欧氏空间超曲面曲率流的Harnack不等式研究
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摘要
本文分成三章。第一章中我们得到平均曲率流中局部Harnack不等式和nonconic估计。这个估计也许可以应用于平均曲率流的手术。第二章我们研究了欧氏空间中H~k流中的Harnack不等式。在第三章中，我们应用H~k流的Harnack不等式，得到了H~k流第Ⅱ类和第Ⅲ类奇点分别是translating soliton和扩张soliton的定理。
·平均曲率流的局部Harnack不等式
首先我们介绍一下在欧氏空间中超曲面的平均曲率流的定义和历史。
设M~n是一个黎曼流形。是一个R~(n+1)中浸入超曲而。若是一族单参数光滑超曲面的浸入映射，且满足方程：
其中v是超曲面的内法向量。则我们称(M，g(t))是平均曲率流的一个解。
平均曲率流的研究始于Huisken[22]。在这篇文章中，他证明在欧氏空间R~(n+1)中的闭超曲面能够在平均曲率流下收缩成一个点，并且这一点是一个圆形的球点。继而他在[23]中考虑了在一般黎曼流形N~(n+1)中的超曲面M~n中的平均曲率流。在[24]中研究了欧氏空间中平均曲率流的奇点的渐进性质，并对奇点进行了分类。在文[25]中，Huisken研究得到了外围空间是一般黎曼流形的第一类奇点的渐进自相似性质。最近在[21]中Huisken与Sinestrari对2-凸的欧氏空间超曲面进行了平均曲率流下的手术，得到了2-凸超曲面的拓扑分类。
正如Ricci流解决了Poincaré猜想，平均曲率流有望解决有Schoenflies猜想。下面陈述一下Schoenflies猜想。
猜想1(Schoenflies猜想)在欧氏空间R~(n+1)中，n≥3，任何微分同胚于S~n光滑的嵌入闭超曲面所包围的区域一定微分同胚于嵌入的欧氏空间R~(n+1)中标准单位球B~(n+1)。
在[21]中，Huisken与Sinestrari得到了关于2-凸超曲面的Scheonflies型定理。
定N1([21]中推论1．3)在欧氏空间R~(n+1)中，n≥3，任何单连通嵌入的n维2-凸闭超曲面一定微分同胚于S~n，并且其包围的区域一定微分同胚于嵌入的欧氏空间R~(n+1)中标准单位球B~(n+1)。
在[21]中，Huisken与Sinestrari运用了Hamilton最近提出的一个用来证明邻域正则性的思想完成了对2-凸超曲面的手术，从而得到了定理1。而在第一章中，我们将运用Hamilton的这一思想得到一个关于平均曲率流的平均曲率导数的估计，我们称之为局部Harnack估计或者nonconic估计。在这里，我们并没有对超曲面进行2-凸的限制，而是对第二基本型进行了另一种pinching的限制。因此，我们的估计也许可以应用于一般情形超曲面的手术和Schoenflies猜想的证明。
设U是M~n上的一个连通开集，且在U×[0，t_0]，t_0 在第一章中，我们令如下曲率条件为(
)：
则我们有如下定理：
定理1．3(平均曲率流的局部Harnack估计)：如果在B_R(O，t)×[0，R~2]上面满足曲率条件(
)，则在(?)(x，t)∈B_(R/2)(O，t)×[0，R~2]，(?)V∈T_pM~n，我们可以找到一个只与n和C_0有关的正常数B，使局部Harnack不等式成立：
定理1．4(平均曲率流中的nonconic估计)在定理1．3相同的条件下，在点(O，R~2)，我们有如下平均曲率的导数估计：其中C只依赖于n和C_0。
·H~k流的Harnack不等式
第二章中我们讨论了欧氏空间中的H~k流上的Harnack不等式，并且到了一些推论。
我们首先来介绍一下H~k流的定义。
设M~n是一个黎曼流形，是一个R~(n+1)中浸入超曲面。若是一族单参数光滑超曲面的浸入映射，且满足方程：
其中v是超曲面的内法向量。则我们称上述浸入映射F(·，t)是一个H~k流的解。
F．Schulze在[30]，仔细的研究了H~k流的短时间存在性，得到了如下结论
定N2．1([30])：令是一个光滑的浸入映射，并且H(F_0(M~n))>0。则存在唯一以F_0为初值的光滑H~k流解，存在时间为[0，T)，其中T是极大存在时间。对于k≥1，我们有T≥C(k，n)~(-1)(max_(p∈M)|A|(p，0))~(-(k+1))。并且在
1)当0<后<1时，F_0(M~n)是严格凸的；
2)当k≥1时，F_0(M~n)是弱凸的；
则曲面F(M~n，t)在t>0时是严格凸的，并且当t→T时，收缩成R~(n+1)中的一点。
在第二章中，M_t代表H~k流的解，当t在解存在时间[0，T)内。在第二章和第三章中，我们假设H~k流的解满足以下条件。这里A是M_t的第二基本形式。基于以上的假设，我们得到了H~k流的Harnack不等式。
定理2．4对任意满足条件(
)的H~k流的严格凸解，当t>0时，我们有其中V为任意的切向量。
这就是H~k流的微分Harnack不等式。我们可以通过在时空上的道路积分得到积分形式的Harnack不等式。
推论2．1对任意满足条件(
)的H~k流的严格凸解，t>0。对任意的0 推论2．2(tH~(k+1)的递增性)．如果M(x，t)是满足条件(
)的H~k流的严格凸解，对任意两个时刻0 推论2．3如果M(x，t)是在(-∞，0]定义的满足条件(
)的H~k流的严格凸古典解，则
·H~k流的奇点模型分类
第三章中，我们利用第二章的Harnack不等式，证明了H~k流第Ⅱ类和第Ⅲ类奇点分别是translating soliton和扩张soliton的定理。
在H~k流中，盛为民和吴超在[32]在紧流形的情况下，得到了第Ⅰ类奇点的结构定理。并且通过Blow up argument指出并第Ⅱ类奇点可以得到一族解，他们的极限收敛于一个永恒解，即满足下面定义3．4。也就是说，这是第Ⅱ类奇点的另外一种方式的定义。
