On Generalized Hopf Differentials

Transcription

1 On 1 Ruhr-Universität Bochum Bochum Germany abresch@math.rub.de August 2006 Joint work with: Harold Rosenberg (Univ. Paris VII) 1 Supported by CNRS and DFG SPP Table of 1. for Cmc Surfaces in Space Forms Theorem Hopf s Theorem 2. Rotationally-Invariant Cmc Spheres in M 2 κ R Adapting Theorem What about Extending Hopf s Theorem? 3. for Cmc Spheres in M 2 κ R Quad. This Equivariant in Nil(3) Examples of in Nil(3) Half-Space Theorems Frame 2 of 25/ Theorem 1.2. Hopf s Theorem Theorem ([ Alexandrov, 1955 ]) Let Σ 2 be a closed embedded cmc surface in R 3, in H 3, or in a hemi-sphere S 3 +. Then Σ 2 is a distance sphere. Theorem ([ Hopf, 1956 ]) Let S 2 be an immersed cmc sphere in R 3, H 3, or S 3. Then S 2 is a distance sphere. ν Σ 2 ρ(σ2 ) Idea of Proof. Consider reflections through a family of (parallel) inward moving planes. By the maximum principle, Σ 2 = ρ(σ 2 ) upon first contact. Thus Σ 2 = ρ(σ 2 ) for all reflections ρ preserving the center of Σ 2. i) Each distance sphere S 2 S 3 is contained in a closed hemi-sphere. ii) In S 3 there are Clifford tori and many other cmc surfaces of higher genus [ cf. Kapouleas, 1997 ]. Theorem Hopf s Theorem Ingredients. i) The Codazzi equations for h Σ =., A. imply: on any immersed cmc surface, Q H := π 2,0 (h Σ ) is a holomorphic quadratic differential. ii) { hol. quad. differentials on S 2 = CP 1} = 0, hence: h Σ 1 2 tr(a) g = 2 Re Q H = 0. iii) Complete, totally-umbilical surfaces Σ 2 in R 3, in H 3, or in S 3 are distance spheres. Remark The identity Q H = 0 can be understood as a first integral of the cmc equation. Theorem Hopf s Theorem Frame 3 of 25/35 Frame 4 of 25/35

2 2.1. Rotationally-Invariant Cmc Spheres in M 2 κ R 2.1. Rotationally-Invariant Cmc Spheres in M 2 κ R These spheres will serve as model surfaces later on! Construction of S 2 H M 2 κ R. 1 κ 1 κ H=1.075 κ H=0.121 κ H=0.033 κ 0 r π κ H=0.375 κ i) The relevant ODE-system: s r = sin θ s = s θ = ii) A first integral [ cf. Hsiang, 1989 ]: cos θ 2H cos(θ) ct κ(r) Convention: (cos θ, sin θ) is the exterior unit normal vector field of the meridian curve c(s) = (r(s), (s)). L := cos(θ) sn κ (r) 4H sn κ ( 1 2 r)2 iii) The curve c(s) intersects the fixed point set L = 0 ( or, in case κ > 0, iff L = 4H /κ ). Rotationally-Invariant Cmc Spheres in M 2 κ R Adapting Theorem What about Extending Hopf s Theorem? Frame 5 of 25/35 Explicit Solution for κ > 0. 1 = (κ + 4H 2 ) 1 κ sin2( 1 2 r κ ) + 4H 2 1 κ sinh2( 1 2 κ 1 2H κ + 4H 2 ) Principal Curvatures. ( ) H + κ 4H h Σ = cos2 (θ) 0 0 H κ 4H cos2 (θ) i) If 0 < 4H 2 < κ, the model spheres S 2 H constructed above do not project into closed hemi-spheres. ii) The spheres S 2 H are not totally-umbilical. iii) The bilinear forms q := 2H h Σ κ d 2, however, are again multiples of the induced metric ι g. Rotationally-Invariant Cmc Spheres in M 2 κ R Adapting Theorem What about Extending Hopf s Theorem? Frame 6 of 25/ Adapting Theorem 2.3. What about Extending Hopf s Theorem? Theorem Any closed embedded cmc surface Σ 2 in H 2 R or S 2 + R is a rotationally-invariant vertical bigraph. Such a bigraph Σ 2 is necessarily congruent to some S 2 H. Idea of the Proof. moving planes argument. Caveats. i) Closed embedded cmc surfaces Σ 2 S 2 R that do not project into some hemi-sphere S 2 + are only guaranteed to be vertical bigraphs. ii) Not all of the rotationally-invariant cmc spheres S 2 H S 2 R do project into hemi-spheres. iii) In S 2 R itself, there again exist embedded cmc tori and embedded cmc surfaces of higher genus. Rotationally-Invariant Cmc Spheres in M 2 κ R Adapting Theorem What about Extending Hopf s Theorem? Immediate Obstacles. For target manifolds other than space forms, the r.h.s. of the Codazzi equations does not vanish anymore: X A Y Y A X, Z = R(X, Y ) ν, Z = X Y, G (ν Z), Here A = D ν and XY = (D XY ) tan, and G Ric 1 Sc 1l denotes the Einstein tensor. 2 Conclusion: QH (π 2,0 (h Σ )) 0. The model spheres S 2 H are not totally-umbilical. Encouraging Facts. i) The fields q = 2H h Σ κ ι (d 2 ) are linear combinations of h Σ and ι (d 2 ) with constant coefficients. ii) Their (2, 0)-parts vanish on S 2 H, and so Q := π 2,0 (q) may be holomorphic on all cmc surfaces Σ 2 M 2 κ R. Rotationally-Invariant Cmc Spheres in M 2 κ R Adapting Theorem What about Extending Hopf s Theorem? Frame 7 of 25/35 Frame 8 of 25/35

