A mean curvature type flow with capillary boundary in a unit ball

In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional E. We show that it has the longtime existence and subconverges to spherical caps. As an application, we solve an isoperimetric problem for hypersurfaces with capillary boundary.


Introduction
In this paper, we are interested in a mean curvature type flow in the unit ball B n+1 ⊂ R n+1 with capillary boundary.Roughly speaking, given a Riemannian manifold N n+1 with a smooth boundary ∂N , a hypersurface with capillary boundary in N is an immersed hypersurface which intersects ∂N at a constant contact angle θ ∈ (0, π).
For closed hypersurfaces, the mean curvature flow plays an important role in geometric analysis and has been extensively studied.One of classical results proved by Huisken [19] states that it contracts a closed convex hypersurface into a round point.Mean curvature type flows with a constraint play an important role in the study of isoperimetric problems.The following curve-shortening (and area-preserving) flow was studied by Gage [10].Let γ : S 1 × [0, T ) → R 2 satisfy ∂ t γ = (κ − 2π L )ν, where κ is the geodesic curvature of γ, L is the length of the curve at scale t, and ν is the outward unit normal vector of curve γ(•, t).In a higher dimensional Euclidean space, Huisken introduced a non-local type mean curvature flow in [21]: Given a closed, connected hypersurface M , consider a family of embeddings x : M × [0, T ) → R n+1 satisfies where c(t) := M t Hdµ |Mt| is the average of the mean curvature H of M t := x(M, t) and ν is the unit outward normal vector field of M t .Huisken proved that such a volume preserving flow converges to a round sphere if the initial hypersurface is uniformly convex.There has been a lot of work on such geometric flows.Here we just mention further [2] for studying such kind of flow in the case where the ambient space is Riemannian manifold and [30] for the extension to a general mixed volume preserving mean curvature flow.As one of applications, such a volume (or area)-preserving flow could be used to prove optimal geometric inequalities.In order to establish optimal geometric inequalities, there is another type mean curvature flow, which is first introduced by Guan and Li [16] inspired by the Minkowski formulas.A flow x : M × [0, T ) → M n+1 k satisfies ∂ t x = (nφ (ρ) − Hu)ν, where u is the support function of hypersurface x(M, •) and M n+1 k is the space form with constant sectional curvature k and metric ds 2 := dρ 2 + φ 2 (ρ)g S n .This flow is also volume preserving and area decreasing by the Minkowski formulas.They obtained the longtime existence of this flow and proved that it smoothly converges to a round sphere if the initial hypersurface is star-shaped.As a result, this yields a flow proof of classical Alexandrov-Fenchel inequalities of quermassintegrals in convex geometry.Recently, they obtained that a similar phenomenon also holds for the general warped produced manifold in [17] jointed with Wang.For the methods which use a fully nonlinear flow to establish geometric inequalities, we refer also to [18].Last but not least, we recommend the literature [4], [5], [6], [7], [32] and references therein for extensions to general anisotropic and fully nonlinear curvature flows in various ambient spaces.
There has been a great interest in the investigation of hypersurfaces with non-empty boundaries in the last thirty years.For instance, Stahl [34] considered the mean curvature flow with free boundary in the Euclidean space, and he showed that the solution either has the longtime existence or the curvature and its derivatives blow up at the maximal time.Later, Marquardt [29] considered the inverse mean curvature flow for hypersurfaces with boundary perpendicular to a convex cone, and proved that it has the long time existence and converges to a piece of round sphere, if the initial hypersurface is starshaped and strictly mean convex.Recently Lambert-Scheuer [25] studied the same flow as [29] but with the supporting hypersurface being a sphere instead of a cone.They proved that a convex hypersurface which is perpendicular to a sphere along the boundary converges to a flat disk in certain sense.As a nice geometric application of this flow they proved in [24] a Willmore type inequality.We also would like to mention the recent articles [36] and [33] for a mean curvature type flow and a fully nonlinear inverse curvature type flow respectively in the unit ball with free boundary, where new geometric inequalities were proved as applications.For the study of a nonparametric mean curvature flow with free or capillary type boundaries, we refer to [3], [11], [27] and [20].
