The surface diffusion and the Willmore flow for uniformly regular hypersurfaces
Abstract.
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 –regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are –close to a sphere, and we prove that these solutions become spherical as time goes to infinity.
Key words and phrases:
Surface diffusion flow, Willmore flow, uniformly regular manifolds, geometric evolution equations, continuous maximal regularity, critical spaces, stability of spheres2010 Mathematics Subject Classification:
35K55, 53C44, 54C35, 35B65, 35B351. 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.
These flows have been studied by several authors for compact (closed) hypersurfaces. In this setting, existence, regularity, and qualitative behavior of solutions have been analyzed in [13, 14, 20, 27, 33, 36, 37] for the surface diffusion flow, and in [9, 17, 18, 19, 24, 25, 26, 32, 35] for the Willmore flow, to mention just a few publications.
In this paper, we consider uniformly regular hypersurfaces. It should be emphasized that these surfaces may be noncompact. 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 , 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 –regular and parameterized over a uniformly regular hypersurface. Moreover, we show that solutions instantaneously regularize and become smooth, and even analytic in case 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 . 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 is an infinitely long cylinder embedded in , 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 is a compact (smooth, closed) surface. Previous results impose more regularity on the initial surface, for instance in [35].
Theorem 5.2, where global existence and convergence to a sphere is shown for surfaces that are –close to a sphere, also seems to be new. A corresponding result was obtained in [35] for surfaces close to a sphere in the –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 –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 . In Section 5, we likewise establish well–posedness properties for solutions to the Willmore flow over (URT)–hypersurfaces in . Additionally, we show stability of Euclidean spheres under perturbations in the –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 and , means that they are equal in the sense of equivalent norms. denotes the set of all bounded linear maps from to and is the subset of consisting of all bounded linear isomorphisms from to . For , denotes the (open) ball in with radius and center . We sometimes write , in lieu of , in case the setting is clear, and we write when . We denote by the Euclidean metric in . Given an embedded hypersurface in , means the metric on induced by . Finally, we set .
2. 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 –dimensional Riemannian manifold 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 pullback metric of in every local coordinate is comparable to the Euclidean metric .
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 the Willmore flow in subsequent sections.
An oriented –manifold of dimension and without boundary is uniformly regular if it admits an orientationpreserving atlas , with a countable index set , satisfying the following conditions.

There exists such that any intersection of more than coordinate patches is empty.

where is the unit Euclidean ball centered at the origin in . Moreover, is uniformly shrinkable; by which we mean that there exists some such that forms a cover for , where .

for all , and such that .

for all and .

for all . Here is the Euclidean metric in and denotes the pullback metric of by .
Here (R5) means that there exists some number such that
Given an open subset , a Banach space , and a mapping ,
is the norm of the space , which consists of all functions such that .
Any uniformly regular manifold possesses a localization system subordinate to , by which we mean a family satisfying:

and is a partition of unity subordinate to the cover .

with satisfying , .

, for , .
Given , the concept of –uniformly regular manifold is defined by modifying (R3), (R4), (L1)(L3) in an obvious way, where is the symbol for real analyticity.
Remark 2.1.
In [12], the authors showed that a –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 , we define the –tensor bundle of as
where and are the tangent and the cotangent bundle of , respectively. Let denote the –module of all smooth sections of .
Throughout the rest of this paper, we will adopt the following convention. {mdframed}

always denotes a point on a uniformly regular manifold.

and .

, , .
Setting for , we define
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
where . Set
endowed with the conventional projective topology. Then
the closure of in .
Letting , the Hölder space is defined by
Here is the real interpolation method, see [1, Example I.2.4.1]. For , we define the little Hölder spaces by
the closure of in .
The spaces and are defined in a similar manner. When , we can give an alternative characterization of these spaces on . For and , we define a seminorm by
For , the space can be equivalently defined as
where ; and
belongs to iff .
For , we put with . We denote by the linear subspace of consisting of all such that
We define as the linear subspace of consisting of all such that is uniformly continuous on for , uniformly with respect to . For , we define as the linear subspace of of all such that
(2.1) 
The following properties of little Hölder spaces were first established in [3, 4]. We also refer to [34, Theorem 2.1 and Proposition 2.2].
Proposition 2.2.
Let . Then is a retraction from onto with as a coretraction. Similarly,
Let denote the continuous interpolation method, c.f. [1, Example I.2.4.4].
Proposition 2.3.
Suppose that , and with . Then
2.3. Continuous maximal regularity
For a fixed interval , , and a given Banach space , we define
where ; and
If is a half open interval, then
We equip these two spaces with the natural Fréchet topology induced by the topology of and , respectively.
Assume that is a pair of densely embedded Banach spaces.
An operator is said to belong to the class , if generates a strongly continuous analytic semigroup on with .
We define
(2.2) 
which are themselves Banach spaces when equipped with the norms
respectively. For , we say is a pair of maximal regularity of if
where is the evaluation map at , i.e., , and . In this case, we use the notation
2.4. Quasilinear equations with singular nonlinearity
Consider the following abstract quasilinear parabolic evolution equation
(2.3) 
We assume that is an open subset of the continuous interpolation space and the operators satisfy the following conditions.

Local Lipschitz continuity of :

Structural regularity of :
There exists a number such that . Moreover, there are numbers , , and with
(2.4) so that for each and there is a constant for which the estimate
(2.5) holds for all .
Following the convention in [30] and [22], we call the index subcritical if (2.5) is a strict inequality and critical in case equality holds in (2.5).
Theorem 2.4.
[22, Theorem 2.2] Suppose satisfies (H1)–(H2).