3 3. for Cmc Spheres in M 2 κ R 3.1. Quad. Theorem 1 ([ A & Rosenberg, 2004 ]) Any cmc surface Σ 2 M 2 κ R comes with a natural holomorphic quadratic differential given by Q := 2H π 2,0 (h Σ ) κ π 2,0 (ι (d 2 )). This result is proved by direct computation. As in H. Hopf s work, Theorem 1 is the key to Theorem 2 ([ A & Rosenberg, 2004 ]) Any immersed cmc sphere S 2 M 2 κ R is one of the rotationally-invariant model spheres S 2 H M 2 κ R. Ingredients in the Proof of Theorem 2. i) { hol. quad. differentials on S 2 = CP 1} = 0. Quadratic This Basic Ingredients for the Proof of Theorem 1. On oriented surfaces (Σ 2, ι g), the almost complex structure J is parallel, and the -operator is given by Q(X; Y 1, Y 2 ) = 1 2( X Q + i JX Q ) (Y 1, Y 2 ) =: 1 2 (1+iJ)X Q (Y 1, Y 2 ). A = H 1l + A 0, and, on surfaces, traceless symmetric endomorphisms like A 0 anti-commute with J. The Codazzi equations for surfaces Σ 2 in 3-manifolds: X A Y Y A X, Z = X Y, G (ν Z) = (X Y ) Z, G ν. Quadratic This ii) Given κ 0 and H R, one can use ODE techniques in order to classify the cmc surfaces ι: Σ 2 M 2 κ R with mean curvature H and with Q 0. Here A = D ν, and G denotes the Einstein tensor. The final expression follows, since ν X, Y, Z and G (ν Z) = tr(g) ν Z (Gν) Z ν (GZ ). Frame 9 of 25/35 Frame 10 of 25/ Quad Quad. Key Steps in the Proof of Theorem 1. i) The Codazzi equations imply that ( π 2,0 (h Σ ) ) (X; Y 1, Y 2 ) = ψ(x; Y 1, Y 2 ), G ν where [ ψ(x; Y 1, Y 2) := 1 2 X, Y + 1 Y X, Y + 2 Y + ] 1, X := 1 (1 + ij) X, and Y + 2 µ := 1 (1 ij) Yµ. 2 Quadratic Getting some Conceptual Understanding. i) Since DL = 0 and G = κ L, it follows from the basic structure of Wirtinger calculus that the terms (π 2,0 (h Σ ))(X; Y 1, Y 2 ) and (π 2,0 (ι L))(X; Y 1, Y 2 ) must both be multiples of ψ(x; Y 1, Y 2 ), L ν. Hence there exist universal constants a, b C such that Quadratic ii) Computing in terms of D and ν, it follows that the vertical projectors L := d 2 satisfy ( π 2,0 (ι L) ) (X; Y 1, Y 2 ) This ( a π 2,0 (h Σ ) b π 2,0 (ι L) ) (X; Y 1, Y 2 ) = 0 for any immersed cmc surface Σ 2 M 2 κ R. This = Y + 1, D (X ) L Y + 2 2H ψ(x; Y 1, Y 2 ), L ν iii) The product structure of the targets M 2 κ R implies that DL = 0 and, moreover, that G = κ L. Thus ( π 2,0 (2H h Σ κ ι L) ) (X; Y 1, Y 2 ) = 0. ii) On the rot.-invariant model spheres S 2 H M 2 κ R, the quadratic differential Q = π 2,0 (2H h Σ κ ι L) vanishes identically, and hence Q 0, too. This particular case now fixes the universal constants a and b above to the claimed values. Frame 11 of 25/35 Frame 12 of 25/35

