Mean curvature flow with free boundary on smooth hypersurfaces

Abstract The classical mean curvature flow of hypersurfaces with boundary satisfying a Neumann condition on an arbitrary, fixed, smooth hypersurface in Euclidean space is examined. In particular, the problem of singularity formation on the free-boundary and the classification of the limiting behaviour thereof is focused on. A monotonicity formula is developed and used to show that any smooth blow up centred about a boundary point is self-similar, with smoothness of the blow up being shown to necessarily follow in the case of Type I singularities. This leads to a classification of boundary singularities for mean convex evolving hypersurfaces.


Introduction
Throughout this work, we let G denote an ðn þ 1Þ-dimensional subset of R nþ1 whose boundary, qG ¼: S, represents a smooth, embedded hypersurface satisfying a uniform interior/exterior sphere condition with ball of maximal radius 1 k S . It is, moreover, assumed throughout that the curvature of S satisfies where A S ¼ fh S ij g denotes the second fundamental form of S and 'A S ¼ f' k h S ij g denotes the 3-tensor of covariant derivatives of the second fundamental form. Here jA S j 2 ¼ h ij S h S ij and, for any integer p f 1, j' p A S j 2 denotes the squared norm of the ð p þ 2Þ-tensor of p-th covariant derivatives of A S . Geometric quantities pertaining to S are indicated with a subor superscript, as convenient.
We let M n denote a smooth, orientable n-dimensional manifold with smooth boundary qM n and set M 0 :¼ F 0 ðM n Þ, where F 0 : M n ! R nþ1 is a smooth embedding satisfying qM 0 :¼ F 0 ðqM n Þ ¼ M 0 X S; hn 0 ; n S F 0 ið pÞ ¼ 0; p A qM n ; ð2Þ for unit normal fields n 0 and n S to M 0 and S, respectively. hn; n S F ið p; tÞ ¼ 0; ð p; tÞ A qM n Â I : HereH Hð p; tÞ ¼ ÀHð p; tÞnð p; tÞ denotes the mean curvature vector of the embeddings M t at F ð p; tÞ, for a choice of unit normal n for M t .
It will be assumed throughout that, for x 0 A R nþ1 , the family of embeddings ðM t Þ have polynomial area-growth towards infinity, for some p and all R f R 0 , and that M t also has locally finite area, H n ðM t X KÞ < y for all KHH R nþ1 and t A I : ð5Þ In particular, this allows compactly supported test functions to be integrated over M t and further implies Ð where C is any function of rapid-decay (that is, C decays faster than any polynomial as jxj ! yÞ.
Wherever possible, explicit indication of the embedding map will be suppressed and the point F ð p; tÞ will be identified simply with its position vector x A R nþ1 . Thus, the above definition of mean curvature flow with Neumann free-boundary on S may be interpreted as saying that, for all t A I , we have Moreover, we henceforth assign the convention that the unit normal n S to S be chosen to coincide with the unit inner co-normal to qM t at all intersection points.
In the boundaryless setting, solutions of the mean curvature flow typically develop singularities in finite time, whereby their curvature becomes unbounded, or else exist for all time. This is true also for solutions of the Neumann boundary problem: [10]). For any smooth hypersurface S and any initial hypersurface M 0 satisfying (2) there exists a unique solution to (3) on a maximal time interval ½0; TÞ which is smooth for t > 0 and in the class C 2þa; 1þa=2 for t f 0, for any a A ð0; 1Þ. Moreover, if T < y then The question concerning the behaviour of solutions as this critical time T is approached, in this setting, has only been directly addressed for a very restrictive class of surfaces. In particular, Stahl [11] has shown that strictly convex, embedded initial data evolving in the interior of a ball or half-space develop a Type I singularity (see section 6.2) and, after appropriate rescaling, asymptotically becoming hemispherical as t ! T.
Away from the boundary one should naturally expect solutions of (3) to behave locally like their boundaryless counterparts. Indeed, the recently developed local monotonicity formula of Ecker, [1], yields that any smooth blow-up of an interior singularity must be self-similar.
In this paper we focus on the remaining task of understanding the behaviour of singularities which develop on the free-boundary of M t . To this end, a special monotonicity formula is derived, which is based largely on Huisken's result [7], Theorem 3.1, but incorporates an idea of Grü ter-Jost [5] to deal with the additional boundary term. To ensure smoothness of an appropriately rescaled limit flow, a Type I singularity assumption-which imposes a natural blow-up rate on the curvature of the evolving surface-is made. The monotonicity formula then allows a characterization of the singular behaviour at boundaries of this limit flow to be made, leading to a complete classification of possible limit surfaces, via a result of Huisken [8], Theorem 5.1, in the special case of weak meanconvexity.
This work is the result of the author's PhD thesis, the research upon which was completed under the supervision of Klaus Ecker. Accordingly, the author is indebted to Klaus Ecker for his insight and useful advice throughout these years, as well as to Maria Athanasenas for her general support and encouragement. The author is also very thankful to the referee for the many comments and suggestions made that have greatly improved the exposition of this paper.