我们现在给出H~k流translating soliton，H~k流扩张soliton以及H~k流的第Ⅱ类奇点和第Ⅲ类奇点的定义。
定义3．2 H~k-translating soliton是H~k流的一个解，同时其度量g(t)又是在M~n某个光滑向量场V的李导数下运动。即即满足
定义3．3 H~k扩张soliton是H~k流的一个解，同时其度量g(t)一边沿着M~n某个光滑向量场V的李导数下运动，一边以1/(k+1)tgij的速率在膨胀。即即满足
定义3．4若(M，g(t))是H~k流的一个定义在-∞ 定义3．5若(M，g(t))是H~k流的一个定义在0 定理3．2：若M是单连通的n维非紧完备流形。则任何H~k流的第Ⅱ类奇点(M，g(t))，且满足曲率条件(
)，则一定是translating soliton。
定理3．3若M是单连通的n维非紧完备流形。则任何H~k流的第Ⅲ类奇点(M，g(t))，且满足曲率条件(
)，则一定是扩张梯度soliton。
This article is separated into three chapters. In chapter one, we get the local Harnack estimate and so called nonconic estimate in mean curvature flow in Euclidean space. The estimate maybe can be used in the surgery of the mean curvature flow. In chapter two, we study the Harnack inequality of H~k-flow in Euclidean space. In chapter three, we apply the Harnack estimate to claim that typeⅡand typeⅢsigularity modle of H~k-flow are H~k-translating soliton and H~k-expending soliton respectively.
·Local Harnack estimate of mean curvature flow
First, we should introduce the definition and history of mean curvature flow in Euclidean space.
Let M~n is a Riemannian manifold. Setis a hypersurface immersion into R~(n+1). Ifis a one-parameter family of smooth hypersurface immersions, and satisfies the equation and initial data:
where v is the inward unit normal respectively, then we say F(·,t) is a solution of mean curvature flow.
Mean curvature flow was first studied by Huisken[22]. In that article, he proved that the convex closed hypersurface flowed by mean curvature flow in Euclidean space R~(n+1) will contract to a round point. And he considered the mean curvature flow of hypersurface in Riemannian manifolds in [23]. In [24] he study the asymptotic behavior for singularities of the mean curvature flow in Euclidean space, and classified the singularities. In [25],Huisken study and conclude that the singularity of mean curvature in Riemannian manifold is asymptotic self-similar. Recently, Huisken and Sinestrari [21] studied the two-convex hypersurfaces in R~(n+1), and do surgery on it, then they conclude the topology of all two-convex surfaces.
As Ricci flow has solved the Poincaréconjecture, mean curvature flow was expected to solve the Schoenflies conjecture. Now we introduce the Schoenflies conjecture first.
Schoenflies Conjecture: If an embedded closed hypersurface M~n (?) R~(n+1) with n≥3 is diffeomorphic to S~n, then it bounds a region whose closure is diffeomorphic to smoothly embedded (n + 1)-dimensional standard closed ball.
