞ and the operators (A, F 1 , F 2 ) satisfy the following conditions. (H1) Local Lipschitz continuity of (A, F 1 ): There exists a number γ ∈ (µ, 1) such that . Following the convention in Prüss, Wilke [30] and [22], we call the index j subcritical if (2.5) is a strict inequality and critical in case equality holds in (2.5).
) and enjoys the regularity (c) If the solution u(·, x 0 ) satisfies the conditions: then it holds that t + (x 0 ) = ∞ and so u(·, x 0 ) is a global solution of (2.3) Moreover, if the embedding E 1 → E 0 is compact, then condition (i) may be replaced by the assumption: and some τ ∈ (0, t + (x 0 )).

URT-hypersurfaces
Suppose Σ is an oriented smooth hypersurface without boundary which is embedded in R m+1 . Let a > 0. Then Σ is said to have a tubular neighborhood of radius a if the map is a diffeomorphism onto its image U a := X((−a, a) × Σ). Here ν Σ is the normal unit vector field compatible with the orientation of Σ. We refer to U a as the tubular neighborhood of Σ of width 2a and note that U a = {x ∈ R m+1 : dist (x, Σ) < a}.
Finally, we say that Σ has a tubular neighborhood if there exists a number a > 0 such that the above property holds.
Remarks 3.1. (a) We lose no generality in assuming Σ is oriented, as any smooth embedded hypersurface without boundary is orientable, cf. [31]. (b) Any smooth (in fact, C 2 ) compact embedded hypersurface without boundary has a tubular neighborhood, see for instance [15,Exercise 2.11].
(c) Suppose Σ is a smooth (oriented) embedded hypersurface with unit normal field ν Σ . Then Σ is said to satisfy the uniform ball condition of radius a > 0 if at each point p ∈ Σ, the open balls B(p ± aν Σ (p), a) do not intersect Σ.
The following assertions are equivalent: (i) Σ has a tubular neighborhood of radius a.
(ii) Σ satisfies the uniform ball condition of radius a. For the reader's convenience, we include a proof of this equivalence.
In the following, we say that Σ is a (URT)-hypersurface in R m+1 if (T1) Σ is a smooth oriented hypersurface without boundary embedded in R m+1 . (T2) (Σ, g) is uniformly regular, where g = g m+1 | Σ denotes the metric induced by the Euclidean metric g m+1 . (T3) Σ has a tubular neighborhood.
Examples 3.2. (a) Every smooth compact hypersurface without boundary embedded in R m+1 is a (URT)-hypersurface. (b) All of the manifolds considered in [20,21] are (URT)-hypersurfaces. In particular, the infinite cylinder with radius r > 0, is a (URT)-hypersurface with tubular neighborhood of radius a = r.
Then the graph of f has a tubular neighborhood of radius a for some a > 0.
Proof. By the inverse function theorem, there exist uniform constants η > 0 and ε > 0 such that, at every point where the supremum is taken over the ball B Txgr(f ) (0, ε). We refer to the proof of Claim 1 in Proposition A.1(b) in the Appendix for a more general situation. Further, we have h x (0) = 0 and ∇h x (0) = 0. Due to (3.2), after Taylor expansion of h x around 0 ∈ T x gr(f ), we have y h x ∞ such that 1/2C ≤ ε, we define a := 1/2C. It follows that the ball B m+1 (aν x , a) lies above the graph where ν x is the upwards pointing unit normal of gr(f ) at the point (x, f (x)). An analogous argument shows that the ball B m+1 (−aν x , a) lies below the graph.
Since the constants ε and a are independent of x, combining with Remark 3.1(c), this proves that gr(f ) has a tubular neighborhood of radius a.
Based on part (b), the manifold Σ = k C k , endowed with the metric induced by g 3 , is uniformly regular. But it is obvious that (Σ, g) does not have a tubular neighborhood.
(f ) There also exist connected uniformly regular hypersurfaces that are not (URT). For instance, we can construct a smooth connected curve C in {(x, y) : y > 0} such that C ∩ {(x, y) : x ≥ 0} is compact and Then (C, g 2 | C ) is a uniformly regular hypersurface that is not (URT). One can take the product of C with R m to produce higher dimensional examples.
Additionally, one can rotate the curve C around the x-axis to obtain a connected rotationally symmetric uniformly regular hypersurface which is not (URT).

The surface diffusion flow
In solving the surface diffusion flow, one seeks to find a family of (oriented) closed hypersurfaces {Γ(t) : t ≥ 0} satisfying the evolution equation for an initial hypersurface Γ 0 .
