Mean Curvature Type Flow with Perpendicular Neumann Boundary Condition inside a Convex Cone

Abstract

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 𝑡 tends to infinity.

1. Introduction

Let ℕ^𝑛+1(𝐾) be a space form of sectional curvature 𝐾=1,0, or −1. It is well known that the Riemannian metric of ℕ^𝑛+1(𝐾) can be defined as 𝑑𝑠²=𝑑𝜌²+𝜙²𝜌𝑑𝑧², (1) where 𝑑𝑧² is the standard induced metric of unit sphere 𝕊^𝑛 in Euclidean space ℝ^𝑛+1, and 𝜙𝜌=sin𝜌𝐾=1𝜌𝐾=0sinh𝜌𝐾=−1. (2)

Recently, Guan and Li [1] introduced a new type flow in the above space form which was called mean curvature type flow, as follows: 𝜕𝐹=𝜕𝑡𝑛𝜙^′𝜌−𝐻𝑢𝜈, (3) where 𝐻 and 𝜈 are the mean curvature and the outward unit normal vector of the evolving hypersurfaces, respectively, and 𝑢 is the support function of the evolving hypersurfaces defined by 𝑢=⟨𝜙(𝜌)(𝜕/𝜕𝜌),𝜈⟩. 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 ℝ^𝑛+1. Precisely, we suppose that Σ⊂ℝ^𝑛+1 is a convex cone with outward unit normal vector 𝜇 and 𝑀^𝑛 is a smooth 𝑛-dimensional hypersurface with boundary 𝜕𝑀 defined by an initial embedding 𝐹₀∶𝑀^𝑛→ℝ^𝑛+1. We will study how 𝑀^𝑛 evolves under the flow (3) with boundary conditions: 𝐹₀(𝜕𝑀)⊂Σ, and ⟨𝜇,𝜈⟩=0, where 𝜈 is the outward unit normal vector to 𝑀^𝑛. Namely, we will consider the following mean curvature type flow with perpendicular Neumann boundary condition: 𝑑𝐹=()𝑑𝑡𝑛−𝐻𝑢𝜈∀𝑦,𝑡∈𝑀^𝑛×[)𝐹()0,𝑇⋅,0=𝑀₀𝐹𝑦,𝑡⊂Σ∀𝑦,𝑡∈𝜕𝑀^𝑛×[0,𝑇)⟨𝜇,𝜈⟩𝑦,𝑡=0∀𝑦,𝑡∈𝜕𝑀^𝑛×[0,𝑇), (4) where 𝑢 is the Euclidean support function to the hypersurface 𝑀_𝑡 in ℝ^𝑛+1 defined by 𝑢=⟨𝐹,𝜈⟩.

We take the origin on ℝ^𝑛+1 at the vertex of the cone, and |𝐹(𝑦)| denotes the norm of the position vector at some point 𝑦∈𝑀^𝑛. In this paper, we obtain the following main result.

Theorem 1. Suppose Σ is a convex cone and 𝑀₀ is the initial hypersurface which can be represented as a graph over the intersection of the interior of the cone Σ and a unit sphere 𝕊^𝑛 centered at the vertex of the cone; then, a solution to (4) exists for all time and stays always between two spheres with radii 𝑅₁=max_𝑦∈𝑀0|𝐹₀(𝑦)| and 𝑅₂=min_𝑦∈𝑀0|𝐹₀(𝑦)|. Furthermore, the solution converges exponentially fast to the intersection of the interior of the cone and the sphere with radius ()|𝑉|𝑅=𝑛+1𝑀_∞, (5) where |𝑀_∞| denotes the area of the intersection of the limit sphere and the interior of the cone and |𝑉| 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 [2] considered the evolution of a graph over a bounded domain Ω⊂ℝ^𝑛 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 [5] founded boundary monotonicity formulae and classified Type I boundary singularities for 𝐻>0 with a perpendicular Neumann boundary condition. Recently, Lambert in [6] 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 [7] 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 𝑀^𝑛 be a tubular hypersurface between two parallel planes in ℝ^𝑛+1, which can be represented as a graph over some cylinder inside it and meets the parallel planes perpendicularly. Recently, Hartley [8] studied the motion of 𝑀^𝑛 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 𝐶^𝑘 topology for any 𝑘∈ℕ as time 𝑡 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 [9] considered the evolution of 2-dimensional graph over compact convex domain in ℝ² by mean curvature flow under this boundary condition and proved this flow exists for all time and converges to translating solutions at last. Guan [10] 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 𝑀^𝑛 be a compact hypersurface inside an 𝑛-dimensional convex cone Σ with boundary condition 𝜕𝑀⊂Σ, given by the embedding 𝐹∶Ω∈𝕊^𝑛→ℝ^𝑛+1, where Ω is the intersection of the interior of the cone and unit sphere 𝕊^𝑛 centered at the vertex 𝑜 of the cone; that is, 𝑀^𝑛 can be expressed as a graph over Ω. Precisely, for any point 𝑥∈Ω⊂𝕊^𝑛, there is only one ray from the vertex 𝑜 through 𝑥 intersecting the hypersurface 𝑀^𝑛 at some point 𝐹(𝑥); the position vector to 𝑀^𝑛 can be expressed as 𝐹(𝑥)=𝜌(𝑥)𝑥,𝜌(𝑥)∈ℝ⁺,𝑥∈Ω⊂𝕊^𝑛. (6)

