极小曲面与平均曲率流

MINIMAL HYPERSURFACES AND MEAN CURVATURE FLOWS LI MA Abstract. In this series of lectures, we discuss the minimal hypersur- faces in an Euclidean space and the mean curvature flows of hyper- surfaces. We introduce monotonicity formula a of normalized volume of minimal surfaces and the Huisken’s monotonicity of mean curvature flows. We shall study Simons equations, various curvature estimates, and Bernstein type theorems. We shall also present Chern’s results about the extension of a result of Heinz. ??? Then we will extend some local estimates of the curvature of Heinz-Chern into estimates for the curvature flows of graphs. Mathematics Subject Classification 2000: 53Jxx Keywords: positive solution, global existence, boundary blow-up 1. Introduction Minimal surfaces and mean curvature flows are important objects in geo- metric analysis. The study of these have a long history and I prefer to other book for the background. The initial goal of this article is for the mem- ory of 100 anniversary of the great geometer Prof. S.S.Chern and I want to write a summary of Chern’s work on geometry of the hypersurfaces in the Euclidean space via the use of the moving frame methods. As I want to let young graduate students understand these material, i have to change my original plan and write the paper as an elementary introduction to the minimal surfaces and mean curvature flows based on my previous lectures. The new plan of the paper is below. First, we review the moving frame method for hypersurfaces in Rn+1. Next, we recall the classical results in 2-d surfaces in R3. Then I introduce minimal surfaces and mean curvature flow. I present the Bernstein type theorem for the 2-d graphic minimal surfaces.I shall pay more attention to various estimates of the area and curvature of minimal surfaces and mean curvature flow. I also present the monotonicity formulae for both of them. I give the proof of Hamilton’s compactness theorem of the convex pinched hypersurface based on the mean curvature flow. In the last part of paper, we follow Chern’s method to derive some new stability results of the mean curvature flow of convex hypersurfaces in Rn+1. The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20090002110019. 1 2 LI MA 2. Chern’s work on hypersurfaces and minimal surfaces Prof. S.S.Chern studied many rigidity properties of hypersurfaces by using the moving frame method of Darboux and Cartan. I have read these papers of S.S.Chern about the uniqueness and rigidity of the hypersurfaces in the whole euclidean spaces. In this paper, I shall review some of his results and consider related extensions. We shall consider the moving frame method. We use the moving frame method to study the geometry of the hyper-surface in n+ 1 dimensional Euclidean space E. I should say that the geometrical objects in my mind are from conformal geometry and general relativity of Einstein and my question about how we can show a hypersurface in an Euclidean space have a positive Yamabe constant. I want to make a remark about a related method, which is the method of moving planes, which was invented by I.A.Alexandrov, who found the method of moving planes in the study of the rigidity problems of convex surfaces. Some years before, Profs. Chen, Li, and Ou (2005-2006) found an integral form of the moving plane method. Then my students and I tried to use it to study the symmetry properties of the stationary Schrodinger systems on the whole space. I also considered the question about how to make the method useful for elliptic problems on bounded domains. In recent years, i want to use heat flow method to study rigidity problems. Here is some story. I have studied Perelman’s papers about Ricci flow. Motivated by Perelman’s F-functional for the Ricci flow and optimal trans- port theory, we can extended