In [21], Huisken and Sinestrari get a Scheonflies type theorem about 2-convex surfaces.
Theorem 1 (Corollary 1.3 in [21]) Any smooth closed simply connected n-dimensional two-convex embedded surface M (?) R~(n+1) with n≥3 is diffeomorphicto S~n, and bounds a region whose closure is diffeomorphic to a smoothly embedded (n + 1)-dimensional standard closed ball.
In order to use surgery properly, Huisken and Sinestrari use a sharp estimate initiated by Hamilton in Ricci flow. In chapter one, we will introduce the idea of Hamilton and used the idea to mean curvature flow, and get another estimate of derivatie of mean curvature. Such estimate is called local Harnack estimate or nonconic estimate. In our case, we won't assume the surface is two-convex, but add another pinching condition of second fundamental form which we will introduce later. It is worth to say, our estimate maybe can be use to approach the Schoenflies conjecture in general case.
Set U (?) M~n is a connected open set, and assume on U×[0,t_0],t_0 < T satisfies the curvature condition
Let O∈U, B_R(O, t) is the geodesic ball in M~n around O, R is the radius at time t, set 0≤R≤π/2(?)M, s.t. B_R(O,t) (?)(?) U. We set C_0 = MR. d_t(x) = d_t(x,O) is the distance function on M~n from x to O respect to metricg(t).
In chapter one, we set the curvature condition (
) as follow.
Then we have following theorems.
Theorem 1.3 (Local Harnack inequality of mean curvature flow): Ifon B_R(O,t)×[0,R~2] we have the curvature condition(
), then for (?)(x,t)∈B_(R/2)(O,t)×[0，R~2],(?)V∈T_pM~n,we can find a positive constant B depend onlyon n and C_0, s.t. the Local Harnack inequality holds:
Then we have the nonconic estimate as a corollary.
Theorem 1.4 (Nonconic estimate of mean curvature flow) Under the same condition of theorem 1.3, then on space-time point (O, R~2), we have following estimate.where C depends only on n and C_0。
·Harnack estimate for H~k-flow
In chapter two, we discuss the Harnack estimate of H~k-flow of hypersurface in R~(n+1), and get some corollaries.
We first introduce the definition of H~k-flow.
Let M~n be a smooth manifold without boundary, and letis a smooth immersion which is convex. Letbe a one-parameter family of smooth hypersurface immersions in R~(n+1). We say that it is a solution to H~k-flow ifwhere v is the inward unit normal respectively, then we say F(·,t) is a solution of H~k-flow.
F.Schulze in [30] studied the short time existence of H~k-flow, and get the following conclusion.
Theorem 2.1 ([30]): Setis a smooth immersion, where H(F_0(M~n)) > 0. Then there exists a unique, smooth solution of H~k-flow, finite time interval [0,T). For k≥1 we have the bound T≥C(k,n)~(-1)(max_(p∈M~n)|A|(p,0))~(-(k+1)). In the case that
1) F_0(M~n) is strictly convex for 0 < k < 1,
2) F_0(M~n) is weakly convex for k≥1,
then the surfaces F(M~n, t) are strictly convex for all t > 0 and they contract for t→T to a point in R~(n+1).
In chapter two, the solution of H~k-flow is noted by M_t,t∈[0, T). In chapter two and chapter three, we assume the solution of H~k-flow satisfies the following condition.
where A is the second fundamental form of M_t.
Based on the assumption above, we get the Harnack estimate of H~k-flow in chapter two.
Theorem 2.4 For any strictly convex solution under the condition (
) to H~k-flow for t > 0 we havefor any tangent vectors V.
This is the differential Harnack inequality for the H~k- flow. As usual we integrate it over paths in space-time to get an integral Harnack inequality.
Corollary 2.1 For any strictly convex solution under the condition (
) to the H~k-flow for t > 0. For any 0 < t_1 < t_2 and Y_1, Y_2∈M, we havewhich△=d~2(Y_1,Y_2,t_1)/α(t_2-t_1),α=min_(x∈γ,t_1 Corollary 2.2(Nondecreasing of tH~(k+1)) If M(x,t) is strictly convex solution under the condition (
) to H~k-flow, for any two times 0 < t_1≤t_2. And (?)x∈M at t_2, we have:which x at t_1 evolves normally to x at t_2.
Corollary 2.3 If a strictly convex solution under the condition (
) to H~k-flowexists in (-∞,0], then
·Classification of singularity models of H~k-flow
In chapter three, we use the Harnack estimate which we get in chapter two to prove that the typeⅡand typeⅢsingularity model of H~k-flow is H~k-translating soliton and H~k-expending soliton respectively.