Those results motivate us to consider the following mean curvature type flow for hypersurfaces with capillary boundary.To be more precise, let Σ 0 be a properly embedded compact smooth hypersurface in B n+1 (n ≥ 2) with capillary boundary ∂Σ 0 ⊂ S n := ∂B n+1 , which is given by and M is a compact manifold with smooth boundary ∂M .In other words, int(Σ 0 ) = x 0 (int(M )), and ∂Σ 0 = x 0 (∂M ) intersects ∂B n+1 at a constant contact angle θ ∈ (0, π).Consider a family of embeddings where f := n x, a + n cos θ ν, a − H X a , ν , for a ∈ S n , ν and H are the unit normal vector field and the mean curvature of hypersurface x(•, t) resp., N is the unit outward normal vector field of S n , the contact angle θ ∈ (0, π) is a constant and the vector field X a will be defined and discussed in the next paragraph.Here, for a vector field ξ along a hypersurface x, we define its normal part by ξ ⊥ := ξ, ν ν.The choice of f is motivated by new Minkowski formulas proved in [35].If θ = π 2 , it corresponds to the free boundary problem of parabolic setting, which was studied by Wang and Xia in [36].
Before we state our main results, we clarify the notation X a used above.In this paper X a is a vector field defined by where a is a fixed unit vector in R n+1 .One can easily to check that X a is a conformal Killing vector field in B n+1 .In fact, X a is exactly the pull back of the position vector field under a conformal transformation from the unit ball to the half Euclidean space.See Section 3.2 for the precise discussion.We say that a properly embedded hypersurface Σ in R n+1 is star-shaped with respect to a if Σ intersects each integral curve of X a only once.For simplicity in this paper we define a hypersurface of star-shaped by a stronger condition that X a , ν > 0 holds everywhere in M .Our main theorem is the following If θ = π 2 , i.e., the free boundary case, this theorem was proved recently by Wang and Xia in [36], where they also proved the global convergence.The free boundary case usually corresponds to a homogeneous or linear Neumann boundary value conditions, see [20], [24], [29], [33], [34] and [36] for example.However, the capillary boundary case in general relates to a nonlinear type Neumann boundary value condition, which is more complicated and technically more difficult to handle from the analytic viewpoint.This difficulty usually prevents us to obtain estimates for a full range of θ ∈ (0, π).For instance, Guan obtained the gradient estimate (depending on the time T ) of solution in [15] for a nonparametric curvature flow with capillary boundary for angle satisfying | cos θ| < √ 3 2 .Recently, the authors [11] obtained the uniformly gradient estimate (independent of time T ) for the nonparametric mean curvature flow with capillary boundary for θ in a small neighborhood of π 2 .In this paper we obtain for our flow (1.1) a better range | cos θ| < 3n+1 5n−1 .The reason why we can have a bigger range of the contact angle is due to an observation that equation (3.4) has a good term when we carry out the gradient estimate.See the proof of Proposition 4.3.Also due to this difficulty, we can only prove the subsequence convergence of this flow and are not able to show that the limits are the same spherical cap at the moment.We will consider this problem in the near future.Nevertheless, the limits have the same radius and hence we can provide a flow proof for the isoperimetric problem for hypersurfaces with capillary boundary in the unit ball Corollary 1.2Among star-shaped capillary boundary hypersurfaces with a volume constraint the spherical caps given in Remark 4.1 are the only minimizers of the energy functional E defined in (2.8) below, provided that the contact angle θ satisfies | cos θ| < 3n+1 5n−1 ,.The Corollary follows from Theorem 1.1 and the crucial properties that the flow preserves the enclosed volume and decreases the energy functional E, which are proved in Subsection 2.3 by the new Minkowski formulas established in [35].
This article is organized as follows.In Section 2, we give some preliminaries about hypersurfaces with capillary boundary and our mean curvature type flow.In Section 3, we convert the flow to a scalar equation on semi-sphere with the help of a conformal transformation.In the last Section, we establish a priori estimates and prove the main theorem.