Reilly formulae to weighted formulae. My interest in extending the Reilly formulae makes me read Chern’s papers again and again. Chern’s work mentioned above was also motivated by oth- ers. In two dimension, Heinz proved some interesting local results for graph surface in R3. S.S.Chern extended these results to higher dimensions. Reilly observed that Chern’s argument can be abstractly formulated as Reilly for- mulae, which can be used to give bounds of the first eigen-values of the Laplacian operators on riemannian manifolds. Moving frame method had been introduced by Darboux and E.Cartan. It is Chern S.S. making it a very powerful tool in the study of global differential geometry. We shall use this method to introduce the curvature of a surface in R3. Let M be a complete 2-d surface immersed into the 3-d Euclidean space. We have the following classical Hilbert-Liebman theorem. Theorem 1. When the curvature of M is a non-zero constant, M must be a sphere. We also have the Hartman-Nirenberg theorem. Theorem 2. When the curvature of M is zero, M must be a plane or a cylinder. The above results were extended to higher dimensions in a famous paper done by Cheng-Yau (Math.Ann./1977). I shall recall a heat flow method to this kind of geometric problem. MINIMAL HYPERSURFACES AND MEAN CURVATURE FLOWS 3 We now consider local method for surfaces. We recall the following result of Heinz (1955). Let Σ be a piece of the graph of the function z = z(x, y) defined on the disk x2 + y2 < r2. Theorem 3. We have three types of results. 1. If |H| ≥ c > 0, then r ≤ 1/c. 2. If K ≥ c > 0, then r ≤ 1/√c. 3. If K ≤ −c < 0, then r ≤ e√3/c. These results are extend to higher dimensions by S.S.Chern (1959) via the use of moving frame methods. We remark that for z = F (x1, ..., xn), x 2 1 + ...+ x 2 n ≤ r2, and a = (0, ...0, 1), we have that nVa = rLa. This then implies that for such a hypersurface with H ≥ c > 0, we have cr ≤ 1, i.e., r ≤ c−1. The most interesting case when scalar.curv. = s2 ≤ −c < 0 was also considered by Chern. We now recall one of Chern’s results below. Theorem 4. Let Σ be a compact hypersurface with boundary ∂Σ. Assume that s1 := H ≥ c > 0 and there exists a fixed unit vector a such that the angle made by a and any normal direction of Σ is less than pi2 . Then ncVa ≤ La where Va and La are the volume and area of the projections of Σ and ∂Σ in the hyperplane perpendicular to a respectively. 3. basic formulation of the moving frame method Here let me start from the elementary material. We recall the definition and basic properties of exterior forms on any open sets in Rn. The restriction of these beings to hypersurfaces is routine. Let (x1, ...xn) be the coordinates in R n. Define, for any vectors v =  v1 . . . vn  and w =  w1 . . . wn  , 4 LI MA the form dxi ∧ dxj(v, w) = det [ vi vj wi wj ] . Here we have used dxj(v) = vj ,etc. Then one can see that dxi ∧ dxj = −dxj ∧ dxi. For any w = ajdxj , define dw = daj ∧ dxj and we can define such an operation by induction. Then we can see that d(dw) = 0. Let me recall the classical Frenet formula for curves in R3 and show how to use the curvature and ODE to give characterization of curves in spaces. Let γ be the regular curve in R3 such that we can define the unit tangent vector T and principal normal N . Then we define B = T × N . In our notation above, we let e1 = T , e2 = N , and e3 = B. Then we have dei = ωijej for some ωij = −ωji. We define k = ω12(T ), the curvature of γ, and τ = ω23(T ) the torsion curvature of γ. We know that ω13 = 0 because de1 = ω12e2 from the definition of e2 from e1. Assume that the torsion curvature τ = 0. Note that ∇e1e3 := de3(e1) = ω32(e1)e2 = 0. This implies that γ is a planar curve. Assume further that k is a positive constant. Write γ = (x(s), y(s)) in the arc-length parameter. Then using e1 = (x ′, y′), e2 = (−y′, x′), and ∇e1e1 = ke2, we know that x ′′ = ky′, y ′′ = −kx′. Then we know that x′ = k(y + c) and y′ = −k(x+ c). Solve these we know that γ is a circle. 