Outline of main ideas
In view of the identity D M t F ð p; tÞ ¼H Hð p; tÞ; where D M t denotes the Laplace-Beltrami operator on M t , one may re-write the governing equation of (3) in the form qF qt ð p; tÞ ¼ D M t F ð p; tÞ; ð8Þ and thus it is not surprising that many results about mean curvature flow are motivated by the linear heat equation. One pertinent example of such a result is Huisken's monotonicity formula, which states how the area of M t evolves when weighted with the function which is the standard linear backward heat kernel on R nþ1 , centred at the point x 0 A R nþ1 , with time-scaling appropriate to an n-dimensional spatial domain: (Huisken,[7]). Let M ¼ fM t g t<T be a family of hypersurfaces evolving by mean curvature flow. Then for all t < T we have d dt This formula leads one to rescale the flow parabolically about the point ðx 0 ; TÞ by defining, for l > 0, the family of rescaled surfaces Under the Type I hypothesis, a natural curvature bound that implies a uniform, local C 2 bound on the rescaled surfaces (see section 6.2), Huisken has shown [7] that there exist subsequences fl i g & 0 such that fM l i t g t A ½ÀT=l 2 i ; 0Þ converges locally on R nþ1 Â ðÀy; 0Þ to a smooth solution of (8) (a so-called limit flow) fM 0 t g t<0 , which is self-similarly (or homothetically) shrinking; that is, which satisfies This is equivalent to the conditioñ which arises exactly when the rescaled weighted area Ð In particular, one obtains the second order elliptic equatioñ which, upon solving for the case of weakly mean-convex, embedded hypersurfaces, yields the following classification theorem.
Theorem 2.2 (Huisken, [8]). If M ¼ fM 0 t g t<0 is a smooth, embedded limit flow in R nþ1 satisfying (11) with nonnegative mean curvature, H f 0, then either M is a homothetically shrinking n-sphere, S n ffiffiffiffiffiffiffi ffi À2nt p , or cylinder, S nÀm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The crucial ingredient in showing the analogous behaviour is true also of singularities which develop on the free-boundary of solutions to (3), is a modified version of Huisken's monotonicity formula. The first step in deriving this formula is the following general expansion result, which applies to su‰ciently smooth functions u : R nþ1 Â R ! R restricted to M, for which the total time derivative is given by where D stands for ordinary di¤erentiation in R nþ1 .
where here and henceforth the operator Q is defined by Proof. The assumptions (4) and (5) combined with those imposed on the functions f and g ensure the interchange of the derivative with the integral is allowed.
Pointwise, one then computes From [6], Corollary 3.6, the surface measure on M t satisfies and therefore, by the Divergence Theorem, we have d dt from which the result follows. r Remark 2.4. 1. For the case of compact surfaces without boundary, we obtain (10) by taking g ¼ r and f ¼ 1, upon noting (by direct computation) that QðrÞ 1 0. (The same result is true for noncompact surfaces, though the argument is a little more subtle, requiring the use of a compactly supported test function and standard convergence theorems for integrals-see [9], Lemma 7.) 2. For the case where qM t 3 j, we obtain d dt which means that the quantity Ð M t r dm t is, in general, no longer monotonically nonincreasing in time.
Corollary 2.5. Let M ¼ fM t g t<T be a family of hypersurfaces satisfying (3) for some weakly convex (with respect to n S ) support surface S, and let x 0 A S. Then for all t < T we have d dt Proof. Recalling the convention for n S , convexity of S implies hx À x 0 ; n S i e 0 for all x A S, and so the boundary term of (17) has the right sign. r In general the boundary term above is di‰cult to deal with and so the approach undertaken for the general case, motivated by the treatment of the corresponding stationary problem for varifolds by Grü ter and Jost [5], is to nullify it using reflections. That is, for the function g above we seek a modified version of the backward heat kernel r, with spatial dependence on some function of x-depending on the distance to, and the curvature of, the support surface S-such that the boundary integrals of (14) are identically zero.
However, the identity QðrÞ 1 0 can no longer be expected to hold for a modified version of the backward heat kernel, and thus one challenge then becomes obtaining suf-ficient control of this term. Secondly, since the distance function will, in general, only be well-defined within a tubular neighbourhood of the support surface, it will be necessary to choose f to be an appropriate localization function; again, to ensure the corresponding boundary integrals vanish identically, this function should depend spatially upon reflections. Thus, a second issue is to find such a localization function for which the term d dt À D M t f be su‰ciently well-behaved.
The resolution of these two problems forms the basis of this work, the culmination of which appears in Proposition 5.1 and Lemma 5.4. A statement of the corresponding monotonicity formula then follows in Theorem 5.6.
The majority of the rest of this work is then concerned with a rescaling analysis, as outlined above, and the extraction and classification of the possible behaviour of limit surfaces.

