Embedded constant mean curvature tori in the three-sphere Haizhong - - PowerPoint PPT Presentation

Embedded constant mean curvature tori in the three-sphere

Embedded constant mean curvature tori in the three-sphere

Haizhong Li (Tsinghua University)

Joint works with Ben Andrews (National Australia University)

PADGE2012, August 27-30, KU Leuven

Embedded constant mean curvature tori in the three-sphere Contents

Contents

1

Backgrounds

2

Lawson Conjecture

3

Pinkal-Sterling Conjecture and our Theorem

4

Outline of proof of Theroem

5

Reference

Embedded constant mean curvature tori in the three-sphere Backgrounds

1

Backgrounds

2

Lawson Conjecture

3

Pinkal-Sterling Conjecture and our Theorem

4

Outline of proof of Theroem

5

Reference

Embedded constant mean curvature tori in the three-sphere Backgrounds

Let x : M → R3 be a compact surface, with two principal curvatures k1 and k2. Then The Gauss curvature and mean curvature are defined by K = k1k2 H = 1 2(k1 + k2)

Embedded constant mean curvature tori in the three-sphere Backgrounds

Gauss-Bonnet Theorem Let M be a compact surface in R3, then

• M

K dA = 2πχ(M), where χ(M) is the Euler characteristic of M, χ(M) = 2(1 − g), g is genus of M.

Embedded constant mean curvature tori in the three-sphere Backgrounds

Gauss-Bonnet Theorem Let M be a compact surface in R3, then

• M

K dA = 2πχ(M), where χ(M) is the Euler characteristic of M, χ(M) = 2(1 − g), g is genus of M. Liebmann Theorem, 1899 Let M be a compact surface with K = constant,then M is a round sphere.

Embedded constant mean curvature tori in the three-sphere Backgrounds

In 1950s, by constructing a holomorphic quadratic differential for CMC surfaces, H. Hopf proved Hopf Theorem Let M be a compact surface with H = constant and g(M) = 0, then M is a round sphere.

Embedded constant mean curvature tori in the three-sphere Backgrounds

In 1950s, by constructing a holomorphic quadratic differential for CMC surfaces, H. Hopf proved Hopf Theorem Let M be a compact surface with H = constant and g(M) = 0, then M is a round sphere.

• S. S. Chern extended Hopf’s result to CMC two-spheres in

3-dimensional space forms.

Embedded constant mean curvature tori in the three-sphere Backgrounds

Hopf proposed in 1950s: Hopf Conjecture Any compact surfaces with H = constant in R3 must be a round sphere.

Embedded constant mean curvature tori in the three-sphere Backgrounds

Hopf proposed in 1950s: Hopf Conjecture Any compact surfaces with H = constant in R3 must be a round sphere. In 1956, Alexsandrov checked Hopf’s conjecture under extra condition “embeddedness". Alexsandrov’s uniqueness Theorem If a compact CMC surface is embedded in R3, H3 or a hemisphere S3

+, then it must be totally umbilical.

Embedded constant mean curvature tori in the three-sphere Backgrounds

In 1984, Wente constructed counterexamples (non-trivial CMC tori) for Hopf’s conjecture by use of integrable systems. The following are Wente’s CMC tori Wente’s paper was followed by a series of papers by Bobenko, Pinkal-Sterling and many others. In particular, they constructed CMC tori in R3, S3 and H3.

Embedded constant mean curvature tori in the three-sphere Lawson Conjecture

1

Backgrounds

2

Lawson Conjecture

3

Pinkal-Sterling Conjecture and our Theorem

4

Outline of proof of Theroem

5

Reference

Embedded constant mean curvature tori in the three-sphere Lawson Conjecture

In 1970, H. B. Lawson conjectured that Lawson conjecture, 1970 The only embedded minimal torus in S3 is the Clifford torus S1( 1

√ 2) × S1( 1 √ 2).

