SURFACES MINIMALES : THEORIE´ VARIATIONNELLE ET APPLICATIONS FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) Contents 1. Introduction 1 2. First variation formula 1 3. Examples 4 4. Maximum principle 5 5. Calibration: area-minimizing surfaces 6 6. Second Variation Formula 8 7. Monotonicity Formula 12 8. Bernstein Theorem 16 9. The Stability Condition 18 10. Simons' Equation 29 11. Schoen-Simon-Yau Theorem 33 12. Pointwise curvature estimates 38 13. Plateau problem 43 14. Harmonic maps 51 15. Positive Mass Theorem 52 16. Harmonic Maps - Part 2 59 17. Ricci Flow and Poincar´eConjecture 60 18. The proof of Willmore Conjecture 65 1. Introduction 2. First variation formula Let (M n; g) be a Riemannian manifold and Σk ⊂ M be a submanifold. We use the notation jΣj for the volume of Σ. Consider (x1; : : : ; xk) local coordinates on Σ and let @ @ gij(x) = g ; ; for 1 ≤ i; j ≤ k; @xi @xj 1 2 FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) be the components of gjΣ. The volume element of Σ is defined to be the p differential k-form dΣ = det gij(x). The volume of Σ is given by Z Vol(Σ) = dΣ: Σ Consider the variation of Σ given by a smooth map F :Σ × (−"; ") ! M. Use Ft(x) = F (x; t) and Σt = Ft(Σ). In this section we are interested in the first derivative of Vol(Σt). Definition 2.1 (Divergence). Let X be a arbitrary vector field on Σk ⊂ M. We define its divergence as k X (1) divΣX(p) = hrei X; eii; i=1 where fe1; : : : ; ekg ⊂ TpΣ is an orthonormal basis and r is the Levi-Civita connection with respect to the Riemannian metric g. Lemma 2.2. @ @F (2) dΣ = div dΣ : @t t Σt @t t Proof. Note that @ detg = tr(g−1@ g)detg; @t t −1 ij −1 where g = (g ) = (gij) . Then @ X detg = (gij@ g )detg: @t t ij i;j We can calculate the first derivative of the metric using the compatibility of r with respect to g @tgij = g(r@F=@t@iF; @jF ) + g(@iF; r@F=@t@jF ); where @iF = @F=@xi. Use the symmetry of r to commute r@F=@t@iF = r@iF @F=@t. Put everything together to obtain @ X detg = 2 gijg(r @F=@t; @ F )detg @t @iF j i;j @F = 2div detg: Σt @t We have then: Theorem 2.3 (First Variation Formula I). d Z @F jΣtj = divΣt dΣt: dt Σt @t SURFACES MINIMALES : THEORIE´ VARIATIONNELLE ET APPLICATIONS 3 In order to characterize the submanifolds that are critical values for area, we need to introduce a important geometric object and some calculation. 2nd fundamental form and mean curvature: Let (M n; g) be a Riemannian manifold and Σk ⊂ M be a submanifold. For each p 2 Σ, define the second fundamental form of Σ ⊂ M as T N B(V; W ) = rV W − (rV W ) = (rV W ) ; where V; W are tangent to Σ. B is a symmetric tensor. If fe1; : : : ; ekg ⊂ TpΣ is an orthonormal basis, we define the mean curvature vector to be k −! X N H = tr B = rei ei : i=1 Lemma 2.4. T N −! divΣX = divΣX − hX ; Hi: Proof. Decompose X in tangent and normal parts X = XT + XN and write k T X N divΣX = divΣX + hrei X ; eii: i=1 N Since X is normal to Σ and ei is tangent, we have N N N (3) 0 = eihX ; eii = hrei X ; eii + hX ; rei eii: Then k T X N divΣX = divΣX − hX ; rei eii i=1 T N −! = divΣX − hX ; Hi: Theorem 2.5 (Divergence Theorem). Let Σ be a compact submanifold of M and X be a tangential vector field on Σ. Then Z Z divΣXdΣ = g(X; ν)dσ; Σ @Σ where ν 2 T Σ is the unique unit outward normal to @Σ and dσ is the volume element of @Σ. Theorem 2.6 (First variation formula II). d Z −! Z jΣtj = − h@F=@t; HidΣt + h@F=@t; νidσt: dt Σt @Σt Moreover, if X = @F=@t vanishes on @Σ at t = 0, then Z d −! jΣtj = − hX; HidΣ: dt t=0 Σ 4 FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) Corollary 2.7. d −! jΣtj = 0; for any X; with X = 0 on @Σ; iff H = 0: dt t=0 −! Definition 2.8. Σk ⊂ M is a minimal submanifold of M if H = 0. 3. Examples (Geodesic). For k = 1, Σ is a smooth curve γ : I ! M satisfying −! γ0 N γ0 0 0 0 = H = r γ 0 = r γ 0 : jγ0j jγ j jγ0j jγ j Geodesics are minimal surfaces of dimension 1. (Minimal surfaces of euclidean space). Start with the particular case of a minimal surfaces given as the graph of a differentiable function u :Ω ! R, k where Ω ⊂ R . Let F :Ω ! graph(u) be given by F (x) = (x; u(x)). Use also the notation @F @ @u = ; : @xi @xi @xi k+1 The euclidean metric of R restricted to graph(u) can be written as @F @F @u @u gij = g ; = δij + ; @xi @xj @xi @xj where δij = 1, if i = j, and equal zero otherwise. 