Let {𝑥₁,…,𝑥_𝑛} be the local normal coordinates on 𝕊^𝑛; 𝑒_𝑖𝑗 denotes the standard spherical metric under the coordinates; the covariant derivative and divergent operator on 𝕊^𝑛 with respect to the metric 𝑒_𝑖𝑗 are denoted by ∇ and div, respectively. Then, tangent vectors and the outward unit normal vector on 𝑀^𝑛 can be expressed as in [11] (see also P28 in [12]): ∇_𝑖𝐹=∇_𝑖𝜌𝑥+𝜌∇_𝑖1𝑥,𝜈=𝜌²+∇𝜌²𝜌𝑥−𝑒^𝑖𝑗∇_𝑖𝜌∇_𝑗𝑥, (7) where 𝑒^𝑖𝑗 is the inverse of 𝑒_𝑖𝑗. Thus, the support function, induced metric, and second fundamental form can be given by straightforward calculation as following: 𝜌𝑢=²𝜌²+∇𝜌²,𝑔_𝑖𝑗=𝜌²𝑒_𝑖𝑗+∇_𝑖𝜌∇_𝑗𝜌,𝑔^𝑖𝑗=1𝜌²𝑒^𝑖𝑗−∇^𝑖𝜌∇^𝑗𝜌𝜌²+∇𝜌²,ℎ_𝑖𝑗=1𝜌²+∇𝜌²−𝜌∇_𝑖∇_𝑗𝜌+2∇_𝑖𝜌∇_𝑗𝜌+𝜌²𝑒_𝑖𝑗. (8)

Furthermore, the system (4) is equivalent to the following parabolic PDE defined on Ω×[0,𝑇): 𝑑𝜌=()𝜌𝑑𝑡𝑛−𝐻𝑢𝑢1=div𝜎+𝑛∇𝜌𝜌𝜎∇𝜌²∀()[),()[),𝜌()𝑥,𝑡∈Ω×0,𝑇∇𝜌⋅𝜇=0∀𝑥,𝑡∈𝜕Ω×0,𝑇𝑥,0=𝜌₀(𝑥), (9) where 𝜎=𝜌²+|∇𝜌|².

The system (9) is a quasi-linear parabolic equation in divergence form, whose long-time existence is equivalent to the uniform parabolicity, and 𝐶¹ bound on 𝜌 by the classical theory of nonlinear parabolic equations (see, e.g., Chapter 12 in [13]).