Embedded constant mean curvature tori in the three-sphere Lawson Conjecture

In 1970, H. B. Lawson conjectured that Lawson conjecture, 1970 The only embedded minimal torus in S3 is the Clifford torus S1( 1

√ 2) × S1( 1 √ 2).

In March 2012, Simon Brendle of Stanford University solved this

• conjecture. See his paper:

“Embedded minimal tori in S3 and the Lawson conjecture.” arXiv: 1203.6596

Embedded constant mean curvature tori in the three-sphere Pinkal-Sterling Conjecture and our Theorem

1

Backgrounds

2

Lawson Conjecture

3

Pinkal-Sterling Conjecture and our Theorem

4

Outline of proof of Theroem

5

Reference

Embedded constant mean curvature tori in the three-sphere Pinkal-Sterling Conjecture and our Theorem

In 1989, Pinkall and Sterling conjectured that Pinkall-Sterling conjecture, 1989 All embedded CMC tori in S3 are surfaces of revolution.

Embedded constant mean curvature tori in the three-sphere Pinkal-Sterling Conjecture and our Theorem

In 1989, Pinkall and Sterling conjectured that Pinkall-Sterling conjecture, 1989 All embedded CMC tori in S3 are surfaces of revolution. In April 2012, Ben Andrews and Haizhong Li confirm this conjecture. Moreover we gave a complete classification of such embedded tori. See their paper: “Embedded constant mean curvature tori in the three-sphere arXiv: 1204.5007

Embedded constant mean curvature tori in the three-sphere Pinkal-Sterling Conjecture and our Theorem

Main Theorem (Andrews-Li,2012) (1) Every embedded CMC torus Σ in S3 is a surface of rotation. (2) If H ∈ {0,

1 √ 3, − 1 √ 3} then every embedded torus with mean

curvature H is congruent to the Clifford torus. (3) If Σ is an embedded CMC torus which is not congruent to a Clifford torus, then there exists a maximal integer m ≥ 2 such that Σ has m-fold symmetry. (4) For given m ≥ 2, there exists at most one such CMC torus (up to congruence). (5) For given m ≥ 2, there exists an embedded CMC torus with mean curvature H and maximal symmetry S1 × Zm if |H| lies strictly between cot π

m and m2−2 2√ m2−1.

Embedded constant mean curvature tori in the three-sphere Pinkal-Sterling Conjecture and our Theorem

Remark (1) The case H = 0 is the Lawson conjecture which was proved by

• Brendle. The rigidity appearing for H = ± 1

√ 3 is unexpected.

(2) For H = {0,

1 √ 3, − 1 √ 3} and is not Clifford torus, CMC embedded

tori are the analogues of Delaunay Surface in R3. Number of these CMC embedded tori depends on the value of H. (3) The embeddedness assumption in Main Theorem is crucial: There exists an infinite family of non-rotationally symmetric immersed CMC tori in S3.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

1

Backgrounds

2

Lawson Conjecture

3

Pinkal-Sterling Conjecture and our Theorem

4

Outline of proof of Theroem

5

Reference

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The simplest examples of CMC surfaces in S3 are: Totally umbilic 2-spheres, Clifford torus Tr ≡ S1(r) × S1( √ 1 − r 2), 0 < r < 1: Tr ≡