Geometric properties of distance function
In this section we introduce the Euclidean distance function, which measures the distance to S of points in R nþ1 , and establish some links between its derivatives and the geometry of S. We then define the signed distance function by The following standard result concerning the regularity of the distance function can be found in [4].
Proposition 3.2. Let G be a subset of R nþ1 and let S ¼ qG be of class C k , for some k f 2, and satisfy a uniform interior/exterior sphere condition. Then there exists an e > 0 such that d A C k ðS e Þ, where S e is the e-tubular neighbourhood of S given by S e :¼ fx A R nþ1 : d S ðxÞ < eg: As we shall see in the following section, the reflection of points across the surface S depends on zero and first order derivatives of the distance function. Hence, to compute the heat-and Q-operator of functions depending spatially upon reflected points-as will be required in the expansion formula (14)-we will firstly need estimates on derivatives of the distance function up to order three. Proposition 3.3. Let S A C k for k f 3 and suppose x 0 A S e and y 0 A S are such that x 0 ¼ y 0 þ dn S ðy o Þ with d A ðÀe; eÞ. Then in terms of a principal coordinate system at y 0 , we have Here n S out is normal to S and points in the direction of increasing signed distance, k i ¼ k i ðy 0 Þ is the principal curvature of S in the direction of the i-th ( principal ) coordinate, and ' S denotes the covariant derivative on S.

Nullifying the boundary integrand
The following definition is inspired by the work of Grü ter and Jost [5] in their treatment of the corresponding free-boundary problem for stationary varifold solutions of (3). Furthermore, for any x 0 A S we define the translates r S; x 0 ðxÞ :¼ jx À x 0 j 2 þ jx À x 0 À 2dðxÞDdðxÞj 2 : Remark 4.2. 1. By construction, we note that r S; x 0 ðxÞ ¼ 0 i¤ x ¼ x 0 .
2. In the special case that S is a hyperplane and x 0 A S, we have hDdðxÞ; x À x 0 i ¼ dðxÞ and so We now establish estimates on the derivatives of r S that will be used in the following section in computing the Q-and heat-operator of functions depending on r S .
Additionally, for every x A S we have hDr S; x 0 ; n S iðxÞ ¼ 0: Proof. We may assume x 0 ¼ 0 A S.

The monotonicity formula
In this section we introduce and prove crucial estimates for the two functions that form the basis of our monotonicity formula. To nullify the boundary integrals in the expansion formula (14), both functions depend spatially on the function r of the previous sections. The first function is therefore required, essentially, to localize the result to a tubular neighbourhood of S within which all quantities involving the distance function are well-defined.
where r S; x 0 ðxÞ ¼ jx À x 0 j 2 þ jx À x 0 À 2dðxÞDdðxÞj 2 and t ¼ T À t, and set and t 0 :¼ maxðT À t 0 ; 0Þ: Then for each t A ½t 0 ; TÞ (or, equivalently, t A ð0; T À t 0 Þ we have h x 0 e 256; ð30Þ H fx A R nþ1 : jx À x 0 jk S e ðk 2 S tÞ d g; Proof. Since for all t e t 0 and d e 2=5 we have ðk 2 S tÞ 1À2d e 3=160n, we estimate h x 0 ðx; tÞ e À 1 þ 160nðk 2 S tÞ 1À2d Á 4 e ð1 þ 3Þ 4 ¼ 256: To estimate the support of h x 0 ðx; tÞ from below, we note that jx À x 0 À 2dDdj 2 e 9jx À x 0 j 2 ; and so for all t A ð0; To estimate the support of h x 0 ðx; tÞ from above it is instructive to firstly set þ , and then estimate, for each : As above, we note that ðk 2 S tÞ 1À2d e ð3=160nÞ for all d e 2=5 and t e t 0 , and thus spt h x 0 H fx A R nþ1 : jx À x 0 jk S e ðk 2 S tÞ d g: Since x 0 A S (which implies dðxÞ e jx À x 0 j), the above result implies that, for all t e t 0 , over the support of h x 0 we have from which (32) follows.
To show the intrinsic heat operator acting on h x 0 ðx; tÞ is non-positive, we proceed as follows: noting h 0 x 0 e 0, d e 2=5, (32) and Lemma 4.3.
Hence, since jx À x 0 jk S e ðk 2 S tÞ d e ð3=160nÞ 2 for all t e t 0 and x A spt h x 0 , we have e 0: r 5.2. The modified backward heat kernel. We now turn our attention to the task of determining an appropriate weighting function for our monotonicity formula. The basis for this is the standard backward heat kernel, r, used by Huisken in (10).
Furthermore, for any x 0 A S and t < T, we define also the translates j a; x 0 ; T ðx; tÞ : where r S; x 0 ðxÞ ¼ jx À x 0 j 2 þ jx À x 0 À 2dðxÞDdðxÞj 2 .