In the case of H~k, Sheng and Wu consider the compact manifolds, and obtain some property for typeⅠsingularity. Moreover, by blow up argument, they show that the typeⅡsingularity can derive a eternal solution, i.e. definition 3.4. Hence we can consider the eternal solution as another approach to define typeⅡsingularity.
Definition 3.2 H~k-translating soliton is a solution of H~k-flow which moves on the direction of some smooth vector field V on M~n. It satisfies the equation
Definition 3.3 H~k-expanding soliton is a solution of H~k-flow which moves on the direction of some smooth vector field V on M~n and expanding by the rate of 1/(k+1)tgij.It satisfies the equation
Definition 3.4 If (M,g(t)) is a strictly convex complete solution of H~k-flow defined on -∞≤t≤∞, and the mean curvature attains its maximum at some point (x_0,t_0), then we (M,g(t)) is the typeⅡsingularity model of H~k-fiow.
Definition 3.5 If (M,g(t)) is a strictly convex complete solution of H~k-flow defined on 0≤t≤∞, and the tH~(k+1) attains its maximum at some point (x_0,t_0), then we (M,g(t)) is the typeⅢsingularity model of H~k-flow.
Theorem 3.2: If M is a simply connected complete n dimensional manifold, then any typeⅡsingularity model of H~k-flow, which satisfies the condition (
) must be the H~k-translating soliton.
Theorem 3.3: If M is a simply connected complete n dimensional manifold, then any typeⅢsingularity model of H~k-flow, which satisfies the condition (
) must be the H~k-expanding soliton.
·平均曲率流的局部Harnack不等式
首先我们介绍一下在欧氏空间中超曲面的平均曲率流的定义和历史。
设M~n是一个黎曼流形。是一个R~(n+1)中浸入超曲而。若是一族单参数光滑超曲面的浸入映射，且满足方程：
其中v是超曲面的内法向量。则我们称(M，g(t))是平均曲率流的一个解。
平均曲率流的研究始于Huisken[22]。在这篇文章中，他证明在欧氏空间R~(n+1)中的闭超曲面能够在平均曲率流下收缩成一个点，并且这一点是一个圆形的球点。继而他在[23]中考虑了在一般黎曼流形N~(n+1)中的超曲面M~n中的平均曲率流。在[24]中研究了欧氏空间中平均曲率流的奇点的渐进性质，并对奇点进行了分类。在文[25]中，Huisken研究得到了外围空间是一般黎曼流形的第一类奇点的渐进自相似性质。最近在[21]中Huisken与Sinestrari对2-凸的欧氏空间超曲面进行了平均曲率流下的手术，得到了2-凸超曲面的拓扑分类。
正如Ricci流解决了Poincaré猜想，平均曲率流有望解决有Schoenflies猜想。下面陈述一下Schoenflies猜想。
猜想1(Schoenflies猜想)在欧氏空间R~(n+1)中，n≥3，任何微分同胚于S~n光滑的嵌入闭超曲面所包围的区域一定微分同胚于嵌入的欧氏空间R~(n+1)中标准单位球B~(n+1)。
在[21]中，Huisken与Sinestrari得到了关于2-凸超曲面的Scheonflies型定理。
定N1([21]中推论1．3)在欧氏空间R~(n+1)中，n≥3，任何单连通嵌入的n维2-凸闭超曲面一定微分同胚于S~n，并且其包围的区域一定微分同胚于嵌入的欧氏空间R~(n+1)中标准单位球B~(n+1)。
在[21]中，Huisken与Sinestrari运用了Hamilton最近提出的一个用来证明邻域正则性的思想完成了对2-凸超曲面的手术，从而得到了定理1。而在第一章中，我们将运用Hamilton的这一思想得到一个关于平均曲率流的平均曲率导数的估计，我们称之为局部Harnack估计或者nonconic估计。在这里，我们并没有对超曲面进行2-凸的限制，而是对第二基本型进行了另一种pinching的限制。因此，我们的估计也许可以应用于一般情形超曲面的手术和Schoenflies猜想的证明。