4 Theorem 3 ([ A & Rosenberg, 2004 ]) Let ι: Σ 2 M 2 κ R be a complete immersed surface with constant mean curvature H and with Q 0. Suppose that (κ, H ) 0. Then the following holds: if 4H 2 + κ > 0, then Σ 2 is congruent to a rot.-inv. model sphere S 2 H M 2 κ R. if 4H 2 + κ 0, then Σ 2 is a complete open surface of type D 2 H, P 2 H, or C 2 H, respectively. The three cases can be distinguished by the sign of 4H 2 + κ cos 2 θ where θ := arcsin(d ν) denotes the Gauß angle. i) Here D 2 H and C 2 H denote rotationally-inv. cmc surfaces that are homeomorphic to disks or annuli (catenoids). ii) The P 2 H are orbits under 2-dim. solvable subgroups A N SO(2, 1) + R. Quadratic This Frame 13 of 25/35 κ > 0 : Meridians for S 2 H 1 κ 1 κ H=1.075 κ 0 π κ r H=0.375 κ H=0.121 κ κ < 0 : Meridians for C 2 H 1 κ 1 κ H=0.033 κ 0 3 r κ H=0.275 κ H=0.033 κ H=0.121 κ κ < 0 : Meridians for S 2 H and D 2 H 2 κ H=0.87 κ H=1.15 κ H=0.66 κ S 2 H H=0.5 κ D 2 H H=0.33 κ H=0.12 κ 0 3 r κ κ < 0 : Meridians of P 2 H are limits P 2 H D 2 H C 2 H Quadratic This Frame 14 of 25/ This 4. The unit normal field ν of an immersion ι: Σ 2 M 2 κ R provides a lift of ι into the total space of the unit tangent bundle π : N 5 κ := T 1 (M 2 κ R) M 2 κ R. Proposition (Prolongation) Immersed surfaces ι: Σ 2 M 2 κ R with constant mean curvature H and Q 0 lift to integral surfaces ν : Σ 2 N 5 κ of an explicit 2-dimensional distribution E H TN 5 κ. Properties of E H. i) E H is invariant under the action of Iso 0 (M 2 κ R). This action has 4-dim. orbits that are separated by Θ: N 5 κ [ 1 2 π, 1 2 π ], ii) The Gauß map θ : s Θ c(s) of any meridian solves iii) E H is integrable. s θ = 1 ( 4H 4H 2 + κ cos 2 θ ). Quadratic This Frame 15 of 25/35 Is it possible to replace the product spaces M 2 κ R by more general oriented Riemannian manifolds (M 3, g)? Theorem 4 ([ A, 2006 ]) Let L 0 be a C-valued, traceless, symmetric bilinear form on (M 3, g). Then the expression Q := π 2,0 (h Σ + ι L 0 ) defines a holomorphic quadratic differential on any surface ι: Σ 2 (M 3, g) with constant mean curvature H, if and only if L 0 solves the differential equation D X L 0 = 1 2 i [ X, G 2H L 0 ]. ( ) Remark The ODE-system ( ) is overdetermined. The integrability condition even required for local solutions imposes serious restrictions on the geometry of (M 3, g). Frame 16 of 25/35