For simplicity, let 𝑟=ln𝜌; then, ∇_𝑖𝑟𝜌=∇_𝑖𝜌,|∇𝑟|²𝜌²=∇𝜌²𝜌,𝑢=𝜔, (10) where 𝜔=1+|∇𝑟|². Then, the geometric quantities in (8) can be represented as 1𝜈=𝜔𝑥−𝑒^𝑖𝑗∇_𝑖𝑟∇_𝑗𝑥,𝑔_𝑖𝑗=𝜌²𝑒_𝑖𝑗+∇_𝑖𝑟∇_𝑗𝑟,𝑔^𝑖𝑗=1𝜌²𝑒^𝑖𝑗−∇^𝑖𝑟∇^𝑗𝑟𝜔²,ℎ_𝑖𝑗=𝜌𝜔−∇_𝑖∇_𝑗𝑟+∇_𝑖𝑟∇_𝑗𝑟+𝑒_𝑖𝑗, (11) and (9) can be rewritten as 𝑑𝑟=𝑛𝑑𝑡𝑢1−𝐻=div+()||𝜌𝜔∇𝑟𝑛+1∇𝑟²∀()[),()[),𝑟()𝜌𝜔𝑥,𝑡∈Ω×0,𝑇∇𝑟⋅𝜇=0∀𝑥,𝑡∈𝜕Ω×0,𝑇𝑥,0=𝑟₀(𝑥). (12)

System (12) is also a quasi-linear parabolic equation in divergence form, and the related estimates will be derived in Section 4.

For convenient calculation, we also parametrize the boundary cone as Lambert [6]. Let 𝑆∶𝕊^𝑛−1→𝔹^𝑛(0)⊂ℝ^𝑛 be a smooth embedding of a sphere into a topological ball centered at the origin with outward unit normal vector 𝐧. Then, we can define the boundary cone Σ by embedding ℝ^𝑛 into ℝ^𝑛+1 at height 1, defining Σ to be the set of all rays going through the origin and some point (𝑆(𝑧),1)∈ℝ^𝑛+1, where {𝑧₁,…,𝑧_𝑛−1} denotes the (𝑛−1)-dimensional coordinate for 𝑆. So we can parametrize the cone by 𝐹_Σ𝑆(𝑧)=𝑙𝑙,1+𝑆²𝑆(𝑧)+𝑒_𝑛+1, (13) where 𝑒_𝑛+1 is the (𝑛+1)th standard coordinate vector in ℝ^𝑛+1. 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)𝐴^Σ(𝜕/𝜕𝑙,⋅)=0,(ii)𝐴^Σ(𝜕𝑆/𝜕𝑧_𝑖,𝜕𝑆/𝜕𝑧_𝑗)=(𝑙/(1+|𝑆|²))𝐴^𝑆(𝜕𝑆/𝜕𝑧_𝑖,𝜕𝑆/𝜕𝑧_𝑗), 𝑖=1,…,𝑛−1,where 𝐴^Σ and 𝐴^𝑆 denote the second fundamental forms, respectively, of Σ in ℝ^𝑛+1 and 𝑆 in ℝ^𝑛.

Proof. From the parametrization for the cone, it is easy to check that 𝜕𝑆𝜕𝑧_𝑖,𝜇=0,𝑆+𝑒_𝑛+1,𝜇=0. (14)
Since 𝜇 is the outward unit normal vector to the boundary cone, calculating by Gauss equation directly we have 𝐴^Σ𝜕,𝜕𝜕𝑙𝜕𝑙=ℎ^Σ_𝑙𝑙𝜕=−²𝐹_Σ𝜕𝑙²=𝐴,𝜇0,𝜇=0,_Σ𝜕,𝜕𝑆𝜕𝑙𝜕𝑧_𝑖=ℎ^Σ_𝑙𝑖1=−1+𝑆²𝜕𝑆𝜕𝑧_𝑖−11+𝑆²_𝑖×𝑆(𝑧)+𝑒_𝑛+1,𝜇=0. (15) This proves the first identity.
Similarly, using (14) again we have 𝐴^Σ𝜕𝑆𝜕𝑧_𝑖,𝜕𝑆𝜕𝑧_𝑗𝜕=−²𝐹_Σ𝜕𝑧_𝑖𝜕𝑧_𝑗𝜕,𝜇=−²𝑆𝜕𝑧_𝑖𝜕𝑧_𝑗𝑙,𝜇1+𝑆². (16)
Observing that 𝜇 can be decomposed as 𝜇=⟨𝜇,𝐧⟩𝐧+⟨𝜇,𝑒_𝑛+1⟩𝑒_𝑛+1 (17) and ⟨𝜕²𝑆/𝜕𝑧_𝑖𝜕𝑧_𝑗,𝑒_𝑛+1⟩=0, we, then, have 𝐴^Σ𝜕𝑆𝜕𝑧_𝑖,𝜕𝑆𝜕𝑧_𝑗=−𝑙cos∠𝜇,𝐧1+𝑆²𝜕𝐧,²𝑆𝜕𝑧_𝑖𝜕𝑧_𝑗=𝑙1+𝑆²𝐴^𝑆𝜕𝑆𝜕𝑧_𝑖,𝜕𝑆𝜕𝑧_𝑗, (18) where we use the fact that cos∠(𝜇,𝐧)=1/1+|𝑆|².