• (x1, x2, x3, x4) ∈ S3 : x2

1 + x2 2 = r 2, x2 3 + x2 4 = 1 − r 2

.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

In 2011, Ben Andrews gave a direct proof of the non-collapsing result for mean-convex hypersurface in Rn+1 moving under the mean curvature flow: Non-collapsing result For any embedded compact mean-convex hypersurface M ⊂ Rn+1 moving under the mean curvature flow, there is a positive constant δ such that at every point x of M there is a sphere of radius δ/H(x) enclosed by M which touches M at x.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Ben Andrews observed that the noncollapsing condition is equivalent to that Z : M × M → R satisfies Z(x, y) = H(x) δ F(y) − F(x)2 + F(y) − F(x), ν(x) ≥ 0 (5.1) for (x, y) ∈ M × M and ν(x) is an unit outward normal vector of F(x). This function was shown to admit a maximum principle argument to preserve initial non-negativity. The idea of working with functions of pairs of points was in turn inspired by earlier work of Huisken and Hamilton for the curve shortening flow and for Ricci flow on surfaces.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The key geometric idea in the non-collapsing argument is to compare the curvature of enclosed balls touching the surface to a suitable function at the touching point. Let Mn = F(Σn) be an embedded hypersurface in Sn+1 ⊂ Rn+2 given by an embedding F, and bounding a region Ω ⊂ Sn+1. The ball in Sn+1 with boundary curvature Φ which is tangent to F(Σ) at the point F(x) is B = BΦ−1(p), where p = F(x) − Φ−1ν(x), and ν is the unit normal to F(Σ) at F(x) in Sn+1 which points out of Ω.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The statement that this ball lies entirely in Ω is equivalent to the statement that for any y ∈ Σ, F(y) − p2 ≥ Φ−2, which can be written as follows: F(y) − (F(x) − Φ−1ν(x))2 − Φ−2 ≥ 0. This is equivalent to Z(Φ, x, y) := Φ(x) 2 F(y) − F(x)2 + F(y) − F(x), ν(x) ≥ 0. (5.2)

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Since F(x), F(y) ∈ Sn+1 we have F(x)2 = F(y)2 = 1 and F(x), ν(x) = 0, so that Z(Φ, x, y) = Φ(x)(1 − F(x) · F(y)) + F(y), ν(x). (5.3) We call the smallest Φ(x) for Z(Φ, x, y) ≥ 0 the interior ball curvature

• f the surface at x, and denote it by ¯

Φ(x).

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The first step of the proof of the main theorem is that for an embedded CMC torus in S3 we always has ¯ Φ(x) = λ(x). The case H = 0 was proved by Brendle, so we assume that H > 0. We denote by λ(x) = λ1(x) the largest principal curvature at x, and by µ(x) = λ(x) − H. Then we choose Φ(x) = κµ(x) + H where κ is a positive constant. We require the following variant of Simons’ identity:

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Proposition 1 Suppose that F : Σ → S3 is an embedded CMC torus in S3. Then the function µ is strictly positive and satisfies ∆µ − |∇µ|2 µ + 2(µ2 − 1 − H2)µ = 0.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Since Σ is compact and embedded, for sufficiently large κ, then Z(κµ + H, x, y) is non-negative. Along any geodesic in Σ through x we have Z(κµ + H, x, γ(s)) = 1 2[κµ + H − hx(γ′, γ′)]s2 + O(s3). Choose γ′(0) to be in the direction of the largest principal curvature, so that hx(γ′, γ′) = λ = H + µ. Then Z(κµ + H, x, γ(s)) = 1 2(κ − 1)µs2 + O(s3)

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

If κ < 1, Z takes negative values for small s. If κ > 1, Z is positive in a neighbourhood of the diagonal {(x, x) : x ∈ Σ} in Σ × Σ. We choose ¯ κ = inf{κ > 0 : Z(κµ + H, x, y) ≥ 0 for all x, y ∈ Σ} Then 1 ≤ ¯ κ < ∞. If ¯ κ > 1, then there must exist (¯ x, ¯ y) in Σ × Σ with ¯ x = ¯ y such that Z(¯ x, ¯ y) = 0, while Z(x, y) ≥ 0 for every (x, y) ∈ Σ × Σ.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Then the second derivatives of Z are non-negative at (¯ x, ¯ y). However, by a careful calculation and using Proposition 1, we have

2

• i=1

∂ ∂xi + ∂ ∂yi 2 Z

• (¯

x,¯ y) ≤ −(κ2 − 1)d2µ2H < 0,

(5.4) which is a contradiction. So we conclude that ¯ κ = 1 and ¯ Φ(x) = λ(x), that is, for any x ∈ Σ and y ∈ Σ Z(λ, x, y) = λ(x)(1 − F(x) · F(y)) + F(y), ν(x) ≥ 0. (5.5)