By part 3 of Lemma 4.3, we have
The following lemma paves the way for our monotonicity formula. Remark 5.5. We note here the importance of the factor d, which appears in the localization function, h, in such a way as to control the growth-rate of its support-see (31).
Namely that, for any d > 0, the factor ðT À tÞ dÀ1 on the right-hand side of (37) becomes integrable. With reference to the expansion formula (14), this means that the second term on the right may be incorporated into the left hand side via an integrating factor, with the third, forth and fifth terms either having the correct sign or else vanishing identically by previous results. See also proof of Theorem 5.6.
Proof. Without loss of generality, we take We show that the modified heat kernel r k S ; x 0 ; T arises naturally from the class of perturbed heat kernels (35) as follows: from the definition of j a; 0; T 1 j and the operator Q we compute, for any a ¼ aðtÞ, Working over the support of h x 0 and using results (1) and (2)   Here r x 0 ðxÞ ¼ jx À x 0 j 2 þ jx À x 0 À 2dðxÞDdðxÞj 2 , t ¼ T À t and d is a fixed but arbitrary constant, 0 < d e 2=5.
Theorem 5.6 (monotonicity formula). Let S be a hypersurface smoothly embedded in R nþ1 that satisfies an inner/outer sphere condition with sphere of radius 1=k S and also the curvature bound jA S j 2 þ j'A S j e k 2 S . Let M t be a family of hypersurfaces evolving by mean curvature flow with Neumann free-boundary on the hypersurface S for all t A ½0; TÞ, as in (3), Then for all t A ½t 0 ; TÞ and any x 0 A S we have where r k S 1 r k S ; x 0 ; T , h 1 h x 0 and C ¼ CðnÞ > 0.
Proof. Taking for each t A ½t 0 ; TÞ, from which the result follows. r Remark 5.7. 1. By Remarks 4.2 and 5.3 and the definition of h, in the case that k S ¼ 0 bounds the curvature of S-that is, S is a hyperplane-the above formula is valid for all t A ½0; TÞ and reduces to Huisken's result (10).
2. Explicitly, the integrand on the right hand side of (38) is given bỹ

Classification of possible limit surfaces
In this section we carry out a rescaling analysis of our evolving surfaces and use the monotonicity formula of the previous section to classify the limiting behaviour of boundary singularities.
6.1. Parabolic rescaling. Let M ¼ ðM t Þ t A ½0; TÞ be a solution of (3) and define, for any x A M t W S and fixed point x 0 A S, the change of variables ðx; tÞ 7 ! ðy; sÞ by Definition 6.5 (limit point). For any point p A M n , we define the limit point function 1 : M n ! R nþ1 by 1ð pÞ ¼ lim t!T F ð p; tÞ: The existence of this limit exists follows directly from (47) and (3). Theorem 6.6 (existence of smooth limiting surface). Let M ¼ ðM t Þ t A ½0; TÞ be a smooth, embedded solution of (3) satisfying the Type I curvature assumption (47), and let x 0 ¼ 1ð pÞ, for some p A M n . Then for every sequence l j & 0 there is a subsequence fl j k g such that the rescaled surfaces M ðx 0 ; TÞ; l j k s converge smoothly on compact subsets of R nþ1 Â ðÀy; 0Þ to a non-empty, embedded limit-surface, M 0 ¼ ðM 0 s Þ s<0 such that: (i) ðM 0 s Þ evolves by mean curvature flow for s < 0.
(ii) If p B qM n then M 0 s has no boundary.
(iii) If p A qM n then M 0 s has boundary qM 0 s , an ðn À 1Þ-plane through the origin (corresponding to T x 0 S), and hn n;n n S 0 i ¼ 0 on qM 0 s .
Proof. The smooth convergence on compact subsets of R nþ1 Â ðÀy; 0Þ of a subsequence of the rescaled surfaces follows precisely as in [2], Remark 4.16 (3), upon using the interior estimates of Stahl [10] in place of those of Ecker and Huisken from [3].
The subsequent claims regarding boundaries are then a direct consequence of the rescaling procedure. r The monotonicity formula then allows the behaviour of this limit flow to be characterized.
Theorem 6.7 (characterization of limit surface). The limiting hypersurfaces M 0 s as obtained in Theorem 6.6 satisfy the equatioñ for all y A M 0 s and s < 0.
Proof. If the limit surface has been obtained by rescaling about a boundary limit point, then we firstly note from the monotonicity formula, Theorem 5.6, that since the weighted area functional