3. Evolution Equations and Boundary Derivatives

In this section, we will derive evolution equations for some useful geometric quantities by straightforward calculation. Let {𝑦₁,…,𝑦_𝑛} be the local normal coordinates of the evolving hyperserface and let 𝑔_𝑖𝑗 be the corresponding induced metric; let ∇ and Δ be, respectively, the covariant derivative and Laplace operator with respect to the induced metric 𝑔_𝑖𝑗; and |𝐹| and |𝐴| denote the norm of the position vector and the second fundamental form to evolving hypersurfaces 𝑀^𝑛 in ℝ^𝑛+1, respectively.

Lemma 3. Under the mean curvature type flow (4), we have (i)(𝑑/𝑑𝑡−𝑢Δ)|𝐹|²=0,(ii)𝑑𝜈/𝑑𝑡=−∇(𝑛−𝐻𝑢),(iii)(𝑑/𝑑𝑡−𝑢Δ)𝑢=𝑛−2𝐻𝑢+|𝐴|²𝑢²+𝐻⟨∇𝑢,𝐹⟩.

Proof. (i) We can calculate directly as in [14] 𝑑|𝐹|²𝑑𝑡=2𝑑𝐹()()𝑑𝑡,𝐹=2⟨𝑛−𝐻𝑢𝜈,𝐹⟩=2𝑛−𝐻𝑢𝑢. (19)
On the other hand, Δ|𝐹|²=𝑔^𝑖𝑗∇_𝑖∇_𝑗⟨𝐹,𝐹⟩=2𝑔^𝑖𝑗∇_𝑖𝜕𝐹𝜕𝑦_𝑗,𝐹=2𝑛−2𝑔^𝑖𝑗⟨ℎ_𝑖𝑗𝜈,𝐹⟩=2(𝑛−𝐻𝑢). (20) This proves (i).
(ii) As ⟨𝜈,𝜈⟩=1, we have ⟨𝑑𝜈/𝑑𝑡,𝜈⟩=0, and then, 𝑑𝜈=𝑑𝑡𝑑𝜈,𝑑𝑡𝜕𝐹𝜕𝑦_𝑖𝑔^𝑖𝑗𝜕𝐹𝜕𝑦_𝑗𝜕=−𝜈,𝜕𝑦_𝑖𝜕𝐹𝑔𝜕𝑡^𝑖𝑗𝜕𝐹𝜕𝑦_𝑗𝜕=−𝜕𝑦_𝑖(𝑛−𝐻𝑢)𝑔^𝑖𝑗𝜕𝐹𝜕𝑦_𝑗=−∇(𝑛−𝐻𝑢). (21)
(iii) Similarly, as in (i), 𝑑𝑢=𝑑𝑑𝑡⟨⟩=𝑑𝑡𝐹,𝜈𝑑𝐹+𝑑𝑡,𝜈𝐹,𝑑𝜈=()−𝑑𝑡𝑛−𝐻𝑢𝐹,∇(𝑛−𝐻𝑢),Δ𝑢=𝑔^𝑖𝑗∇_𝑖∇_𝑗⟨𝐹,𝜈⟩=𝑔^𝑖𝑗∇_𝑖𝜕𝐹𝜕𝑦_𝑗,𝜈+𝐹,ℎ_𝑗𝑘𝜕𝐹𝜕𝑦_𝑘=𝑔^𝑖𝑗𝑔_𝑖𝑘ℎ_𝑗𝑘+𝐹,∇𝐻−ℎ_𝑗𝑘ℎ_𝑖𝑙𝑔^𝑖𝑗𝑔^𝑘𝑙𝑢=𝐻+⟨𝐹,∇𝐻⟩−|𝐴|²𝑢. (22) Then, (iii) follows by combining the above two formulae.