设U是M~n上的一个连通开集，且在U×[0，t_0]，t_0
)：
则我们有如下定理：
定理1．3(平均曲率流的局部Harnack估计)：如果在B_R(O，t)×[0，R~2]上面满足曲率条件(
)，则在(?)(x，t)∈B_(R/2)(O，t)×[0，R~2]，(?)V∈T_pM~n，我们可以找到一个只与n和C_0有关的正常数B，使局部Harnack不等式成立：
定理1．4(平均曲率流中的nonconic估计)在定理1．3相同的条件下，在点(O，R~2)，我们有如下平均曲率的导数估计：其中C只依赖于n和C_0。
·H~k流的Harnack不等式
第二章中我们讨论了欧氏空间中的H~k流上的Harnack不等式，并且到了一些推论。
我们首先来介绍一下H~k流的定义。
设M~n是一个黎曼流形，是一个R~(n+1)中浸入超曲面。若是一族单参数光滑超曲面的浸入映射，且满足方程：
其中v是超曲面的内法向量。则我们称上述浸入映射F(·，t)是一个H~k流的解。
F．Schulze在[30]，仔细的研究了H~k流的短时间存在性，得到了如下结论
定N2．1([30])：令是一个光滑的浸入映射，并且H(F_0(M~n))>0。则存在唯一以F_0为初值的光滑H~k流解，存在时间为[0，T)，其中T是极大存在时间。对于k≥1，我们有T≥C(k，n)~(-1)(max_(p∈M)|A|(p，0))~(-(k+1))。并且在
1)当0<后<1时，F_0(M~n)是严格凸的；
2)当k≥1时，F_0(M~n)是弱凸的；
则曲面F(M~n，t)在t>0时是严格凸的，并且当t→T时，收缩成R~(n+1)中的一点。
在第二章中，M_t代表H~k流的解，当t在解存在时间[0，T)内。在第二章和第三章中，我们假设H~k流的解满足以下条件。这里A是M_t的第二基本形式。基于以上的假设，我们得到了H~k流的Harnack不等式。
定理2．4对任意满足条件(
)的H~k流的严格凸解，当t>0时，我们有其中V为任意的切向量。
这就是H~k流的微分Harnack不等式。我们可以通过在时空上的道路积分得到积分形式的Harnack不等式。
推论2．1对任意满足条件(
)的H~k流的严格凸解，t>0。对任意的0
)的H~k流的严格凸解，对任意两个时刻0
)的H~k流的严格凸古典解，则
·H~k流的奇点模型分类
第三章中，我们利用第二章的Harnack不等式，证明了H~k流第Ⅱ类和第Ⅲ类奇点分别是translating soliton和扩张soliton的定理。
在H~k流中，盛为民和吴超在[32]在紧流形的情况下，得到了第Ⅰ类奇点的结构定理。并且通过Blow up argument指出并第Ⅱ类奇点可以得到一族解，他们的极限收敛于一个永恒解，即满足下面定义3．4。也就是说，这是第Ⅱ类奇点的另外一种方式的定义。
我们现在给出H~k流translating soliton，H~k流扩张soliton以及H~k流的第Ⅱ类奇点和第Ⅲ类奇点的定义。
定义3．2 H~k-translating soliton是H~k流的一个解，同时其度量g(t)又是在M~n某个光滑向量场V的李导数下运动。即即满足
定义3．3 H~k扩张soliton是H~k流的一个解，同时其度量g(t)一边沿着M~n某个光滑向量场V的李导数下运动，一边以1/(k+1)tgij的速率在膨胀。即即满足
定义3．4若(M，g(t))是H~k流的一个定义在-∞
)，则一定是translating soliton。
定理3．3若M是单连通的n维非紧完备流形。则任何H~k流的第Ⅲ类奇点(M，g(t))，且满足曲率条件(
)，则一定是扩张梯度soliton。
This article is separated into three chapters. In chapter one, we get the local Harnack estimate and so called nonconic estimate in mean curvature flow in Euclidean space. The estimate maybe can be used in the surgery of the mean curvature flow. In chapter two, we study the Harnack inequality of H~k-flow in Euclidean space. In chapter three, we apply the Harnack estimate to claim that typeⅡand typeⅢsigularity modle of H~k-flow are H~k-translating soliton and H~k-expending soliton respectively.
·Local Harnack estimate of mean curvature flow
First, we should introduce the definition and history of mean curvature flow in Euclidean space.
Let M~n is a Riemannian manifold. Setis a hypersurface immersion into R~(n+1). Ifis a one-parameter family of smooth hypersurface immersions, and satisfies the equation and initial data:
where v is the inward unit normal respectively, then we say F(·,t) is a solution of mean curvature flow.