Here, V (t) denotes the velocity in the normal direction of Γ at time t, H Γ(t) is the mean curvature of Γ(t) (i.e., the average of the principal curvatures), and ∆ Γ(t) is the Laplace-Beltrami operator on Γ(t). We use the convention that a sphere has negative mean curvature. We note that this convention is in agreement with [29,32,33], but differs from [13,20,22].
In the following, we assume that Σ is a (URT)-hypersurface in R m+1 with tubular neighborhood U a and with an orientation-preserving atlas A : In the following, we assume that Σ carries the metric induced by the Euclidean metric g m+1 . Finally, we assume that Γ 0 lies in U a .
Given ρ ∈ E µ with ρ ∞ < a, it follows, by assumption that Σ is (URT) with tubular neighborhood U a , that is a diffeomorphism from Σ onto the C 1 -manifold Γ ρ := im(Ψ ρ ); see also Proposition A.1 for additional properties of Γ ρ . When the temporal variable t is included in ρ, i.e.
we can also extend Ψ ρ to Ψ ρ : [0, T ) × Σ → R m+1 . In the sequel, we will omit the temporal variable t in ρ, Ψ ρ and Γ ρ when the dependence on t is clear from context. Let us fix some notation. We denote by g m+1 | Γρ the metric induced on Γ ρ by the Euclidean metric g m+1 of R m+1 . Let g(ρ) := Ψ * ρ (g m+1 | Γρ ) be the pull-back metric of g m+1 | Γρ on Σ.
In local coordinates with respect to the atlas A, ∆ ρ is given by where Γ k ij (ρ) are the Christoffel symbols of (Σ, g(ρ)). Here we note that the terms Γ k ij (ρ) depend on ρ and up to its second-order derivatives. More precisely, where p k ij is a polynomial of ρ and its derivatives up to second order and q k ij is a polynomial of ρ and its first-order derivatives (both polynomials having BC ∞coefficients).

By defining
we obtain an equivalent formulation of (4.1) as We note that for each ρ ∈ C 1 (Σ, R) with ρ ∞ < a, the mapping gives rise to a differential operator of order 4. A linear operator of order l, acting on scalar functions, is said to be uniformly strongly elliptic if there exist positive constants r, R > 0 such that the principal symbol of A, Remark 4.1. It is not difficult to see that in the scalar case, the notion of uniformly strongly elliptic is equivalent to the notion of uniformly normally elliptic introduced in [34, Section 3], see also [6].
In the following computations,C denotes a generic constant depending only on R and ρ 0 1+α,∞ . In every patch (O κ , ϕ κ ), by the discussion in [22,Section 4.1], we have the following estimate.
Proof. We have already proved part (a) above. Part (b) follows directly from the argument in [33,Sections 3 and 5]. For part (c), we first note that Lipschitz continuity of the semiflow follows from [22,Corollary 2.3]. Regarding additional regularity of the semiflow; for any τ > 0, we note that and so the result holds in V γ because of [10, Theorem 6.1] and the mapping properties of A(·) and F (·). Regularity of the semiflow in V µ then follows by embedding.
Treating (5.1) as a lower-order perturbation of (4.1), we again define for all ρ ∈ V µ , and we introduce the mapping Q : We thus arrive at the following expression for (5.1) in our current setting: By Proposition 4.2, we know that A ∈ C ω (V µ , M µ (E 1 , E 0 )) so we focus on showing regularity and structural properties for Q(ρ). By (4.11) and the definition of Q(ρ), we note that and it follows that the local expression for Q(ρ) is of the form To confirm this local expression for Q(ρ), we first note that all third-order derivatives of ρ appear in F (ρ), while the terms 2 β(ρ) H 3 ρ and 2 β(ρ) H ρ K ρ depend only on up to second-order derivatives. With the structure for F (ρ) already established in (4.15), it suffices to confirm that Q(ρ) only contributes additional terms of the form |η|,|σ|,|τ |≤2 d η,σ,τ (ρ, ∂ρ) ∂ η ρ ∂ σ ρ ∂ τ ρ.