The following relationship between 𝐴^Σ and 𝐴 was proved by Stahl in [3].

Lemma 4 (see [3]). For 𝑝∈𝜕𝑀×[0,𝑇), one has 𝐴^Σ(𝜉,𝜈)=−𝐴𝜉,𝜇, (23) where 𝜉∈𝑇_𝑝𝑀∩𝑇_𝑝Σ.

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

Lemma 5. For (𝑦,𝑡)∈𝜕𝑀×[0,𝑇), one has(i)⟨∇|𝐹|²,𝜇⟩=0,(ii)⟨∇𝑢,𝜇⟩=𝑢𝐴^Σ(𝜈,𝜈).

Proof. (i) Denote by ∇ the Euclidean covariant derivative in ℝ^𝑛+1. Obviously, ∇|𝐹|²∈𝑇_𝑝Σ; combining the boundary condition in (4), we have ⟨∇|𝐹|²,𝜇⟩=⟨∇|𝐹|²−⟨∇|𝐹|²,𝜈⟩𝜈,𝜇⟩=⟨∇|𝐹|²,𝜇⟩=0. (24)
(ii) Calculating directly, we obtain ∇𝑢=𝐹,𝜕𝜈𝜕𝑦_𝑖𝑔^𝑖𝑗𝜕𝜕𝑦_𝑗=𝐴𝐹^⊤,𝜕𝜕𝑦_𝑖𝑔^𝑖𝑗𝜕𝜕𝑦_𝑗,⟨∇𝑢,𝜇⟩=𝐴𝐹^⊤,𝜕𝜕𝑦_𝑖𝑔^𝑖𝑗𝜕𝜕𝑦_𝑗,𝜇=𝐴𝐹^⊤,𝜇. (25) Observing 𝐹^⊤=𝐹−𝑢𝜈, combination of Lemma 4 and (i) in Proposition 2 yields ∇𝑢,𝜇=−𝐴^Σ(𝐹−𝑢𝜈,𝜈)=−𝐴^Σ(𝐹,𝜈)+𝑢𝐴^Σ(𝜈,𝜈)=𝑢𝐴^Σ(𝜈,𝜈). (26)

4. Gradient Estimate

Let 𝑎_𝑖𝑗(𝑦,𝑡)∈𝐿^∞(𝑀^𝑛×[0,𝑇)) be a positive definite matrix such that ℒ𝑓=𝑑𝑓𝑑𝑡−𝑎^𝑖𝑗∇_𝑖∇_𝑗𝑓 (27) is a parabolic operator. We have the following maximal principle from Hopf Lemma [15] or Stahl’s corresponding result in [4].

Theorem 6. Suppose 𝑓∶𝑀^𝑛×[0,𝑇)→ℝ satisfies 𝑑𝑓𝑑𝑡−𝑎^𝑖𝑗∇_𝑖∇_𝑗𝑓∀𝑦,𝑡≤0𝑦,𝑡∈𝑀^𝑛×[0,𝑇)𝑠.𝑡.𝑦⟨∇𝑓=0,∇𝑓,𝜇⟩≤0∀𝑦,𝑡∈𝜕𝑀^𝑛×[0,𝑇), (28) and then 𝑓(𝑦,𝑡)≤sup_{𝑦∈𝑀𝑛}𝑓(𝑦,0) for all (𝑦,𝑡)∈𝑀^𝑛×[0,𝑇).

Now we apply the above theorem to |𝐹|². Combining (i) in Lemma 3 and (i) in Lemma 5, we immediately have the following estimates.

Corollary 7. Let 𝐹(𝑦,𝑡) be a solution to system (4); then, one has min_{𝑦∈𝑀𝑛0}𝐹₀𝑦≤𝐹𝑦,𝑡≤max_{𝑦∈𝑀𝑛0}𝐹₀𝑦 (29) for any (𝑦,𝑡)∈𝑀^𝑛×[0,𝑇).

This result means that the evolving hypersurface always stays between two spheres with radii 𝑅₁=max_𝑦∈𝑀^𝑛₀|𝐹₀(𝑦)| and 𝑅₂=min_𝑦∈𝑀^𝑛₀|𝐹₀(𝑦)|.