Preliminaries
In this Section we provide basic facts of capillary hypersurfaces and prove the crucial facts of our flow in Proposition 2.4 by using the new Minkowski formulas obtained in [35].For convenience of the reader we provide complete proofs.For more information about capillary hypersurfaces we refer to the wonderful exposition book [9].

Integral identities
In this paper we consider hypersurfaces Σ ⊂ B n+1 with capillary boundary ∂Σ on ∂B n+1 which will be precisely defined below.Since we will use a flow to study such hypersurfaces, it will convenience to use the parametrization: Let x : M → B n+1 be an isometric immersion of an orientable n-dimensional compact manifold M with smooth boundary ∂M such that Σ := x(M ) and ∂Σ := x(∂M ).However, we will identity M with Σ and ∂M with ∂Σ, if there is no confusion.
Let N be the unit outward normal N of the unit sphere ∂B n+1 .Let Σ ⊂ B n+1 be a smooth oriented hypersurface with boundary ∂Σ satisfying int(Σ) ⊂ B n+1 and ∂Σ ⊂ ∂B n+1 .Σ divides the unit ball into two parts.We denote one part by Ω and define ν the unit outward normal vector field of Σ w.r.t.Ω.
Let µ be the unit outward conormal vector field along ∂Σ and ν be the unit normal to ∂Σ in ∂B n+1 such that {ν, µ} and {ν, N } have the same orientation in the normal bundle of ∂Σ ⊂ B n+1 .See Figure 1.
We call the angle between −ν and N contact angle and denote it by θ.It follows N = sin θµ − cos θν, ν = cos θµ + sin θν.
We denote D and ∇ derivatives on (B n+1 , δ B n+1 ) and (M, g) resp., where δ B n+1 is the standard Euclidean metric and g is the induced metric on M .
Recall that X a is the conformal vector field defined by (1.2).Decompose X a into X a := X T a + X a , ν ν, where X T a is the tangential projection of X a on Σ.It is clear to see that X a := x, a x − a on ∂Σ and N = x on ∂B n+1 , which follows that Let h be the second fundamental form of the hypersurface Σ given by h(X, Y ) := D X ν, Y for any X, Y ∈ T Σ with Σ := x(M ) and H is the mean curvature of Σ.Note that for any e ∈ T (∂Σ) and (see Lemma 3.1 in [26] or Proposition 2.1 in [35] for a proof).These two simple facts are important in the study of capillary hypersurfaces.From these two face we have where we have used the fact µ = sin θN + cos θν in the last equality.
The following proposition was proved for hypersurfaces with free boundary recently in [35].For completeness, we provide a proof here for hypersurfaces with capillary boundary.
where dA and dσ are the area element of M and ∂M respectively with respect to the induced metric g, κ := (κ 1 , . . ., κ n ) are the principal curvatures of the Weingarten tensor (g −1 h) and σ 2 (κ) is the 2nd elementary symmetric function acting on the principal curvatures.
Proof Let {e i } n i=1 be the orthonormal frame on M and e n+1 = N .By using equation (3.5) in [35], we have This follows easily from the conformality of the vector field X a .It follows that div Denote the Newton tensor by T 1 (κ) := ∂σ2 ∂(g −1 h) .In local coordinates, we have Multiplying the both side of the above identity by T ij 1 := ∂σ2 ∂h i j and integrating, we have Since X T a is the tangential projection of X a on M , integrating by parts we have Hence the proof is complete.
The following property is also crucial for us.
Proposition 2.3 Under the same conditions as in Proposition 2.2, it holds that Proof Set P a := ν, a x − x, ν a.By a direct computation, we have and div P T a = n ν, a . (2.7) , we obtain Integrating by parts we conclude that where we have used equation (2.2) in the fifth equality.Therefore we complete the proof.

The first variation formulas
Let x : (M, ∂M ) → (B n+1 , ∂B n+1 ) be an isometric embedded of an orientable n-dimensional compact manifold M with smooth boundary ∂M such that Σ := x(M ) and ∂Σ := x(∂M ).We define the volume functional of x as the usual volume of the n + 1-dimensional domain Ω enclosed by Σ and ∂B n+1 as in Figure 1.The so-called wetting area W (Σ) is just the area of the region T := ∂Ω ∩ ∂B n+1 , which is also bounded by ∂Σ on ∂B n+1 .The energy functional is defined as Next we present the first variational formula for the energy functional E.An admissible variation of x is a differential map n+1 is an immersion with x(int(M ), t) ⊂ B n+1 and x(∂M, t) ⊂ ∂B n+1 , and x(•, 0) = x 0 (•).Denote the corresponding hypersurfaces by Σ t = x(M, t), its enclosed domain Ω t and the "wet" part by T t .It is well-known that the first variations of volume functional and area functional are given by where dV B n+1 is the volume element of B n+1 and Y := ∂ ∂t x t (•) t=0 .Moreover the variation of the area of T t is given by For a proof, see [31] (See Section 4 Appendix there) for instance.Now, the variation of the enery functional E is given by (2.9)