4. moving frame in higher dimensions We now give another useful definition of the second fundamental form on Σ by using the moving frames on Σ. Introduce ei, i = 1, ..., n, the local moving frames in Σ forming an or- thonormal basis at a region in Σ. We can find a unit normal vector en+1 to Σ such that {eA}, A = 1, ..., n, n+ 1, as a local orthonormal basis on Σ. Then the differential mapping dp can be written as dp = ωiei on Σ, where ωi’s are one-forms on in the region D. Since dp(ej) = ej = ωi(ej)ei, we have ωi(ej) = δij and then we may consider {ωi} as the dual frame to {ej} on Σ. Hence ωj ’s are linearly independent. We can extend {ωi} into linearly independent system {ωA}n+1A=1 such that ωn+1 is the dual to en+1. MINIMAL HYPERSURFACES AND MEAN CURVATURE FLOWS 5 Note that ωn+1 = 0 on Σ. Let I = ds2 = dp · dp = ∑ (ωj) 2 be the metric and let dv = ω1 ∧ ... ∧ ωn be the volume form. From the relation (eA, eB) = δAB, we know that there are one-forms ωAB = −ωBA, deA = ωABeB. From the relation d2 = 0 we know that 0 = dωABeB + deB ∧ ωAB. Then dωAB = ωAC ∧ ωCB. We shall call these the structure equations of ωAB. Restricted to Σ, we have dej = ωjBeB, den+1 = ωn+1kek. We may assume that dωi = µij ∧ ωj . Then we deine, for any tangent vector field X, ∇Xej = ωji(X)ei, ∇Xωi = µij(X)ωj Since ωi(ej) = δij , then we have ∇Xωi(ej) + ωi(∇Xej) = 0. This implies that µik(X)ωk(ej) + ωi(ωjk(X)ek) = 0 and then µjk(X) = ωjk(X). Furthermore, ∇Xωi = ωij(X)ωj and using ωn+1 = 0 (and dωn+1 = 0), ωn+1j ∧ ωj = 0. By Cartan’s lemma we have ωn+1j = hjkωk, hjk = hkj . Define the second fundamental form II = dp · den+1 = ωjωn+1j = hjkωjωk. 6 LI MA Following Gauss (see also [3]) we define the invariants sk by the relation det(λδij + hij) = ∑ Cknskλ n−k. We call sk the k-th curvature of Σ. In particular, for the diagonal matrix (hjk) = ( ⊕ λk), s1 = 1 n (λ1 + ...+ λn), s2 = ∑ i 0, ..., sk > 0}. Set s0 = 0 = sk for k > n. Let sk,j = sk|λj=0. Then ∂λjsk = sk−1,j . We have the following properties sk = sk,j − λjsk−1,j ,∑ j sk,j = (n− k)sk, sk−1,n ≥ ... ≥ sk−1,j > 0, λk ≥ 0, and sk ≤ Cknλ1...λk, sksk−2 ≤ Ckn[sk−1]2 sk ≤ Cknsk/ll , 1 ≤ l < k, λ1sk−1,1 ≥ Cknsk, sk−1,k ≥ Cknsk−1,∏ j sk,j ≥ Cknsk, (∂2λsk) ≤ 0, in Γk. MINIMAL HYPERSURFACES AND MEAN CURVATURE FLOWS 7 Hence, for functions in Γk, the equation sk = f(x) is an elliptic equation. Then we can have local gradient estimate, which will imply some Liouville type theorems. The important results for higher derivatives of II are the Gauss equation, Coddazi equation, Simons equation (see Chern-DoCarmo-Kobayashi, 1967, J.Simons, 1968). The method of Chern is to introduce some interesting n forms and n − 1 forms by using p, e := en+1, dp, den+1. For example, he defined, the determinant of the (n+ 1)× (n+ 1) matrix Ar = (p, e, de, ..., de, dp, ..., dp), Cr = (p, de, de, ..., de, dp, ..., dp), and Dr = (e, de, de, ..., de, dp, ..., dp), where r (1 ≤ r ≤ n) means that we have r factors of dp. One can image that these forms have close relationships with the principal curvature of the hypersurfaces. Let h = p·e be the support function. Then one has Cr = hDr and dAr = Cr −Dr+1 = hDr −Dr+1. Chern found the uniqueness results by using the mixed forms between two hypersurfaces and viewing the two surfaces as the mapping from the unit sphere. 