In order to obtain the gradient estimate for system (12), we only need to estimate the lower bound for 𝑢=𝜌/𝜔=𝜌/1+|∇𝑟|². From (iii) in Lemma 3, 𝑑𝑑𝑡−𝑢Δ𝑢=𝑛−2𝐻𝑢+|𝐴|²𝑢²+𝐻≥()−∇𝑢,𝐹𝑛−𝐻𝑢𝐻𝑢𝑛(𝑛−𝐻𝑢)+𝐻=1∇𝑢,𝐹𝑛(𝑛−𝐻𝑢)²+𝐻∇𝑢,𝐹. (30)

On the other hand, noticing ⟨𝜇,𝜈⟩=0 by assumption, 𝜈 must be in the tangent space of the boundary cone; that is, 𝜈∈𝑇Σ. Combining (ii) in Lemma 5, Proposition 2, and the convexity of the boundary cone, we have ⟨∇𝑢,𝜇⟩=𝑢𝐴^Σ(𝜈,𝜈)≥0. (31) Equivalently, 𝑑𝑑𝑡−𝑢Δ()1−𝑢≤−𝑛(𝑛−𝐻𝑢)²∀≤0𝑦,𝑡∈𝑀^𝑛×[0,𝑇)s.t.⟨∇𝑢𝑦,𝑡=0,∇(−𝑢),𝜇⟩=−𝑢𝐴^Σ(𝜈,𝜈)≤0∀𝑦,𝑡∈𝜕𝑀^𝑛×[0,𝑇). (32)

By the maximal principle we have the following.

Corollary 8. For all (𝑦,𝑡)∈𝑀^𝑛×[0,𝑇), the support function 𝑢(𝑦,𝑡) satisfies 𝑢𝑦,𝑡≥min_{𝑦∈𝑀𝑛}𝑢𝑦,0. (33)

Combining Corollaries 7 and 8, we obtain long time existence for the system (12) by the standard argument for divergence PDE (cf. [13]), and then, the first part of Theorem 1 follows by the equivalence of (4) and (12).

Let 𝑉 be the domain enclosed by the interior of the cone and the evolving hypersurface. The volume element for 𝑉 is denoted by 𝑑𝑉 and 𝑑𝑉_𝜕 for its boundary. We will also denote the area elements for the evolving hypersurface and its boundary by 𝑑𝑣 and 𝑑𝑣_𝜕, respectively. Flow (4) has the following interesting property.

Proposition 9. Along flow (4), the volume of 𝑉, denoted by |𝑉|, is a constant.

Proof. For the function Φ(𝑦,𝑡)=(1/2)|𝐹(𝑦,𝑡)|² defined on the evolving hypersurface 𝑀_𝑡, we have ΔΦ=𝑛−𝐻𝑢 (34) from the proof of Lemma 3(i). Integrating the above equation on 𝑀_𝑡 and taking into consideration Lemma 5(i) yield _𝑀𝑡(𝑛−𝐻𝑢)𝑑𝑣=_𝑀𝑡ΔΦ𝑑𝑣=_𝜕𝑀𝑡∇Φ,𝜇𝑑𝑣_𝜕=12_𝜕𝑀𝑡⟨∇|𝐹|²,𝜇⟩𝑑𝑣_𝜕=0. (35)
Denote by div𝑋 the divergence of the position vector 𝑋 for 𝑉 in ℝ^𝑛+1; we have div𝑋𝑑𝑉=(_𝑉𝑛+1)|𝑉|. (36)
Taking derivative with respect to time 𝑡 and combining divergence theorem yield ()𝑑𝑛+1|𝑉|=𝑑𝑡_𝑉div𝑑𝑋𝑑𝑡𝑑𝑉=_𝜕𝑉𝑑𝑋𝑑𝑡,𝜂𝑑𝑉_𝜕=_Σ𝑑𝑋𝑑𝑡,𝜇𝑑𝑣_Σ+_𝑀𝑡𝑑𝑋𝑑𝑡,𝜈𝑑𝑣=_𝑀𝑡(𝑛−𝐻𝑢)𝑑𝑣=0, (37) where 𝜂 is the outward unit normal vector to the (𝑛+1)-dimensional region 𝑉, and 𝑑𝑣_Σ is the area element on boundary cone Σ. Hence, |𝑉| is a constant and the result follows.