5 Theorem 5 ([ A, 2006 ]) Let ( M 3, g) be a simply-connected, oriented Riemannian manifold. Then there exists a solution L 0 of D X L 0 = 1 2 i [ X, G 2H L 0 ], ( ) iff ( M 3, g) is a hom. space with an at least 4-dimensional isometry group, or, equivalently, iff it is a space form or a homogeneous bundle M 3 κ,τ N 2 κ. New target spaces: S 3 Berger, Nil(3), and Sl(2, R). Remark The hom. bundles Mκ,τ 3 Nκ 2 are the products S 2 R and H 2 R, the Berger spheres S 3 η, the Heisenberg group Nil(3), and Sl(2, R), and explicit solutions of ( ) are L 0 := κ τ 2 2H iτ (P 1 3 1l). Here P denotes the field of vertical projectors. Frame 17 of 25/35 The holomorphic quad. differentials Q := π 2,0 (h Σ + ι L 0 ) that come with these solutions L 0 are the key to: Theorem 6 ([ A, 2006 ]) Any immersed cmc sphere S 2 in a homogeneous bundle Mκ,τ 3 Nκ 2 is in fact embedded and rotationally-invariant. Thus its shape is determined by the mean curvature H. Of couse, we have to refine Theorem 3 appropriately, too. i) Thus we have extended H. Hopf s result to immersed cmc spheres in homogeneous spaces representing 7 of the 8 maximal homogeneous structures [ cf. Thurston ]. ii) On Solv(3), however, the cmc equation has due to lack of symmetry no first integrals like our holomorphic quad. differentials. More precisely, there is no 1-dim. isotropy group. Frame 18 of 25/ The dimension of G := Iso( M 3, g) is either 3, 4, or 6. We ll discuss each case for simply-connected spaces. a) dim G = 6 : These spaces have constant curvature κ. Up to scaling, they are the standard spaces S 3, R 3, and H 3. Their Einstein tensor is G = κ 1l. Evidently, DG 0. b) dim G = 4 : These spaces are homogeneous bundles π κ,τ : M 3 κ,τ Ñ 2 κ over simply-connected surfaces of constant curvature κ. They have tot.-geod. fibers and const. bundle curvature τ. Convention: [X, Y ] vert = τ X Y for all hor. vector fields. Range of (κ, τ) : the curve κ = τ 2 must be excluded, as it yields spaces of constant curvature. Complete list: S 2 R R 3 H 2 R S 3 Berger Nil(3) Sl(2, R) b) dim G = 4 (cont.): properties of the spaces M 3 κ,τ : i) Their Einstein tensor is G = 1 4 τ 2 1l (κ τ 2 ) P. ii) Moreover, D X G = 1 2 τ [ X, G ]. So they are symmetric spaces, iff τ = 0. iii) Finally, their Cotton tensor turns out to be 3 2 τ G 0. So they are locally conformally flat, iff τ = 0. iv) For any pair κ, τ, the isotropy group G p of any point p is bigger than SO(2). It contains S(O(2) O(1)). v) Hence, for any horizontal geodesic γ : R M 3 κ,τ, there is a 180 o -rotation φ γ containing γ in its fixed point set. In fact, G is generated by these rotations. Frame 19 of 25/35 Frame 20 of 25/35