Local expressions for β(ρ) and H ρ are given in (4.4) and (4.8), respectively. Since β(ρ) depends on at most first-order derivatives of ρ and H ρ depends linearly on second-order derivatives, we see that at most cubic powers of ∂ 2 ρ appear in 2 β(ρ) H 3 ρ . Regarding the term (2/β(ρ))H ρ K ρ , we first express Gaussian curvature as derived in [32,Section 2]. Here l ij (ρ) are the components of the pull-back of the second fundamental form of Γ ρ . It follows from (4.8) that Observing that each l ij (ρ) is linear with respect to ∂ 2 ρ, it follows that ∂ 2 ρ appears at most quadratically in det[l ij (ρ)], since it is a 2 × 2 matrix. Therefore, we conclude that K ρ contains at most quadratic factors of ∂ 2 ρ and thus, multiplying with the second-order quasilinear term H ρ , we conclude that the term (2/β(ρ))H ρ K ρ contains at most cubic powers of ∂ 2 ρ.
] defines a semiflow on V µ which is analytic for t > 0 and Lipschitz continuous for t ≥ 0.
Proof. Part (b) follows from [32] and [33,Section 3]. Part (c) follows exactly as in the proof of Theorem 4.3(c) above.

Stability of spheres.
In the case Σ is a Euclidean sphere in R 3 , we apply the generalized principle of linearized stability (c.f. [22,Section 3]) to prove the following result regarding stability of spheres under the Willmore flow, with control on only first-order derivatives of perturbations.
Proof. It is shown in the proof of [ Corollary 5.3. There exist non-convex hypersurfaces Γ 0 such that the solution ρ(·, ρ 0 ) to (5.2) with Γ(ρ 0 ) = Γ 0 , exists globally in time and converges exponentially fast to a sphere.
We note here that Theorem 5.2 also holds true for the surface diffusion flow, as was shown in [22,Section 4.5].
Appendix A.
Suppose Σ is a (URT)-hypersurface with tubular neighborhood of radius a. Given Then Γ ρ enjoys the following properties.
(b) Let r 0 ∈ (0, 1) be the constant related to the uniformly shrinkable property.
Proof of Claim 1. Let κ ∈ K and p ∈ ψ κ (r 0 B m ) be given. Then there exists x p iñ r 0 B m such that p = ψ k (x p ). Let be the orthogonal projection of R m+1 onto T p Σ. In the following we will identify T p Σ with R m . It follows from the boundedness of L Σ ∞ that there is a universal constant b 0 such that P p : ψ κ (B(x p , b 0 Then we obtain for the Fréchet derivative of F κ,p as Dψ κ (x p )ξ ∈ T p Σ for all ξ ∈ R m . We infer from (R5) that 3) for some uniform constant γ 1 ≥ 1. It follows from (A.2) and (A.3) that the spectrum of DF κ,p (x) lies outside the ball B C (0, 1/γ 1 ) for any x ∈ B m . Indeed, suppose µv = DF κ,p (x)v for some µ ∈ C and v = ξ + iη ∈ C m with |v| = 1. Then Lemma 4.1 in [7] implies that DF κ,p (x) is invertible with where the constant γ 2 is independent of x ∈ B m and κ, p. By the inverse function theorem, there exists a uniform constant r 1 which is independent of κ and p ∈ ψ κ (r 0 B m ) such that is a diffeomorphism. Next we note that In view of [3,Formula (3.19)], (A.3) and the boundedness of L Σ ∞ , we conclude that It follows from (A.4) and (A.5) that F −1 κ,p 2,∞ ≤ c for some c independent of κ, p. Define Φ κ,p : P p ψ κ (B m (x p , r 1 )) → Σ by Φ κ,p := ψ κ • F −1 κ,p . Note that Φ κ,p (y) = P p • Φ κ,p (y) + (I − P p ) • Φ κ,p (y) = y + (ν Σ (p)|Φ κ,p (y))ν Σ (p)) =: y + f κ,p (y)ν Σ (p).
We can now conclude that (A.1) holds.
In the following, we assume that By Claim 1, we can find L ∈ N such that, in every O κ , there exist x κ,i ∈ r 0B m with i = 1, · · · , L such that L i=1 ψ κ (B m (x κ,i , r 1 /4)) covers ψ κ (r 0B m ).
Proof of Claim 2. We set We now show that dist(p, ∂D κ ) is uniformly positive.
Because of (A.7), we can realize p as a point on the graph of f κ,q , cf. the following figure. q q p T q Σ ν Σ (q) Σ By (A.8), we observe that |p − q| ≥ η 0 . Let d = |p − q|. Using (A.1), we have for some uniform constant η 1 > 0. Thus we can take c 1 = min{a/4, η 1 } independent of p and κ.
Proof of Claim 3. The proof is basically the same as that of Claim 1, as Γ ρ is a C 2 -hypersurface and C 1 -uniformly regular by part (a) of the proposition; and all we need for the proof of Claim 1 is this property.