Mean Curvature Type Flow with Perpendicular Neumann Boundary Condition inside a Convex Cone
We investigate the evolution of hypersurfaces with perpendicular Neumann boundary condition under mean curvature type flow, where the boundary manifold is a convex cone. We find that the volume enclosed by the cone and the evolving hypersurface is invariant. By maximal principle, we prove that the solutions of this flow exist for all time and converge to some part of a sphere exponentially as tends to infinity.
Let be a space form of sectional curvature , or . It is well known that the Riemannian metric of can be defined as where is the standard induced metric of unit sphere in Euclidean space , and
Recently, Guan and Li  introduced a new type flow in the above space form which was called mean curvature type flow, as follows: where and are the mean curvature and the outward unit normal vector of the evolving hypersurfaces, respectively, and is the support function of the evolving hypersurfaces defined by . They proved that this flow evolves closed star-shaped hypersurfaces in space form into some sphere. A natural feature of this flow is that, along the mean curvature type flow, the volume enclosed by the evolving hypersurface is a constant and its area is always decreasing as long as the solution exists.
Inspired by this result, we focus on the corresponding problem with perpendicular Neumann boundary condition inside a convex cone in . Precisely, we suppose that is a convex cone with outward unit normal vector and is a smooth -dimensional hypersurface with boundary defined by an initial embedding . We will study how evolves under the flow (3) with boundary conditions: , and , where is the outward unit normal vector to . Namely, we will consider the following mean curvature type flow with perpendicular Neumann boundary condition: where is the Euclidean support function to the hypersurface in defined by .
We take the origin on at the vertex of the cone, and denotes the norm of the position vector at some point . In this paper, we obtain the following main result.
Theorem 1. Suppose is a convex cone and is the initial hypersurface which can be represented as a graph over the intersection of the interior of the cone and a unit sphere centered at the vertex of the cone; then, a solution to (4) exists for all time and stays always between two spheres with radii and . Furthermore, the solution converges exponentially fast to the intersection of the interior of the cone and the sphere with radius where denotes the area of the intersection of the limit sphere and the interior of the cone and denotes the volume of the domain enclosed by the cone and the evolving hypersurface which is a constant under this flow.
The MCF with boundary conditions has been extensively studied by many mathematicians. Huisken in  considered the evolution of a graph over a bounded domain with perpendicular Neumann boundary condition and proved that the solution exists for all time and converges to a plane domain at last. More generally, for the hypersurfaces not necessarily represented as graphs, Stahl [3, 4] studied this problem and proved that the flow converges to a round point on the condition that the boundary manifold was umbilic and the initial surface was convex. Buckland in  founded boundary monotonicity formulae and classified Type I boundary singularities for with a perpendicular Neumann boundary condition. Recently, Lambert in  considered this problem in a Minkowski space with a timelike cone boundary condition and proved that this flow converges to a homothetically expanding hyperbolic solution. Subsequently, he  also considered this problem inside a rotational tori.
However, little is known about the modified MCF, such as volume or area preserving MCF, with perpendicular Neumann boundary condition. Let be a tubular hypersurface between two parallel planes in , which can be represented as a graph over some cylinder inside it and meets the parallel planes perpendicularly. Recently, Hartley  studied the motion of under the volume preserving mean curvature flow by the center manifold analysis and proved that the solution exists for all time and converges exponentially fast to a cylinder in the topology for any as time tends to infinity.
Generally, for the prescribed contact angle which is not necessarily a right angle, this problem seems more hard. Altschuler and Wu in  considered the evolution of 2-dimensional graph over compact convex domain in by mean curvature flow under this boundary condition and proved this flow exists for all time and converges to translating solutions at last. Guan  extended Altschuler-Wu’s result to graphs of high dimensions.
The rest of this paper is organized as follows. In Section 2, we reparametrize the system (4) as a graph and give some primary facts. In Section 3, the evolution equations and boundary derivatives for some useful geometric quantities will be derived. In Section 4, a maximal principle will be introduced and some basic estimates will be given. In the last section, we prove the convergence and complete the proof of the main theorem.
2. Reparametrization and Notations
Let be a compact hypersurface inside an -dimensional convex cone with boundary condition , given by the embedding , where is the intersection of the interior of the cone and unit sphere centered at the vertex of the cone; that is, can be expressed as a graph over . Precisely, for any point , there is only one ray from the vertex through intersecting the hypersurface at some point ; the position vector to can be expressed as
Let be the local normal coordinates on ; denotes the standard spherical metric under the coordinates; the covariant derivative and divergent operator on with respect to the metric are denoted by and div, respectively. Then, tangent vectors and the outward unit normal vector on can be expressed as in  (see also P28 in ): where is the inverse of . Thus, the support function, induced metric, and second fundamental form can be given by straightforward calculation as following:
Furthermore, the system (4) is equivalent to the following parabolic PDE defined on : where .
The system (9) is a quasi-linear parabolic equation in divergence form, whose long-time existence is equivalent to the uniform parabolicity, and bound on by the classical theory of nonlinear parabolic equations (see, e.g., Chapter 12 in ).