6 Equivariant in Nil(3) c) dim G = 3 : These spaces are 3-dimensional Lie groups equipped with left-invariant metrics [ cf. Milnor, 1976 ]. i) There are several isomorphism classes of 3-dimen. real Lie algebras, but only one of them gives raise to a new maximal homogeneous structure: Solv(3). ii) A quotient of Solv(3) is a torus bundle over S 1. iii) The geometry of Solv(3) is also very special: ker(ric) is a 2-dim. integrable distribution. Its Weingarten map has 2 distinct eigenvalues. The Cotton tensor has 3 distinct eigenvalues. G and Cotton commute. Yet, the isotropy groups are finite and, in fact, isomorphic to the dihedral group D 4. The 4 Basic Types [ cf. Figueroa, Mercuri, Pedrosa ] a) Vertical Planes: total preimages of straight lines, invariant w.r.t. vertical translations. b) Catenoids and Horizontal Umbrellas: invariant w.r.t. a group φ t of rotations around some vert. axis. c) Helicoids and Helicoidal Catenoids: invariant w.r.t. a group φ t of screw motions around a vert. axis. d) Saddle-Type Surfaces: invariant w.r.t. a group φ t of isometries that project to translations of R 2. i) The umbrellas and the saddle-type surfaces are graphs w.r.t. the Riem. submersion Nil(3) R 2. ii) Q = 0 on umbrellas and on vertical planes, whereas Q = c dz 2 0 for the saddle-type surfaces. Equivariant Minimal Examples of in Nil(3) Half-Space Theorems Frame 21 of 25/35 Frame 22 of 25/ Examples of in Nil(3) 5.2. Examples of in Nil(3) a) Surfaces. They come as Nitsche graphs over a square in R 2 w.r.t. the submersion Nil(3) R 2. Their boundary consists of the vertical geodesics over the 4 vertices of the square. i) They are invariant w.r.t. the 180 o -rotations around hor. lifts of the diagonals. ( Schwarz reflection principle.) ii) They do not extend to doubly-periodic minimal surfaces in Nil(3). iii) Upon enlarging the square, they converge to saddle-type surfaces not umbrellas. Application (A Weak Bernstein Theorem) Serrin style curvature bounds for (global) minimal graphs. Equivariant Minimal Examples of in Nil(3) Half-Space Theorems b) Triply-Periodic Scherk Surfaces ˆΣ 2. In order to construct these surfaces, fix a triangle γ in the barycentric subdivision of the fundamental square in R 2, and proceed as follows: i) Consider a horizontal lift of γ starting over the vertex of the square, and add a vertical segment to get a closed polygon γ. ii) Solve the Plateau problem Σ 2 = γ and extend Σ 2 to a global minimal surface ˆΣ 2 by means of the Schwarz reflection principle. Remark ˆΣ 2 = Γ Σ 2 where Γ Iso (Nil(3)) is the discrete subgroup generated by the four 180 o -rotations around the edges of γ. Equivariant Minimal Examples of in Nil(3) Half-Space Theorems Frame 23 of 25/35 Frame 24 of 25/35

7 5.3. Half-Space Theorems 6.1. Basic Theorem 7 ([ A & Rosenberg, 2004 ]) Let Σ 2 be a proper, possibly branched minimal surface in the Heisenberg group Nil(3). Suppose that Σ 2 is contained in the complement of some horizontal umbrella. Then Σ 2 is congruent to this umbrella by a vertical translation. Method of Proof. The same argument as in R 3 works, since the catenoids collapse to doubly-covered punctured umbrellas when their necksize is shrunk to 0. i) There is no half-space theorem w.r.t. the level sets H 2 {t 0 } in the product H 2 R. ii) Yet, the horizontal umbrellas in Nil(3) are hyperbolic and not parabolic. Question: Are there also half-space theorems w.r.t. the saddle-type surfaces in Nil(3) rather than the umbrellas? Equivariant Minimal Examples of in Nil(3) Half-Space Theorems U. Abresch, H. Rosenberg: A Hopf Differential for Constant Mean Curvature Surfaces in S 2 R and H 2 R, Acta Math. 193 (2004), U. Abresch: A Hopf Differential for Constant Mean Curvature Surfaces in Riemannian 3-Manifolds, in preparation. A. D. Alexandrov: A Characteristic Property of Spheres, Ann. Mat. Pura Appl. 58 (1962), B. Gidas, W. M. Ni, L. Nirenberg: Symmetry and Related Properties via the Maximum Principle, Comm. Math. Phys. 68 (1979), H. Hopf: Differential Geometry in the Large, LNM 1000, Springer-Verlag, Berlin Basic Frame 25 of 25/35 Frame 26 of 25/ Basic 6.2. W.-Y. Hsiang, W.-T. Hsiang: On the Uniqueness of Isoperimetric Solutions and Imbedded Soap Bubbles in Noncompact Symmetric Spaces I, Invent. Math. 98 (1989), N. Kapouleas: Complete Embedded of Finite Total Curvature, J. Differential Geom. 47 (1997), Ch. B. Figueroa, F. Mercuri, R. Pedrosa: Invariant Surfaces of the Heisenberg s, Ann. Mat. Pura Appl. (4) 177 (1999), Basic U. Abresch: Constant Mean Curvature Tori in Terms of Elliptic Functions, J. Reine Angew. Math. 374 (1987), U. Pinkall, I. Sterling: On the Classification of Constant Mean Curvature Tori, Ann. of Math. (2) 130 (1989), A. I. Bobenko: All Constant Mean Curvature Tori in R 3, S 3, H 3 in Terms of Theta-Functions, Math. Ann. 290 (1991), Basic P. A. Scott: The Geometries of 3-Manifolds, Bull. London Math. Soc. 15 (1983), N. J. Hitchin: Harmonic Maps from a 2-Torus to the 3-Sphere, J. Differential Geom. 31 (1990), W. P. Thurston: Three-Dimensional Geometry and Topology, Princeton Math. Series 35, Princeton Univ. Press, F. E. Burstall, D. Ferus, F. Pedit, U. Pinkall: Harmonic Tori in Symmetric Spaces and Commuting Hamiltonian Systems on Loop Algebras, Ann. of Math. (2) 138 (1993), Frame 27 of 25/35 Frame 28 of 25/35