2 Pk 2 Exercise: Prove that detg = 1 + jruj = 1 + i=1(@u=@xj) and conclude Z p jgraph(u)j = 1 + jruj2dx: Ω Consider the variations of graph(u) given by the graphs of functions ut = u + tv, for any fixed v so that v = 0 on @Ω. Then Z d d p 2 jgraph(ut)j = 1 + jr(u + tv)j dx dt t=0 dt t=0 Ω Z ru Z ru = hrv; idx = − v div k dx: p 2 R p 2 Ω 1 + jruj Ω 1 + jruj The graph of a function u is a minimal surface iff u satisfies the minimal surface equation ru (4) div k = 0: R p1 + jruj2 Equivalently, if u is a solution of the second order elliptic quasilinear P.D.E. given by: X @iu@ju (5) δ − @ @ u = 0: ij 1 + jruj2 i j i;j SURFACES MINIMALES : THEORIE´ VARIATIONNELLE ET APPLICATIONS 5 3 Basic examples of minimal surfaces in R . 3 • The plane: f(x; y; z) 2 R : z = 0g. 3 2 2 • The catenoid: f(x; y; z) 2 R : x + y = cosh zg. A bigraph, it is obtained by the revolution of the curve y = cosh z over the z-axis. • The helicoid: (u; v) 7! (u cos v; u sin v; v). This is a multigraph over 2 R n f0g for the function u(x; y) = arctan(y=x). cos x • The Scherk's surface: graph(u), where u(x; y) = log cos y , for x; y 2 (−π=2; π=2). Scherk's surface is a doubly periodic minimal surface. 3 It is important to classify and understand these minimal surfaces in R . In this direction there is the Colding-Minicozzi Theory, that studies the structure of embedded minimal disks. One important application is Theorem 3.1 (Meeks and Rosenberg, 2005). The only complete embedded 3 simply connected minimal surfaces in R are the plane and the helicoid. 4. Maximum principle In this section, we apply the strong maximum principle for the minimal surface equation. Theorem 4.1 (Strong maximum principle). Let v be a solution to X X aij@i@jv + bi@iv = 0; i;j i where (aij(x))ij is positive definite for each x 2 Ω. If v ≤ 0 and v = 0 at some point, then v ≡ 0. The main result of this section is Theorem 4.2 (Maximum principle). Let u1; u2 :Ω ! R be solutions to minimal surface equation. Suppose Ω is connected, u1 ≤ u2 and u1(p) = u2(p) for some p 2 Ω. Then u1 ≡ u2. Proof. Let v = u1 − u2. We have X @iu1@ju1 X @iu2@ju2 @iu1@ju1 0 = δ − @ @ v + − @ @ u : ij 1 + jru j2 i j 1 + jru j2 1 + jru j2 i j 2 i;j 1 i;j 2 1 Since v = u1 − u2 ≤ 0 and v(p) = 0, the strong maximum principle gives v ≡ 0. Then we conclude u1 = u2. n−1 n−1 n Corollary 4.3. Let Σ1 ; Σ2 ⊂ R be connected minimal hypersurfaces, such that Σ2 is on one side of Σ1. If there exists p 2 Σ1 \ Σ2 6= ;, then Σ1 = Σ2. Remark. This result also holds for minimal hypersufaces Σ1; Σ2 ⊂ M of a Riemannian manifold. 6 FERNANDO CODA´ MARQUES - IMPA (NOTES BY RAFAEL MONTEZUMA) 5. Calibration: area-minimizing surfaces Let u :Ω ! R be a solution to the minimal surface equation, where n Ω ⊂ R . Then ut(x) = u(x) + t is also a solution and Σt = graph(ut) form a foliation of Ω×R by minimal hypersurfaces. This means for every p 2 Ω×R n+1 T we can find t so that p 2 Σt. For each vector V 2 TpR , let V be the component of V in TpΣt. Consider the following differential n-form T T (6) !(p)(X1;:::;Xn) = VolΣt (X1 ;:::;Xn ): Observe we can also write ! as T T !(p)(X1;:::;Xn) = VolRn+1 (X1 ;:::;Xn ;N) n T T = (−1) VolRn+1 (X1 ;:::;Xn ;N) n = (−1) (ιN VolRn+1 )(X1;:::;Xn); where N is the unit normal vector to Σt and ιN VolRn+1 is the contraction of VolRn+1 with the vector field N. Cartan's formula: If LX ! is the Lie derivative of ! with respect to X, then d(ιX !) + ιX (d!) = LX !: Remark: If X is a vector field on (M n; g), then LX dVg = (divgX)dVg: Using Cartan's formula and the previous remark we can calculate d! as n (−1) d! = d(ιN VolRn+1 ) = LN VolRn+1 − ιN d(VolRn+1 ) = divRn+1 N VolRn+1 : Can use that N is explicitly given by N = (1 + jruj2)−1=2(−∇u; 1) to conclude that n n X @N ru (−1) d! = Vol n+1 = −div n Vol n+1 : @x R R p 2 R i=1 i 1 + jruj Corollary 5.1.