Mean curvature flow was first studied by Huisken[22]. In that article, he proved that the convex closed hypersurface flowed by mean curvature flow in Euclidean space R~(n+1) will contract to a round point. And he considered the mean curvature flow of hypersurface in Riemannian manifolds in [23]. In [24] he study the asymptotic behavior for singularities of the mean curvature flow in Euclidean space, and classified the singularities. In [25],Huisken study and conclude that the singularity of mean curvature in Riemannian manifold is asymptotic self-similar. Recently, Huisken and Sinestrari [21] studied the two-convex hypersurfaces in R~(n+1), and do surgery on it, then they conclude the topology of all two-convex surfaces.
As Ricci flow has solved the Poincaréconjecture, mean curvature flow was expected to solve the Schoenflies conjecture. Now we introduce the Schoenflies conjecture first.
Schoenflies Conjecture: If an embedded closed hypersurface M~n (?) R~(n+1) with n≥3 is diffeomorphic to S~n, then it bounds a region whose closure is diffeomorphic to smoothly embedded (n + 1)-dimensional standard closed ball.
In [21], Huisken and Sinestrari get a Scheonflies type theorem about 2-convex surfaces.
Theorem 1 (Corollary 1.3 in [21]) Any smooth closed simply connected n-dimensional two-convex embedded surface M (?) R~(n+1) with n≥3 is diffeomorphicto S~n, and bounds a region whose closure is diffeomorphic to a smoothly embedded (n + 1)-dimensional standard closed ball.
In order to use surgery properly, Huisken and Sinestrari use a sharp estimate initiated by Hamilton in Ricci flow. In chapter one, we will introduce the idea of Hamilton and used the idea to mean curvature flow, and get another estimate of derivatie of mean curvature. Such estimate is called local Harnack estimate or nonconic estimate. In our case, we won't assume the surface is two-convex, but add another pinching condition of second fundamental form which we will introduce later. It is worth to say, our estimate maybe can be use to approach the Schoenflies conjecture in general case.
Set U (?) M~n is a connected open set, and assume on U×[0,t_0],t_0 < T satisfies the curvature condition
Let O∈U, B_R(O, t) is the geodesic ball in M~n around O, R is the radius at time t, set 0≤R≤π/2(?)M, s.t. B_R(O,t) (?)(?) U. We set C_0 = MR. d_t(x) = d_t(x,O) is the distance function on M~n from x to O respect to metricg(t).
In chapter one, we set the curvature condition (
) as follow.
Then we have following theorems.
Theorem 1.3 (Local Harnack inequality of mean curvature flow): Ifon B_R(O,t)×[0,R~2] we have the curvature condition(
), then for (?)(x,t)∈B_(R/2)(O,t)×[0，R~2],(?)V∈T_pM~n,we can find a positive constant B depend onlyon n and C_0, s.t. the Local Harnack inequality holds:
Then we have the nonconic estimate as a corollary.
Theorem 1.4 (Nonconic estimate of mean curvature flow) Under the same condition of theorem 1.3, then on space-time point (O, R~2), we have following estimate.where C depends only on n and C_0。
·Harnack estimate for H~k-flow
In chapter two, we discuss the Harnack estimate of H~k-flow of hypersurface in R~(n+1), and get some corollaries.
We first introduce the definition of H~k-flow.
Let M~n be a smooth manifold without boundary, and letis a smooth immersion which is convex. Letbe a one-parameter family of smooth hypersurface immersions in R~(n+1). We say that it is a solution to H~k-flow ifwhere v is the inward unit normal respectively, then we say F(·,t) is a solution of H~k-flow.
F.Schulze in [30] studied the short time existence of H~k-flow, and get the following conclusion.
Theorem 2.1 ([30]): Setis a smooth immersion, where H(F_0(M~n)) > 0. Then there exists a unique, smooth solution of H~k-flow, finite time interval [0,T). For k≥1 we have the bound T≥C(k,n)~(-1)(max_(p∈M~n)|A|(p,0))~(-(k+1)). In the case that
1) F_0(M~n) is strictly convex for 0 < k < 1,
2) F_0(M~n) is weakly convex for k≥1,
then the surfaces F(M~n, t) are strictly convex for all t > 0 and they contract for t→T to a point in R~(n+1).
In chapter two, the solution of H~k-flow is noted by M_t,t∈[0, T). In chapter two and chapter three, we assume the solution of H~k-flow satisfies the following condition.
where A is the second fundamental form of M_t.
Based on the assumption above, we get the Harnack estimate of H~k-flow in chapter two.
Theorem 2.4 For any strictly convex solution under the condition (
) to H~k-flow for t > 0 we havefor any tangent vectors V.
This is the differential Harnack inequality for the H~k- flow. As usual we integrate it over paths in space-time to get an integral Harnack inequality.