Key properties of flow (1.1)
From the Minkowski type formula in [35] (see Proposition 3.2 and equation (3.4) there), we have the following two important facts of (1.1).
Proposition 2.4 Flow (1.1) preserves the volume functional Vol(Ω t ) and decreases E(M t ).
Proof It is easy to see that this flow preserves the enclosed volume where the last equality is the new Minkowski identity proved in [35].With the above preparation and for the convenience of the reader, we point out that this formula follows from which, in turn, follows from equations (2.5), (2.7) and (2.1).From (2.9) and Proposition 2.2, we have that For the term S 2 , we claim that S 2 ≤ 0. In fact, this follows from facts that X a , ν > 0 in M and the following well-known fact (2.12) For the term S 1 , from equations (2.1) and (2.4), we have Combining with Proposition 2.3 and the fact that θ ≡ const, we have Therefore, we obtain Hence we complete the proof.

A scalar equation
In this section we will reduce flow (1.1) to a scalar flow, provided the initial hypersurface is star-shaped.

Basic facts
In this subsection, we first recall some basic facts and identities for the relevant geometric quantities of a smooth star-shaped hypersurface X : + with respect to the origin.If Σ is star-shaped with respect to the origin, then the position vector X of Σ can be written as in where u ∈ C 2 (Ω) ∩ C 0 (Ω) and ρ := e u .Let {e i } n i=1 be the local frame field on S n + with the round metric σ, and denote ∇ and D the gradient on S n + and R n+1 + respectively.Then in terms of ρ the metric g ij is given by where •, • denotes the standard inner product in R n+1 + , σ ij := e i , e j and ρ i := ∇ ei ρ, ρ ij := ∇ ei ∇ ej ρ.The inverse of metric g is where σ ij denotes the inverse of σ ij and u i := σ ik u k .The unit outer normal vector field to Σ in R n+1 + is given by ν(X(x)) = x − ∇u(x) > 0 which means that ν satisfies the choice of orientation on a radial graph.The second fundamental form of X is and the mean curvature is given by where v := 1 + |∇u| 2 and div is the divergence operator with respect to the canonical metric σ on S n + .Using the same method in [12], we assume that flow equation (1.1) is satisfied by a family of the radial graphs over S n + , that is, x(ξ, t) := X(ξ, t)ρ(X(ξ, t), t) with X ∈ S n + .Then we have (3.1)

+
) are isometric, a proper embedding Σ = x(M ) in For a star-shaped hypersurface ) , where ỹ := ϕ • x, we can write it as In polar coordinates, a direct computation implies that

It gives us
From now on, we always set a := −E n+1 .We have Set w := log ).Then the capillary boundary condition gives us that It follows that By a straightforward computation as above, under the conformal transformation ϕ we have Similarly, we have Note that e −w := ρ 2 +2ρ cos β+1

2
. It then yields that Applying the transformation law for the mean curvature under a conformal metric, we know that the mean curvature H of Σ in R n+1 + , (ϕ −1 ) * (δ R n+1

+
).From the discussion in Section 3.1, in particular, equation (3.1), we know that the first equation in (3.2) is reduced to the following scalar equation where It is easy to see that equation (3.4) is also equivalent to In summary, from the above discussion, flow (1.1) is equivalent to (up to a tangential diffeormphism) the following scalar parabolic equation on where u 0 = log ρ 0 , ρ 0 is related to the initial hypersurface x 0 (M ) under the transformation ϕ and F is defined in the previous equation.