5. Riemannian geometry of hypersurfaces We shall consider the Riemannian geometry of the surfaces and show that Theorem 5. R = n(n − 1)s2 is the scalar curvature of the Riemannian metric ds2 from the construction above. We now recall the differential geometry of the metric ds2. Define ω := (ωij) the Levi-Civita connection on Σ and the curvature forms of the con- nection Ωij = dωij − ωik ∧ ωkj = 1 2 Rijklωk ∧ ωl. We call Rm := (Rijkl) the Riemannian curvature tensor of Σ. Define the Ricci curvature of ds2 by Rjk = Rijki, and the scalar curvature R = Rjj . Of course one can extend above construction for any immersed submani- fold in Rn+1. We now prove theorem 5. Using the relation ωjn+1 = hjiωi and the structure equation for ωij that dωij = ωik ∧ ωkj + ωin+1 ∧ ωn+1j , 8 LI MA we know that Rijkl = hilhjk − hikhjl. This implies that Rjk = hiihjk − hikhij and R = Rjj = hiihjj − hijhij = 2s2. This completes the proof. � In the current study of global differential geometry (or conformal geome- try) we are concerned with the hypersurfaces with scalar-flat curvature, i.e., R = 0. Physicists are also interested in such surfaces. For mathematicians, we would like to set up a theory, which includes some Bernstein type theorems, which are something like, if Σ is the graph of the function z : Rn → R in E and s2 = 0,under which conditions we have z = constant, i.e., Σ is a hyperplane. In the remaining part of this section we compute curvatures of one exam- ple, Σ = M = Snr (x0), the sphere centered at x0 of radius r > 0. Let e = en+1 be the unit outward normal to the sphere M . Then for p ∈M , p− re = x0 is the fixed center. We then have dp = rde = rwn+1iei. Recall that dp = wiei. Hence, we have rwn+1i = wi. This implies that si = 1/r for each 1 ≤ i ≤ n. By this we know that the curvatures of the sphere M , saying, for example, the Gauss-Kronecker curvature K is 1/rn and s2 = 1/r 2. 6. Extensions Assume that n = 2. We first consider some global results. Proposition 6. Assume that K = constant and Σ is compact. Then K > 0. The proof is easy via the use of the maximum principle. In fact, letting f(x) = 12 |x|2, we have 0 < supΣ f(x) < ∞, which can be achieved by some point p in the compact Σ. Recall that dx = eiωi. Then we have at p, D2f ≤ 0 and 0 = df = x · dx = (x, e1)ω1 + (x, e2)ω2, which implies that (x, e1) = 0 = (x, e2), and then x is parallel to e3. Using this and D2f ≤ 0, we obtain that 0 ≥ D2f = d(x, e1) ⊗ ω1 + d(x, e2) ⊗ ω2. Note that d(x, ei) = (dx, ei) + (x, dei) = ωi + ωi3(x, e3). Then we have at p, 0 ≥ D2f = ω21 + ω22 + ωi ⊗ ωi3(x, e3). MINIMAL HYPERSURFACES AND MEAN CURVATURE FLOWS 9 This implies that II = −ωi ⊗ ωi3 is definite and K > 0. Proposition 7. Assume that K = 1 and Σ is compact. Then Σ is a sphere. We can extend the 2-d global results Propositions 6 and 7 to =higher dimensions. Proposition 8. Assume that Σ is a hypersurfaces in Rn+1 with n ≥ 3. Assume that s1 > 0, s2 = constant, and Σ is compact. Then s2 > 0. The proof is easy via the use of the maximum principle. Proposition 9. Assume that Σ is a hypersurfaces in Rn+1 with n ≥ 3. Assume that s1 > 0, s2 = 1, and Σ is compact. Then Σ is a sphere. Proof???????????????? R.Hamilton [5] proves a very interesting result. Theorem 10. Let M be a smooth strictly convex complete hypersurface bounding a region in Euclidean space. Suppose that for some c > 0 its second fundamental form is c-pinched in the sense that Hij ≥ cHgij where gij is the induced metric, Hij the second fundamental form, and H the mean curvature. Then M is compact. Hamilton also indicates how this result can be used to simplify the proof of Huisken’s theorem for the mean curvature flow of a convex hypersurface in Euclidean space. Another problem considered by R.Hamilton is below (see his paper: Hamil- ton, Richard S. , Worn stones with flat sides. A tribute to Ilya Bakelman (College Station, TX, 1993), 69C78, Discourses Math. Appl., 3, Texas A & M Univ., College Station, TX, 1994.). The worn stone problem asks for families or properties of families of convex bodies in 3-space whose bounding surfaces change with time so that each surface point moves in the inward normal direction at a rate equal to the momentary