Corollary 2.1 For any strictly convex solution under the condition (
) to the H~k-flow for t > 0. For any 0 < t_1 < t_2 and Y_1, Y_2∈M, we havewhich△=d~2(Y_1,Y_2,t_1)/α(t_2-t_1),α=min_(x∈γ,t_1
) to H~k-flow, for any two times 0 < t_1≤t_2. And (?)x∈M at t_2, we have:which x at t_1 evolves normally to x at t_2.
Corollary 2.3 If a strictly convex solution under the condition (
) to H~k-flowexists in (-∞,0], then
·Classification of singularity models of H~k-flow
In chapter three, we use the Harnack estimate which we get in chapter two to prove that the typeⅡand typeⅢsingularity model of H~k-flow is H~k-translating soliton and H~k-expending soliton respectively.
In the case of H~k, Sheng and Wu consider the compact manifolds, and obtain some property for typeⅠsingularity. Moreover, by blow up argument, they show that the typeⅡsingularity can derive a eternal solution, i.e. definition 3.4. Hence we can consider the eternal solution as another approach to define typeⅡsingularity.
Definition 3.2 H~k-translating soliton is a solution of H~k-flow which moves on the direction of some smooth vector field V on M~n. It satisfies the equation
Definition 3.3 H~k-expanding soliton is a solution of H~k-flow which moves on the direction of some smooth vector field V on M~n and expanding by the rate of 1/(k+1)tgij.It satisfies the equation
Definition 3.4 If (M,g(t)) is a strictly convex complete solution of H~k-flow defined on -∞≤t≤∞, and the mean curvature attains its maximum at some point (x_0,t_0), then we (M,g(t)) is the typeⅡsingularity model of H~k-fiow.
Definition 3.5 If (M,g(t)) is a strictly convex complete solution of H~k-flow defined on 0≤t≤∞, and the tH~(k+1) attains its maximum at some point (x_0,t_0), then we (M,g(t)) is the typeⅢsingularity model of H~k-flow.
Theorem 3.2: If M is a simply connected complete n dimensional manifold, then any typeⅡsingularity model of H~k-flow, which satisfies the condition (
) must be the H~k-translating soliton.
Theorem 3.3: If M is a simply connected complete n dimensional manifold, then any typeⅢsingularity model of H~k-flow, which satisfies the condition (
) must be the H~k-expanding soliton.
引文
[1] Ben.Andrews, Harnack inequalities for evolving hypersurfaces, Math.Z. 215, 179-197(1994)
[2] S. Brendle, A generalization of Hamilton's Harnack inequality for the Ricci flow, preprint
[3] Cao,H.D., On Harnack's inequalities for the Kahler-Ricci flow, Invent. Math., 109(1992), 247-263
[4] Huai-Dong Cao, Limits of solutions to the Kahler-Ricci Flow, J. Diff. Geom. 45 (1997) 257-272
[5]陈兵龙，平均曲率流的第Ⅲ类奇点，中山大学学报，Vol．39，No．2，Mar.2000．
[6] Ben.Chow, On Harnack's inequality and entropy for the Gaussian curvature flow, Comm.Pure Appl.Math. 44, (1991) 469-483.
[7] Ben.Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm.Pure Appl.Math. 45, (1992)1003-1014
[8] Huai-Dong Cao, Xi-Ping Zhu, A complete proof of the Poincare and Geometrizationconjectures- Application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. Vol. 10, No. 2, pp. 165-492, June 2006.
[9] R.Hamilton, Three-Manifolds with Positive Ricci Curvature, J.Differential Geom., 17, (1982), pp.255-306
[10] R.Hamilton, Four-manifolds with positive curvature operator, J.Differential Geom., 24 (1986), pp.153-179.
[11] R.Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry, 2, pp. 7-136, International Press, 1995
[12] R.Hamilton, Curvature and Volume Bounds II, preprint
[13] R.Hamilton, The Harnack estimate for the mean curvature flow, J.Diff.Geom. 41, (1995) 215-226
[14] R.Hamilton, A matrix Harnack estimate for the heat equation, Comm.Anal.Geom. 1, (1993) 113-126
[15] R.Hamilton, The Ricci flow on surfaces, Contemp.Math. 71, (1988)237-262, Amer.Math.Soc, Providence, RI.
[16] R.Hamilton, The Harnack estimate for the ricci flow, 250-265, Collected papers on Ricci Flow by Cao, Chow
[17] R. S. Hamilton, Eternal solutions to the Ricci flow, J. Diff. Geom. 38 (1993) 1-11.