A priori estimates
The short-time existence of our flow is established by the standard PDE theory, since due to our assumption of star-shaped, for initial hypersurface, the flow is equivalent to the scalar flow (3.5).In this section, we will show the uniform height and gradient estimates for equation (3.5).Then the longtime existence of the flow follow immediately from the standard parabolic PDE theory.
In this section, we use the Einstein summation convention, i.e., if not stated otherwise, the repeated arabic indices i, j, k should be summed from 1 to n.We also use the notations u β := σ(∇u, ∂ β ) = ∇ ∂ β u and |∇u| 2 := σ(∇u, ∇u) in this section.Recall that ρ = e u and 2e −w = 1 + ρ 2 + 2ρ cos β.For the convenience, we introduce the following notations It is easy to check that ∂C r,θ is a static solution to the flow (1.1), that is, and meets the support S n = ∂B n+1 at the contact angle θ.Such a spherical cap is certainly star-shaped and determines a corresponding radial function ψ, which is a stationary solution of flow (3.5).
Now we are ready to show that the radial function u has the following C 0 estimate.Proposition 4.2 Assume that the initial star-shaped hypersurface x 0 (M ) satisfies for some R 2 > R 1 > 0, where C R,θ is defined in Remark 4.1.Then this property is preserved along flow (1.1).In particular, if u(x, t) solves the initial boundary value problem (3.5) on interval [0, +∞), then for any T > 0, , where C is a constant depends only on the initial value and their covariant derivatives with respect to the round metric σ on S n + .
Proof For any T > 0, we want to get the C 0 estimate of u in S n + × [0, T ].Assume that ψ is the radial function of the corresponding upper spherical cap with respect to ∂C R2,θ ∩ B n+1 after the conformal transformation ϕ.Since ψ is a static solution to flow (3.5), we know that where |c|.Applying the maximum principle, we know that e λt (u − ψ) attains its nonnegative maximum value at the parabolic boundary, say (x 0 , t 0 ).That is, {0, e λt (u(x, t) − ψ(x))}.
with either x 0 ∈ ∂S n + or t 0 = 0.If x 0 ∈ ∂S n + , from the Hopf lemma, we have Here we denote ∇ and ∇ n as the tangential and normal part of ∇ on ∂S n + , e n = −∂ β is the inner normal vector field on ∂S n + .From the boundary condition in (3.5) we have a contradiction to the fact that function τ √ 1+s 2 +τ 2 is strictly increasing with respect to τ ∈ R and ∇ n u < ∇ n ψ at (x 0 , t 0 ).Hence we have t 0 = 0, which follows that Hence we obtain the desired upper bound of u.Similarly, we can get the desired lower bound of u.After the conformal transformation we finish the proof of the Proposition.
In order to obtain the gradient estimate, we need to employ the distance function d(x) := dist σ (x, ∂S n + ).It is well-known that d is well-defined and smooth for x near ∂S n + and ∇d = −∂ β on ∂S n + , where ∂ β is the unit outer normal vector field on ∂S n + .In the following, we extend d to be a smooth function on S n + and satisfying that We will use O(s) to denote terms that are bounded by Cs for a constant C > 0, which depends only on the C 0 norm of u.Our choice of test functions are motivated from [11], [14] and [22].Now we show the uniform gradient estimate.This is the key step of this paper.
where C is a constant depends only on the initial values and the covariant derivatives with respect to round metric σ on S n + .
Proof Define a function where K > 0 is the positive constant to be determined later.Assume that Φ attains its maximum value at (x 0 , t 0 ) ∈ S n + × [0, T ].We divide it into the following three cases to complete the proof.
Case 1: (x 0 , t 0 ) ∈ ∂S n + × [0, T ].At x 0 , we choose local coordinates such that ∂ ∂xn be the inner normal direction of ∂S n + , which is exactly equal to −∂ β and corresponds to ∇d.And let {x i } n−1 i=1 be the geodesic coordinate of x 0 ∈ ∂S n + .Along the geodesic x n = t (0 < t ≤ ε), one takes the parallel transport of tangential direction ∂ ∂xi (1 ≤ i ≤ n − 1) to establish the geodesic coordinate in the neighborhood around point x 0 in S n + .
First, we notice that Φ = v + cos θu n on the boundary ∂S n + from the boundary condition in (3.5).We denote ∇ u and u n the tangential and the normal part of ∇u on the boundary by our choice of coordinates above.The boundary condition u n = − cos θv implies that in other words, Moreover we have From the Gauss-Weingarten equation we have where b ij := σ(∇ ei e n , e i ) = 0 is the second fundamental form of ∂S n + in S n + for 1 ≤ i, j ≤ n − 1.Then at x 0 ∈ ∂S n + , from the Hopf lemma, it implies that Since {∂ xi } n−1 i=1 are the tangential vector fields on ∂S n + , for 1 ≤ i ≤ n − 1, we have that