Gauss curvature at that point. The special case of this paper assumes that the initial surface has a cylindrical portion and also has sufficient smoothness to prove that at some later time the surface still contains a cylindrical portion, albeit a smaller one. Thus, if the initial surface contains a flat portion, it will contain a smaller flat portion a little later, but after some finite time the surface will become strictly convex. Even more, the author shows that the result applies under weaker smoothness conditions. Hamilton [6] also considers a smooth compact strictly convex body M in Euclidean space subject to wear under impact at a random angle. The probability of impact at any point P is proportional to the Gauss cur- vature K. Thus the surface evolves in time by the Gauss curvature flow Pt = −KN , where N is the unit outward normal. This equation was first studied by Firey. Later, K. Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867C882; MR0812353 (87e:53009)] showed that the solution exists and stays smooth and strictly convex until t=T for some time T, when the 10 LI MA diameter shrinks to zero. Thus, one can assume M shrinks to the origin. Recently B. Chow [Comm. Pure Appl. Math. 44 (1991), no. 4, 469C483; MR1100812 (93e:58032)] proved an entropy and a Harnack estimate for this flow. Hamilton uses these results to prove that the diameter, as well as the Gauss curvature, satisfies a dilatation-invariant bound. 7. The first and second variational formulae of Minimal surfaces We now consider the stability result of the mean curvature evolution of the hypersurfaces. We firstly consider the graph case. Given the graph G of the function z = f(x). We can easily see that we can choose the unit normal of it as N = (−∇z, 1)√ 1 + |∇z|2 . For any τ > 0 small we can define a new hypersurface surface Σ = Στ = G+ τN. We can compute the curvatures sk of Σ as functions of τ and the curva- tures of G. Given a hyper-surface Σ in the Euclidean space E = Rn+1 with induced metric g and local orthnormal frames (ej) n j=1 on Σ. Let ω j be the dual of the basis (ej). Then g = (ω j)2. For any smooth function F on Rn+1. Let f be the restriction of F on Σ. We define the gradient of f on Σ to be the tangential part of the gradient ∇F , i.e., ∇gf = (∇F )T . Then ∇gf = ∇ekfek = ek(F )ek = (∇F, ek)ek. The divergence of a tangential vector field X on Σ is divgX = g(∇ejX, ej). Define on Σ the Laplacian operator of f to be ∆gf = divg(∇gf). When the meaning is clear we often write ∇f = ∇gf , etc. Assume that p : Σ→ E is the defining inclusion mapping of the surface. We may use the local parametrization of the surface Σ such that locally, p = p(x) is a smooth mapping with the coordinates x = (x1, ..., xn) ∈ D some domain in Rn, where p is the position vector in E and the tangent vectors pj = ∂p ∂xj form a basis at any tangent space TpΣ. Then the induced metric on the surface Σ is I := g = gij(x)dx idxj with gij(x) = pi · pj . We shall denote the inverse matrix of (gij) by (g ij) = (gij) −1. Note that pj = (pj , ek)ek. MINIMAL HYPERSURFACES AND MEAN CURVATURE FLOWS 11 Then gjl = (pj , ek)(pl, ek). Let U = ((pj , ek)). Then the matrix g = UU t. Write ek = b m k pm. Then ek = b m k (pm, el)el and δkl = b m k (pm, el). Let B = (bmk ). Then the above implies that B = U −1. By this we have g−1 = BtB,i.e., gml = bmk b l k. Then ∇ekf = bmk fm and ∇gf = bmk fmek = bmk blkfmpl = gmlfmpl. Similarly, we have divgX = g jkg(∇xjX, pk). We now consider an example. Let p ∈ Σ and F (x) = |x − p|. Then ∇F = (x−p)F and ∇g = (x−p) T |x−p| . Let N = N(p) be the unit normal at the point p ∈ Σ. We consider it as the mapping from Σ to the unit sphere Sn−1, which is called the Gauss mapping from Σ into Sn−1. The differential A = −dN of the negative Gauss mapping is called the Weingarten mapping. One may check