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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

Let

Recently, Guan and Li [

Inspired by this result, we focus on the corresponding problem with perpendicular Neumann boundary condition inside a convex cone in

We take the origin on

Suppose

The MCF with boundary conditions has been extensively studied by many mathematicians. Huisken in [

However, little is known about the modified MCF, such as volume or area preserving MCF, with perpendicular Neumann boundary condition. Let

Generally, for the prescribed contact angle which is not necessarily a right angle, this problem seems more hard. Altschuler and Wu in [

The rest of this paper is organized as follows. In Section

Let

Let

Furthermore, the system (

The system (

For simplicity, let

System (

For convenient calculation, we also parametrize the boundary cone as Lambert [

For the second fundamental form of the boundary cone, one has

From the parametrization for the cone, it is easy to check that

Since

Similarly, using (

Observing that

In this section, we will derive evolution equations for some useful geometric quantities by straightforward calculation. Let

Under the mean curvature type flow (

(i) We can calculate directly as in [

On the other hand,

(ii) As

(iii) Similarly, as in (i),

The following relationship between

For

In order to apply the Hopf maximal principle to obtain the basic estimates, we also need the following boundary derivatives.

For

(i) Denote by

(ii) Calculating directly, we obtain

Let

Suppose

Now we apply the above theorem to

Let

This result means that the evolving hypersurface always stays between two spheres with radii

In order to obtain the gradient estimate for system (

On the other hand, noticing

By the maximal principle we have the following.

For all

Combining Corollaries

Let

Along flow (

For the function

Denote by

Taking derivative with respect to time

In this section, we use an idea of Guan and Li in [

Using Cauchy-Schwartz inequality and similar rearrangement as in [

Therefore,

Recall that

With the assumption of convexity on the boundary cone, we obtain, by Theorem

By the equivalence of (

From the above estimate, there exists a uniform positive constant

Denote by

Then, by Theorem

Assume the radial function

So we have

On the other hand,

Because

Denote by

Let

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).