5. Convergence

In this section, we use an idea of Guan and Li in [1] 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 {𝑥₁,…,𝑥_𝑛} on Ω⊂𝕊^𝑛. ∇ and Δ are again the covariant derivative and Laplace operator with respect to the standard metric 𝑒_𝑖𝑗 on 𝕊^𝑛. Let 𝔏 be a parabolic operator defined by 𝔏𝑓=𝑑𝑓−1𝑑𝑡𝑒𝜌𝜔^𝑖𝑗−∇^𝑖𝑟∇^𝑗𝑟𝜔²𝑓_𝑖𝑗 (38) on Ω×[0,𝑇) for a smooth function 𝑓∶Ω×[0,𝑇)→ℝ. From [1], we have the following evolution equation for |∇𝑟|²/2 at the critical point 𝔏|∇𝑟|²2𝑛=−||𝜌𝜔∇𝑟⁴−𝑛−1||𝜌𝜔∇𝑟²−1||𝜌𝜔Δ𝑟∇𝑟²−1∇𝜌𝜔²𝑟². (39)

Using Cauchy-Schwartz inequality and similar rearrangement as in [10], we have 𝔏|∇𝑟|²≤−3𝑛||2𝜌𝜔∇𝑟⁴−2(𝑛−1)||𝜌𝜔∇𝑟²−2𝑛𝑛𝜌𝜔Δ𝑟+2|∇𝑟|²². (40)

Therefore, 𝔏|∇𝑟|²2()≤−𝑛−1||𝜌𝜔∇𝑟²≤0. (41)

Recall that 𝜌=𝜔𝑢 and 𝜔=1+|∇𝑟|²; we have, by Lemma 5 and the Neumann boundary assumption, for any points (𝑥,𝑡)∈𝜕Ω×[0,𝑇), ∇|∇𝑟|²,𝜇=∇𝜔²=−1,𝜇2𝜌𝑢²−∇𝜌,𝜇2𝜌²𝑢³∇𝑢,𝜇=−2𝜌²𝑢²𝐴^Σ(𝜈,𝜈). (42)

With the assumption of convexity on the boundary cone, we obtain, by Theorem 6, |∇𝑟|²(𝑥,𝑡)≤max_𝑥∈Ω|∇𝑟|²(𝑥,0). (43)

By the equivalence of (4) and (9), we have min_𝑥∈Ω𝜌(𝑥,0)≤𝜌(𝑥,𝑡)≤max_𝑥∈Ω𝜌(𝑥,0) (44) from Corollary 7.

From the above estimate, there exists a uniform positive constant 𝛼≤2(𝑛−1)/𝜌𝜔 depending only on the upper bound of 𝜔 and 𝜌 such that (41) reads as 𝔏|∇𝑟|²≤−𝛼|∇𝑟|². (45)

Denote by 𝑔=𝑒^𝛼𝑡|∇𝑟|²𝔏𝑔=𝔏𝑒^𝛼𝑡|∇𝑟|²=𝛼𝑒^𝛼𝑡|∇𝑟|²+𝑒^𝛼𝑡𝔏|∇𝑟|²≤0,⟨∇𝑔,𝜇⟩=𝑒^𝛼𝑡⟨∇𝜔²𝜌,𝜇⟩=−2²𝑢²𝑒^𝛼𝑡𝐴^Σ(𝜈,𝜈)≤0. (46)

Then, by Theorem 6 again, 𝑔=𝑒^𝛼𝑡|∇𝑟|² has uniform upper bound 𝐶₁=max_𝑥∈Ω𝑔(𝑥,0); that is |∇𝑟|²≤𝐶₁𝑒^−𝛼𝑡,𝑡∈[0,∞). (47) Or, equivalently, for a different constant 𝐶₂=𝜌²_max𝑥∈Ω(𝑥,0)𝐶₁, ∇𝜌²≤𝐶₂𝑒^−𝛼𝑡,𝑡∈[0,∞). (48) This means that 𝑀_𝑡 converges exponentially fast to some part of a sphere.

