Delaunay Surfaces in S 3

Transcription

1 Delaunay Surfaces in S 3 Ryan Hynd ryanhynd@math.berkeley.edu Department of Mathematics University of California, Berkeley Queen Dido Conference Carthage, Tunisia May 28, 2010

2 Part 1: Delaunay Surfaces in R3

3 CMC surfaces of revolution in R 3 u t t Easy computation Plane curve t (t, u(t)) Rotate around z-axis X(t,θ) = (t cos θ,tsinθ,u(t)) Integrate mean curvature equation 2H = 1 t ( tu 1 + u 2) u 1 + u 2 = Ht + c t

4 Solutions

5 Other things to know Theorem (Delaunay 1841) The generating curves of CMC surfaces of revolution are roulettes of conic sections.

6 Other things to know Theorem (Delaunay 1841) The generating curves of CMC surfaces of revolution are roulettes of conic sections. Applications Used to build other CMC surfaces in R 3 (K-noids, Bubbletons) Arise in Capillarity (modeling soap films, Double Bubble problem)

7 Part 2: Delaunay Surfaces in S3

8 Symmetry in S 3 Definition M S 3 is symmetric, with respect to a circle C, if Ψ(M) = M for each conformal Ψ : S 3 S 3 that fixes C pointwise.

9 Symmetry in S 3 Definition M S 3 is symmetric, with respect to a circle C, if Ψ(M) = M for each conformal Ψ : S 3 S 3 that fixes C pointwise. Lemma M S 3 is symmetric iff there is a rotation O such that π O(M) is rotationally symmetric in R 3. Here π : S 3 \ {e 4 } R 3 ;(x 1,x 2,x 3,x 4 ) (x 1,x 2,x 3 ) 1 x 4 is stereographic projection.

10 Stereographic projection, reflection, and inversion Corollary M S 3 is symmetric with respect to a circle C iff there Ψ Q (M) = M for each 2-sphere Q, C Q S 3.

11 A convenient parametrization Theorem M S 3 immersed and symmetric. Then there is a rotation O such that π O(M) admits the (local) parametrization X(θ,φ) = (R(θ,φ)cos θ,r(θ,φ)sin θ,h + r(θ)sinφ) (θ,φ) ( ǫ,ǫ) [0,2π), for some h R and ρ > 0. Here R(θ,φ) := r(θ) 2 + ρ 2 + r(θ)cos φ.

12 Mean curvature calculation Expression for the mean curvature In general H π 1 (X) = 1 + X 2 H X + X N 2 For X = (Rcos θ,rsin θ,h + r sin φ), H π 1 (X) = c 0 + c 1 cos φ + c 2 (cos φ) 2 4rR 2 (r 2 + d 2 ) 3/2 where d = ρ 2 + r 2 and c 0,c 1,c 2 depend on r,r and r. NOTE: H π 1 (X) and H X are even functions of φ.

13 Important lemma Lemma If M S 3 is a non-spherical symmetric surface, we can parametrize M with the convenient parameterization with (h,ρ) = (0,1).

14 Important lemma Lemma If M S 3 is a non-spherical symmetric surface, we can parametrize M with the convenient parameterization with (h,ρ) = (0,1). Proof. Suppose h 0. 0 = φ H M φ=0 = ( ) X X φ H π(m) + X N φ φ=0 ( ) φ=0 r = h rh π(m) ρ 2 r 2 + r 2 + ρ 2. Then (r 2 + ρ 2 )rr + ρ 2 r 2 + (r 2 + ρ 2 ) 2 = 0, which corresponds to a sphere.

15 A few corollaries Corollary M S 3 is a symmetric surface, ( 1 + r 2 r(r 2 + 1)r r 2 + r 4 1 ) H M = 2r(1 + r 2 + r 2 ) 3/2.

16 A few corollaries Corollary M S 3 is a symmetric surface, ( 1 + r 2 r(r 2 + 1)r r 2 + r 4 1 ) H M = 2r(1 + r 2 + r 2 ) 3/2. Corollary M S 3 is a CMC (H M H) symmetric surface, where c is constant r r 2 + r + H r 2 = c

17 Standard Tori parameter values H = c 1/4c r Θ

18 Spheres parameter values H = c r Θ

19 Catenoid types parameter values H 6= 0, c = 0. r Θ

20 Unduloid types parameter values c > 0, c 1/4c < H < c and c < 0, c < H < c 1/4c.

21 Nodoid types parameter values H > 0, 0 < c < H and H < 0, H < c < 0.

22 Period analysis Period of Unduloids and Nodoids T := rmax 2(c(r 2 + 1) H) r min 1 + r 2 r 2 (c(r 2 + 1) H)) 2dr

23 Period analysis Period of Unduloids and Nodoids T := rmax 2(c(r 2 + 1) H) r min 1 + r 2 r 2 (c(r 2 + 1) H)) 2dr Proposition M is an immersed torus iff T 2πQ. For unduloid types, this immersion is an embedding if T = 2π/m, m 2. This happens only for H 0. In this case, there is c = c(m,h) corresponding to an embedded unduloid type with m bulges and necks.

24 Classification theorem Theorem The complete symmetric CMC surfaces in S 3 are: standard tori, spheres, catenoid types, unduloid types, and nodoid types.

25 Classification theorem Theorem The complete symmetric CMC surfaces in S 3 are: standard tori, spheres, catenoid types, unduloid types, and nodoid types. There are (strictly immersed) catenoid, unduloid and nodoid type tori.

26 Classification theorem Theorem The complete symmetric CMC surfaces in S 3 are: standard tori, spheres, catenoid types, unduloid types, and nodoid types. There are (strictly immersed) catenoid, unduloid and nodoid type tori. For H 0, there are embedded unduloid type tori.

27 Two interesting and motivating open problems Sterling and Pinkall s Conjecture (1989) Embedded CMC tori are symmetric.

28 Two interesting and motivating open problems Sterling and Pinkall s Conjecture (1989) Embedded CMC tori are symmetric. Lawson s Conjecture (1970) The unique embedded minimal torus is the Clifford torus C = { (x 1,x 2,x 3,x 4 ) S 3 : x x2 2 = 1/2 = x } x2.

29 Two interesting and motivating open problems Sterling and Pinkall s Conjecture (1989) Embedded CMC tori are symmetric. Lawson s Conjecture (1970) The unique embedded minimal torus is the Clifford torus C = { (x 1,x 2,x 3,x 4 ) S 3 : x x2 2 = 1/2 = x } x2. Recall: Aleksandrov s Theorem (1956) The round sphere is the unique compact embedded CMC surface in R 3.

30 Rolling interpretation? An observation: Suppose M S 3 is symmetric and π(c) is the unit circle in R 2 R 3. Then for each ρ 1 π ρ (π(m) S ρ ) is a plane curve parametrized via ( ) θ 1 + r(θ) 2 ± r(θ) (cos θ,sin θ).

31 Rolling interpretation? An observation: Suppose M S 3 is symmetric and π(c) is the unit circle in R 2 R 3. Then for each ρ 1 π ρ (π(m) S ρ ) is a plane curve parametrized via ( ) θ 1 + r(θ) 2 ± r(θ) (cos θ,sin θ). Conjecture The above curve can also be obtained by trace of the focus of a conic section that rolls without slipping on the unit circle.