This implies that
By differentiating the boundary condition of (3.5) and combining with (4.3) we have that Since | cos θ| < 1, we get for some universial positive constant C 1 .By choosing K large enough, say K := 2C 1 , we get a contradiction.So Case 1 is impossible.
Case 2: (x 0 , t 0 ) ∈ S n + × {0}.In this case we have It yields that sup where C is a positive constant depending only on n and u 0 .
We may also assume that u 1 (x 0 , t 0 ) large enough in the below computation, such that u 1 , v = 1 + u 2 1 , and Φ = (1 + Kd)v + u 1 d 1 cos θ are equivalent to each other at (x 0 , t 0 ).Otherwise, we have completed the proof.All the computation below are done at the point (x 0 , t 0 ).
First it is easy to see and Denote S := (1 + Kd) u1 v + cos θd 1 .It is easy to check that 2 + K ≥ S ≥ C(δ, θ) > 0 if we assume that u 1 ≥ δ > 0, otherwise we have obtained the estimate.Equation (4.8) yields that Substituting equation (4.10) into equation (4.9), we conclude that On the other hand, we have 0 Next we carefully handle these six terms one by one.Differentiating the main equation in (3.5), we get Combining with the communicative formula on S n we have ρe w v 2 := J 11 + J 12 + J 13 + J 14 .Now we tackle the above terms one by one.First, by using equation (4.11), we obtain that It is also not difficult to show that J . J 12 will be considered later, together with J 22 and J 32 , and J 13 with J 23 .See below.For the term J 3 , we have From equation (4.13), we deduce that For these terms, we first notice that J . Furthermore, we get by using the arithmetic-geometric inequality Before continuing, we fix a constant b 0 ∈ (| cos θ|, 3n+1 5n−1 ), for | cos θ| < 3n+1 5n−1 .If which implies that u 1 is uniformly bounded.Therefore, we may assume that Hence we conclude that u 1 ≤ C.
We have completed the proof.
Remark 4.4 We remark that the condition | cos θ| ≤ b 0 < 3n+1 5n−1 was only used in the estimate of term J 13 + J 23 + J 12 + J 22 + J 32 .And the main dominating term is J 13 , which ensures us to obtain the gradient estimate under this contact angle range.
The higher order a priori estimates of u follow from the uniform C 0 and C 1 estimates.Denote j(p) := σ(p, ∂ β − cos θ 1 + |p| 2 for p ∈ R n .It is easy to see that which means that we have a uniformly oblique boundary condition.To be more precise, from the classical parabolic theory for quasi-linear parabolic equations (See [23] for instance), it follows that Therefore any convergent subsequence of x(•, t) must converge to a spherical cap as t → +∞.Moreover the capillary boundary condition implies that this spherical caps intersects with the sphere at a contact angle θ.Hence it should belong to the family given in Remark 4.1.Hence we have completed the proof of our main theorem.

Theorem 1 . 1
If the initial hypersurface is a star-shaped hypersurface with capillary boundary in the unit ball and the contact angle θ satisfies | cos θ| < 3n+1 5n−1 , then flow (1.1) exists globally with uniform C ∞ -estimates.Moreover, x(•, t) subsequently converges to a spherical cap in the C ∞ topology as t → ∞, whose enclosed domain has the same volume as the domain enclosed by Σ 0 .

Proposition 4 . 5
If u(•, t) solves the initial boundary value problem (3.5) on interval [0, T * ) for T * ∈ (0, ∞] with | cos θ| < 3n+1 5n−1 , then for any 0 < T < T * , we haveu(•, t) C k ≤ C, 0 ≤ t ≤ T,where C is a positive constant only depends on k, and the initial values and the covariant derivatives with respect to the round metric on S n + .It follows, in particular, T * = ∞.Proof (Proof of Theorem 1.1) We only need to show that each subsequential limit is a spherical cap.As is shown in the proof of Proposition 2.4, integrating the equation (2.11) over t ∈ [0, +∞) and combining with Proposition 4.5, we have that∞ 0 S n + i<j |κ i − κ j | 2 dµ(y)dt ≤ C,where κ i (y, t) is the principal curvature of radial graph at (y, t) ∈ S n + × [0, ∞).Due to the uniform estimates from Proposition 4.5, one can show that lim t→∞ |κ i − κ j | 2 = 0, ∀1 ≤ i, j ≤ n.