Assume the radial function 𝜌(𝑥,𝑡₀) attains its minimum at a point 𝑝∈Ω for some 𝑡₀∈[0,∞); that is, 𝜌(𝑝,𝑡₀)=𝜌_min𝑥∈Ω(𝑥,𝑡₀). Let 𝜎∶[0,𝑠)→Ω be a geodesic on the unit sphere 𝕊^𝑛 starting from 𝑝 to any point 𝑥∈Ω with 𝜎(𝑠)=𝑥. Integrating both sides of the last inequality on [0,𝑠] and taking into account the boundedness of 𝑠, we have, for some constant 𝐶₃=𝐶₃(Ω,𝐶₂,𝛼) and 𝛽=𝛼/2, ^𝑠₀𝑑𝜌𝑑𝑠𝑑𝑠≤^𝑠₀∇𝜌𝑑𝑠≤𝐶₃𝑒^−𝛽𝑡0, (49) and then, 𝜌𝑥,𝑡₀−𝜌𝑝,𝑡₀=𝜌𝑥,𝑡₀−𝜌_min𝑥∈Ω𝑥,𝑡₀≤𝐶₃𝑒^−𝛽𝑡0. (50)

So we have 𝜌_max𝑥∈Ω(𝑥,𝑡)−𝜌_min𝑥∈Ω(𝑥,𝑡)≤𝐶₃𝑒^−𝛽𝑡,𝑡∈[0,∞). (51)

On the other hand, (𝑛+1)|𝑉|=_𝑉div𝑋𝑑𝑉=_𝜕𝑉𝑋,𝜂𝑑𝑉_𝜕=_Σ𝑋,𝜇𝑑𝑣_Σ+_𝑀𝑡⟨𝑋,𝜈⟩𝑑𝑣=_𝑀𝑡𝑢𝑑𝑣. (52)

Because 𝜌_max𝑥∈Ω(𝑥,𝑡)−𝐶₃𝑒^−𝛽𝑡≤𝜌(𝑥,𝑡)≤𝜌_min𝑥∈Ω(𝑥,𝑡)+𝐶₃𝑒^−𝛽𝑡,1≤𝜔=1+|∇𝑟|²≤1+𝐶₁𝑒^−𝛼𝑡, (53) the support function 𝑢=𝜌/𝜔 satisfies 𝜌_max𝑥∈Ω(𝑥,𝑡)−𝐶₃𝑒^−𝛽𝑡1+𝐶₁𝑒^−𝛼𝑡≤𝑢≤𝜌_min𝑥∈Ω(𝑥,𝑡)+𝐶₃𝑒^−𝛽𝑡. (54)

Denote by |𝑀_𝑡| the area of 𝑀_𝑡; combining (52), we have 𝑀_𝑡𝜌_max𝑥∈Ω(𝑥,𝑡)−𝐶₃𝑒^−𝛽𝑡1+𝐶₁𝑒^−𝛼𝑡≤(𝑉|≤𝑀𝑛+1)|_𝑡𝜌_min𝑥∈Ω(𝑥,𝑡)+𝐶₃𝑒^−𝛽𝑡, (55) or, equivalently, (𝑛+1)|𝑉|𝜌_min𝑥∈Ω(𝑥,𝑡)+𝐶₃𝑒^−𝛽𝑡≤𝑀_𝑡≤(𝑛+1)|𝑉|1+𝐶₁𝑒^−𝛼𝑡𝜌_max𝑥∈Ω(𝑥,𝑡)−𝐶₃𝑒^−𝛽𝑡. (56)

Let lim_𝑡→∞𝜌(𝑥,𝑡)=𝜌(𝑥), and from the above inequality, we obtain 𝜌(𝑥)=𝑅, a constant, and therefore, 𝑀_𝑡⟶(𝑛+1)|𝑉|𝑅,𝑡⟶∞. (57) That is to say, 𝑀_𝑡 converges exponentially to the intersection of the interior of the cone Σ and a sphere centered at the vertex of the cone with radius (𝑉|𝑅=𝑛+1)|𝑀_∞, (58) where |𝑀_∞| denotes the area of the limit sphere 𝑀_∞. 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.

Acknowledgments

The research is partially supported by NSFC (no. 11171096), RFDP (no. 20104208110002), and Funds for Disciplines Leaders of Wuhan (no. Z201051730002).