For convenient calculation, we also parametrize the boundary cone as Lambert . Let be a smooth embedding of a sphere into a topological ball centered at the origin with outward unit normal vector . Then, we can define the boundary cone by embedding into at height 1, defining to be the set of all rays going through the origin and some point , where denotes the -dimensional coordinate for . So we can parametrize the cone by where is the th standard coordinate vector in . The second fundamental form of the boundary cone has the following characterization.
Proposition 2. For the second fundamental form of the boundary cone, one has(i),(ii), ,where and denote the second fundamental forms, respectively, of in and in .
Proof. From the parametrization for the cone, it is easy to check that
Since is the outward unit normal vector to the boundary cone, calculating by Gauss equation directly we have This proves the first identity.
Similarly, using (14) again we have
Observing that can be decomposed as and , we, then, have where we use the fact that .
3. Evolution Equations and Boundary Derivatives
In this section, we will derive evolution equations for some useful geometric quantities by straightforward calculation. Let be the local normal coordinates of the evolving hyperserface and let be the corresponding induced metric; let and be, respectively, the covariant derivative and Laplace operator with respect to the induced metric ; and and denote the norm of the position vector and the second fundamental form to evolving hypersurfaces in , respectively.
Lemma 3. Under the mean curvature type flow (4), we have (i),(ii),(iii).
Proof. (i) We can calculate directly as in 
On the other hand, This proves (i).
(ii) As , we have , and then,
(iii) Similarly, as in (i), Then, (iii) follows by combining the above two formulae.
The following relationship between and was proved by Stahl in .
Lemma 4 (see ). For , one has where .
In order to apply the Hopf maximal principle to obtain the basic estimates, we also need the following boundary derivatives.
Lemma 5. For , one has(i),(ii).
Proof. (i) Denote by the Euclidean covariant derivative in . Obviously, ; combining the boundary condition in (4), we have
(ii) Calculating directly, we obtain Observing , combination of Lemma 4 and (i) in Proposition 2 yields
4. Gradient Estimate
Theorem 6. Suppose satisfies and then for all .
Corollary 7. Let be a solution to system (4); then, one has for any .
This result means that the evolving hypersurface always stays between two spheres with radii and .
On the other hand, noticing by assumption, must be in the tangent space of the boundary cone; that is, . Combining (ii) in Lemma 5, Proposition 2, and the convexity of the boundary cone, we have Equivalently,
By the maximal principle we have the following.
Corollary 8. For all , the support function satisfies
Combining Corollaries 7 and 8, we obtain long time existence for the system (12) by the standard argument for divergence PDE (cf. ), and then, the first part of Theorem 1 follows by the equivalence of (4) and (12).
Let be the domain enclosed by the interior of the cone and the evolving hypersurface. The volume element for is denoted by and for its boundary. We will also denote the area elements for the evolving hypersurface and its boundary by and , respectively. Flow (4) has the following interesting property.
Proposition 9. Along flow (4), the volume of , denoted by , is a constant.
Proof. For the function defined on the evolving hypersurface , we have
from the proof of Lemma 3(i). Integrating the above equation on and taking into consideration Lemma 5(i) yield
Denote by the divergence of the position vector for in ; we have
Taking derivative with respect to time and combining divergence theorem yield where is the outward unit normal vector to the -dimensional region , and is the area element on boundary cone . Hence, is a constant and the result follows.
In this section, we use an idea of Guan and Li in  to obtain our estimate and prove the exponential convergence. For that purpose, the system (9) or (12) is convenient for us. Now, we choose a local coordinate on . and are again the covariant derivative and Laplace operator with respect to the standard metric on . Let be a parabolic operator defined by on for a smooth function . From , we have the following evolution equation for at the critical point
Using Cauchy-Schwartz inequality and similar rearrangement as in , we have
Recall that and ; we have, by Lemma 5 and the Neumann boundary assumption, for any points ,
With the assumption of convexity on the boundary cone, we obtain, by Theorem 6,
From the above estimate, there exists a uniform positive constant depending only on the upper bound of and such that (41) reads as
Then, by Theorem 6 again, has uniform upper bound ; that is Or, equivalently, for a different constant , This means that converges exponentially fast to some part of a sphere.
Assume the radial function attains its minimum at a point for some ; that is, . Let be a geodesic on the unit sphere starting from to any point with . Integrating both sides of the last inequality on and taking into account the boundedness of , we have, for some constant and , and then,
So we have
On the other hand,
Because the support function satisfies
Denote by the area of ; combining (52), we have or, equivalently,
Let , and from the above inequality, we obtain , a constant, and therefore, That is to say, converges exponentially to the intersection of the interior of the cone and a sphere centered at the vertex of the cone with radius where denotes the area of the limit sphere . This finishes the proof of Theorem 1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research is partially supported by NSFC (no. 11171096), RFDP (no. 20104208110002), and Funds for Disciplines Leaders of Wuhan (no. Z201051730002).
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