8 J. Dorfmeister, F. Pedit, H. Wu: Weierstrass Type Representation of Harmonic Maps into Symmetric Spaces, Comm. Anal. Geom. i6 (1998), F. E. Burstall, F. Pedit, U. Pinkall: Schwarzian Derivatives and Flows of Surfaces, Diff. Geometry and (Tokyo, 2000), 39 61, Contemp. Math. 308, Amer. Math. Soc., Providence, RI, J. Dorfmeister, M. Kilian: Dressing Preserving the Fundamental, Differential Geom. Appl. 23 (2005), Basic D. Benoît: Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv., in preparation. D. Benoît: The Gauss map of minimal surfaces group, ArXiv: math.dg/ D. A. Berdinski, I. A. Taimanov: Surfaces in three-dimensional Lie groups, ArXiv: math.dg/ v2, 14. Nov T. H. Colding, W. P. Minicozzi II: Embedded minimal disks, in: Global theory of minimal surfaces, Clay Math. Proc. 2 (2005), , Amer. Math. Soc., Providence, RI. Basic T. H. Colding, W. P. Minicozzi II: The space of embedded minimal surfaces of fixed genus in a 3-manifold, I. Estimates off the axis for disks, Ann. of Math. 160 (2004), 27 68, II. Multi-valued graphs in disks, Ann. of Math. 160 (2004), 69 92, III. Planar domains, Ann. of Math. 160 (2004), , IV. Locally simply connected surfaces, Ann. of Math. 160 (2004), Frame 29 of 25/35 Frame 30 of 25/ P. Collin, R. Kusner, W. H. Meeks III, H. Rosenberg: The topology, geometry and conformal structure of properly embedded minimal surfaces, J. Differential Geom. 67 (2004), W. H. Meeks III, H. Rosenberg: Stable minimal surfaces in M R, J. Differential Geom. 68 (2004), Basic ν W. H. Meeks III: The limit lamination metric or the Colding-Minicozzi minimal lamination, Illinois J. Math. 49 (2005), W. H. Meeks III, H. Rosenberg: The uniqueness of the helicoid, Ann. of Math. 161 (2005), W. H. Meeks III, H. Rosenberg: Minimal surfaces of finite topology, in: Global theory of minimal surfaces, Clay Math. Proc. 2 (2005), , Amer. Math. Soc., Providence, RI. Σ 2 ρ(σ2 ) W. H. Meeks III, H. Rosenberg: The theory of minimal surfaces in M R, Comment. Math. Helv. 80 (2005), Frame 31 of 25/35 Frame 32 of 25/35

9 κ > 0 : Meridians for S 2 H κ < 0 : Meridians for S 2 H and D 2 H 1 κ H=0.121 κ H=0.87 κ H=0.66 κ S 2 H H=0.5 κ H=1.075 κ H=0.033 κ 0 r π κ 2 κ H=1.15 κ D 2 H H=0.33 κ 1 κ H=0.12 κ H=0.375 κ 0 3 r κ Frame 33 of 25/35 Frame 33 of 25/ κ < 0 : Meridians for C 2 H κ < 0 : Meridians of P 2 H are limits 1 κ D 2 H H=0.033 κ 1 κ 0 3 r κ H=0.275 κ H=0.121 κ P 2 H C 2 H Frame 33 of 25/35 Frame 33 of 25/35

10 Frame 34 of 25/35 Frame 35 of 25/35