Given any , there exist positive constants and such that (2.3) has a unique solution
for all initial values . Moreover,

Each solution with initial value exists on a maximal interval and enjoys the regularity

If the solution satisfies the conditions:

and

there exists so that for all ,
then it holds that and so is a global solution of (2.3) Moreover, if the embedding is compact, then condition (i) may be replaced by the assumption:

the orbit is bounded in for some and some .

3. URT–hypersurfaces
Suppose is an oriented smooth hypersurface without boundary which is embedded in . Let . Then is said to have a tubular neighborhood of radius if the map
(3.1) 
is a diffeomorphism onto its image . Here is the normal unit vector field compatible with the orientation of . We refer to as the tubular neighborhood of of width and note that .
Finally, we say that has a tubular neighborhood if there exists a number such that the above property holds.
Remarks 3.1.
(a) We lose no generality in assuming is oriented, as any smooth embedded hypersurface without boundary is orientable, cf. [31].
(b) Any smooth (in fact, ) compact embedded hypersurface without boundary has a tubular neighborhood, see for instance [15, Exercise 2.11].
(c) Suppose is a smooth (oriented) embedded hypersurface with unit normal field . Then is said to satisfy the uniform ball condition of radius if at each point , the open balls do not intersect .
The following assertions are equivalent:

has a tubular neighborhood of radius .

satisfies the uniform ball condition of radius .
For the reader’s convenience, we include a proof of this equivalence.
Proof.
(i) (ii). Suppose there exists such that where . Then and there exists such that . Hence , with , contradicting the assumption that is bijective. The case is treated in the same way.
(ii) (i). We only need to prove the injectivity of . Suppose, by contradiction, that for . Without loss of generality we may assume that , as we can otherwise replace by . Moreover, we may assume that . Let and set
Then we have showing that
Therefore, contradicting the assumption in (ii). ∎
(d) Suppose has a tubular neighborhood of radius . Let be the principal curvatures of , and the Weingarten tensor.
Then it follows from part (c) that and
In the following, we say that is a (URT)–hypersurface in if

is a smooth oriented hypersurface without boundary embedded in .

is uniformly regular, where denotes the metric induced by the Euclidean metric .

has a tubular neighborhood.
Examples 3.2.
(a) Every smooth compact hypersurface without boundary embedded in is a (URT)–hypersurface.
(b) All of the manifolds considered in [20, 21] are (URT)–hypersurfaces. In particular, the infinite cylinder with radius ,
is a (URT)–hypersurface with tubular neighborhood of radius .
(c) Assume that belongs to . Then the graph of has a tubular neighborhood of radius for some .
Proof.
By the inverse function theorem, there exist uniform constants and such that, at every point , can be expressed as the graph of a –function over , the tangent space to the graph of at the point , such that the set is contained in . Moreover, there exists a uniform constant independent of , such that
(3.2) 
where the supremum is taken over the ball . We refer to the proof of Claim 1 in Proposition A.1(b) in the Appendix for a more general situation. Further, we have and . Due to (3.2), after Taylor expansion of around , we have
for sufficiently small . Choosing such that we define . It follows that the ball lies above the graph
where is the upwards pointing unit normal of at the point . An analogous argument shows that the ball lies below the graph.
Since the constants and are independent of , combining with Remark 3.1(c), this proves that has a tubular neighborhood of radius . ∎
(d) We refer to [3, 4, 5] for additional examples of uniformly regular manifolds. In particular, embedded hypersurfaces with tame ends, considered in [5, Theorem 1.2], are (URT)–hypersurfaces. More precisely, given a compact hypersurface without boundary , embedded in , and we define
which we endow with the metric induced by its embedding into . An embedded hypersurface is said to have tame ends if
where is compact and is isometric to . Then, is a (URT)–hypersurface.
In particular, when , has finitely many cylinder ends; when , has finitely many (blunt) cone ends.
(e) Let
Based on part (b), the manifold endowed with the metric induced by , is uniformly regular. But it is obvious that 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 in such that is compact and
Then is a uniformly regular hypersurface that is not (URT). One can take the product of with to produce higher dimensional examples.
Additionally, one can rotate the curve around the –axis to obtain a connected rotationally symmetric uniformly regular hypersurface which is not (URT).
4. The surface diffusion flow
In solving the surface diffusion flow, one seeks to find a family of (oriented) closed hypersurfaces satisfying the evolution equation
(4.1) 
for an initial hypersurface .
Here, denotes the velocity in the normal direction of at time , is the mean curvature of (i.e., the average of the principal curvatures), and is the LaplaceBeltrami operator on . 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 is a (URT)–hypersurface in with tubular neighborhood and with an orientationpreserving atlas with satisfying (R1)–(R5). In the following, we assume that carries the metric induced by the Euclidean metric . Finally, we assume that lies in .
Given with , it follows, by assumption that is (URT) with tubular neighborhood , that
(4.2) 
is a diffeomorphism from onto the –manifold ; see also Proposition A.1 for additional properties of .
When the temporal variable is included in , i.e.
we can also extend to . In the sequel, we will omit the temporal variable in , and when the dependence on is clear from context.
Let us fix some notation. We denote by the metric induced on by the Euclidean metric of . Let be the pullback metric of on .
The following expression for was derived in [28, Formula (23)]:
(4.3) 
where and are the components of the Weingarten tensor and the second fundamental form with respect to ; i.e.,
where is a local basis of at and is the corresponding dual basis, characterized by .
We introduce an open subset of defined by
By Remark 3.1(d), the functions
(4.4) 
are well–defined for all , where is the gradient vector and .
It is easy to verify that , where
Hence, we obtain