| 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 Hk-flow in Euclidean space. In chapter three, we apply the Harnack estimate to claim that typeâ…¡and typeâ…¢sigularity modle of Hk-flow are Hk-translating soliton and Hk-expending soliton respectively.·Local Harnack estimate of mean curvature flowFirst, we should introduce the definition and history of mean curvature flow in Euclidean space.Let Mn is a Riemannian manifold. Setis a hypersurface immersion into Rn+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 Rn+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 Rn+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 Mn (?) Rn+1 with n≥3 is diffeomorphic to Sn, 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 (?) Rn+1 with n≥3 is diffeomorphicto Sn, 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 (?) Mn is a connected open set, and assume on U×[0,t0],t0 < T satisfies the curvature condition Let O∈U, BR(O, t) is the geodesic ball in Mn around O, R is the radius at time t, set 0≤R≤π/2(?)M, s.t. BR(O,t) (?)(?) U. We set C0 = MR. dt(x) = dt(x,O) is the distance function on Mn 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 BR(O,t)×[0,R2] we have the curvature condition(), then for (?)(x,t)∈BR/2(O,t)×[0,R2],(?)V∈TpMn,we can find a positive constant B depend onlyon n and C0, 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, R2), we have following estimate.where C depends only on n and C0。·Harnack estimate for Hk-flow In chapter two, we discuss the Harnack estimate of Hk-flow of hypersurface in Rn+1, and get some corollaries.We first introduce the definition of Hk-flow.Let Mn be a smooth manifold without boundary, and letis a smooth immersion which is convex. Letbe a one-parameter family of smooth hypersurface immersions in Rn+1. We say that it is a solution to Hk-flow ifwhere v is the inward unit normal respectively, then we say F(·,t) is a solution of Hk-flow.F.Schulze in [30] studied the short time existence of Hk-flow, and get the following conclusion.Theorem 2.1 ([30]): Setis a smooth immersion, where H(F0(Mn)) > 0. Then there exists a unique, smooth solution of Hk-flow, finite time interval [0,T). For k≥1 we have the bound T≥C(k,n)-1(maxp∈Mn|A|(p,0))-(k+1). In the case that1) F0(Mn) is strictly convex for 0 < k < 1,2) F0(Mn) is weakly convex for k≥1,then the surfaces F(Mn, t) are strictly convex for all t > 0 and they contract for t→T to a point in Rn+1.In chapter two, the solution of Hk-flow is noted by Mt,t∈[0, T). In chapter two and chapter three, we assume the solution of Hk-flow satisfies the following condition. where A is the second fundamental form of Mt.Based on the assumption above, we get the Harnack estimate of Hk-flow in chapter two.Theorem 2.4 For any strictly convex solution under the condition () to Hk-flow for t > 0 we havefor any tangent vectors V.This is the differential Harnack inequality for the Hk- 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 Hk-flow for t > 0. For any 0 < t1 < t2 and Y1, Y2∈M, we havewhichâ–³=d2(Y1,Y2,t1)/α(t2-t1),α=minx∈γ,t12Hk-1(x,t),γis the geodesic line betweenY1 and Y2 at time t1, such that Y2 evolves normally to Y2 at t2, d(Y1, Y2,t1) is thelength ofγat t1.Corollary 2.2(Nondecreasing of tHk+1) If M(x,t) is strictly convex solution under the condition () to Hk-flow, for any two times 0 < t1≤t2. And (?)x∈M at t2, we have:which x at t1 evolves normally to x at t2.Corollary 2.3 If a strictly convex solution under the condition () to Hk-flowexists in (-∞,0], then·Classification of singularity models of Hk-flowIn chapter three, we use the Harnack estimate which we get in chapter two to prove that the typeâ…¡and typeâ…¢singularity model of Hk-flow is Hk-translating soliton and Hk-expending soliton respectively.In the case of Hk, Sheng and Wu consider the compact manifolds, and obtain some property for typeâ… singularity. Moreover, by blow up argument, they show that the typeâ…¡singularity can derive a eternal solution, i.e. definition 3.4. Hence we can consider the eternal solution as another approach to define typeâ…¡singularity.Definition 3.2 Hk-translating soliton is a solution of Hk-flow which moves on the direction of some smooth vector field V on Mn. It satisfies the equationDefinition 3.3 Hk-expanding soliton is a solution of Hk-flow which moves on the direction of some smooth vector field V on Mn and expanding by the rate of 1/(k+1)tgij.It satisfies the equationDefinition 3.4 If (M,g(t)) is a strictly convex complete solution of Hk-flow defined on -∞≤t≤∞, and the mean curvature attains its maximum at some point (x0,t0), then we (M,g(t)) is the typeâ…¡singularity model of Hk-fiow.Definition 3.5 If (M,g(t)) is a strictly convex complete solution of Hk-flow defined on 0≤t≤∞, and the tHk+1 attains its maximum at some point (x0,t0), then we (M,g(t)) is the typeâ…¢singularity model of Hk-flow.Theorem 3.2: If M is a simply connected complete n dimensional manifold, then any typeâ…¡singularity model of Hk-flow, which satisfies the condition () must be the Hk-translating soliton.Theorem 3.3: If M is a simply connected complete n dimensional manifold, then any typeâ…¢singularity model of Hk-flow, which satisfies the condition () must be the Hk-expanding soliton.
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