[18] Klaus Ecker, Gerhard Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent.math.105, 547-569 (1991)
[19] G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Variations 8 (1999), 1-14.
[20] G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70.
[21] Gerhard Huisken, Carlo Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, preprint
[22] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237-266.
[23] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 84(1986), 463-480
[24] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom., 31(1990), 285-299
[25] G.HUisken, Local and global behavior of hypersurfaces moving by mean curvature, Proceedings of symposia in Pure Mathematics. 54(1993) 175-191
[26] P.Li and S.T.Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156, (1986) 153-201
[27] G Perelmann, The entropy formula for the Ricci flow and its geometric applications, arXiv:math. DG/0211159 vl November 11, 2002, preprint.
[28] G Perelmann, Ricci flow with surgery on three manifolds, arXiv:math.DG/0303109 vl March 10, 2003, preprint.
[29] Peter Petersen, Riemannian Geometry, Second Edition
[30] Felix.Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math.Z. 251, 721-733(2005)
[31] Weimin Sheng, Geometry of Complete Hypersurfaces Evolved by mean curvature flow, Chin. Ann. Math. (2003)
[32] Weimin Sheng, Chao Wu, On the Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow, preprint
[33] Schoen and Yau, Lectures on Differetial Geometry.
[2] S. Brendle, A generalization of Hamilton's Harnack inequality for the Ricci flow, preprint
[3] Cao,H.D., On Harnack's inequalities for the Kahler-Ricci flow, Invent. Math., 109(1992), 247-263
[4] Huai-Dong Cao, Limits of solutions to the Kahler-Ricci Flow, J. Diff. Geom. 45 (1997) 257-272
[5]陈兵龙，平均曲率流的第Ⅲ类奇点，中山大学学报，Vol．39，No．2，Mar.2000．
[6] Ben.Chow, On Harnack's inequality and entropy for the Gaussian curvature flow, Comm.Pure Appl.Math. 44, (1991) 469-483.
[7] Ben.Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm.Pure Appl.Math. 45, (1992)1003-1014
[8] Huai-Dong Cao, Xi-Ping Zhu, A complete proof of the Poincare and Geometrizationconjectures- Application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. Vol. 10, No. 2, pp. 165-492, June 2006.
[9] R.Hamilton, Three-Manifolds with Positive Ricci Curvature, J.Differential Geom., 17, (1982), pp.255-306
[10] R.Hamilton, Four-manifolds with positive curvature operator, J.Differential Geom., 24 (1986), pp.153-179.
[11] R.Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry, 2, pp. 7-136, International Press, 1995
[12] R.Hamilton, Curvature and Volume Bounds II, preprint
[13] R.Hamilton, The Harnack estimate for the mean curvature flow, J.Diff.Geom. 41, (1995) 215-226
[14] R.Hamilton, A matrix Harnack estimate for the heat equation, Comm.Anal.Geom. 1, (1993) 113-126
[15] R.Hamilton, The Ricci flow on surfaces, Contemp.Math. 71, (1988)237-262, Amer.Math.Soc, Providence, RI.
[16] R.Hamilton, The Harnack estimate for the ricci flow, 250-265, Collected papers on Ricci Flow by Cao, Chow
[17] R. S. Hamilton, Eternal solutions to the Ricci flow, J. Diff. Geom. 38 (1993) 1-11.
[18] Klaus Ecker, Gerhard Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent.math.105, 547-569 (1991)
[19] G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Variations 8 (1999), 1-14.
[20] G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70.
[21] Gerhard Huisken, Carlo Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, preprint
[22] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237-266.
[23] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 84(1986), 463-480
[24] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom., 31(1990), 285-299
[25] G.HUisken, Local and global behavior of hypersurfaces moving by mean curvature, Proceedings of symposia in Pure Mathematics. 54(1993) 175-191
[26] P.Li and S.T.Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156, (1986) 153-201
[27] G Perelmann, The entropy formula for the Ricci flow and its geometric applications, arXiv:math. DG/0211159 vl November 11, 2002, preprint.
[28] G Perelmann, Ricci flow with surgery on three manifolds, arXiv:math.DG/0303109 vl March 10, 2003, preprint.
[29] Peter Petersen, Riemannian Geometry, Second Edition
[30] Felix.Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math.Z. 251, 721-733(2005)
[31] Weimin Sheng, Geometry of Complete Hypersurfaces Evolved by mean curvature flow, Chin. Ann. Math. (2003)
[32] Weimin Sheng, Chao Wu, On the Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow, preprint
[33] Schoen and Yau, Lectures on Differetial Geometry.