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The second step of the proof is to prove the following Proposition: Proposition 2 Let F : Σ → S3 be a CMC embedding for which Z(λ(x), x, y) ≥ 0 for every x, y ∈ Σ (equivalently, ¯ Φ(x) = λ(x) everywhere). Then Σ is rotationally symmetric.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Since Σ is a CMC torus and therefore has no umbilical points, we have global smooth eigenvector fields e1 and e2 such that h(e1, e1) = λ1 = λ and h(e2, e2) = λ2 = 2H − λ, and h(e1, e2) = 0. We can deduce that (∇e1h)(e1, e1) = 0, and consequently also (∇e1h)(e2, e2) = 0 everywhere on Σ. So that e1λ = 0 and then e1µ = 0.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

From the Codazzi equations and H = constant, we have ∇e1e1 = e2(µ) 2µ e2, ∇e2e1 = 0. (5.6) ∇e2e2 == 0, ∇e1e2 = −e2(µ) 2µ e1. (5.7) It follows from (5.7) that the flow lines of e2 are geodesic in Σ.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Writing w = µ− 1

2 ,

(5.8) From the Gauss equation, we get 1 w e2(e2(w)) − 1 w4 + H2 + 1 = 0. (5.9) Multiplying by 2we2(w), we obtain [e2(w)]2 + w−2 + (1 + H2)w2 = C1, (5.10) where C1 is a constant.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Let us fix a point x0 ∈ M, and denote by σ(u) the geodesic in Σ such that σ(0) = x0 and σ′(0) = e2(x0). We write g(u) = w(σ(u)). Equation (5.10) implies that (g′)2 + g−2 + (1 + H2)g2 + 2H = C (5.11) where C is a constant greater than 2(H + √ 1 + H2) and C = C1 + 2H. The polynomial ξ(s) = Cs2 − 1 − (1 + H2)s4 − 2Hs2 (5.12) is positive on an interval (t1, t2) with 0 < t1 < t2 and ξ(t1) = ξ(t2) = 0.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The roots can be explicitly calculated: t1 =

• C − 2H −

√ C2 − 4HC − 4 2(1 + H2) , t2 =

• C − 2H +

√ C2 − 4HC − 4 2(1 + H2) . (5.13)

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

We have that g is a periodic function with period T = 2 t2

t1

t

• (C − 2H)t2 − 1 − (1 + H2)t4 dt.

We can solve g(u) from (24) g(u) =

• C − 2H +

√ C2 − 4 − 4HCsin(2 √ 1 + H2u) 2(1 + H2) . From the expression of g(u), we get that its period T =

π

√

1+H2 .

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

Through delicate analysis, we can solve F(u, v) = (r(u) cos v, r(u) sin v,

• 1 − r 2(u) cos θ(u),
• 1 − r 2(u) sin θ(u))

(5.14) where 0 ≤ v < 2π, 0 ≤ u <

mπ

√

1+H2 and m is some positive integer,

where θ(u) = u r(τ)(µ(τ) + H) 1 − r 2(u) dτ) r(u) = µ−1/2 √ C =

• C − 2H +

√ C2 − 4 − 4HCsin(2 √ 1 + H2u) 2(1 + H2)C . where C is the constant in (5.11). Note that g, r only depends on u, since e1(µ) = 0. This completes the proof of Proposition 2.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The third step of the proof is proving that the period function K(H, C) is monotone in C, where K(H, C) =

• t2

2 C t2 1 C

(Hu + C−1) √u(1 − u)

• −u2(1 + H2) + (1 − 2HC−1)u − C−2)

du, where C is a constant greater than 2(H + √ 1 + H2) and t1 and t2 are defined by (5.12).

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The following result due to Otsuki and Perdomo: Proposition 6 Suppose that F : Σ → S3 is a rotational torus in S3, which is not a Clifford torus and is given by (5.14) Then F(Σ) is an embedded torus if and only if K(H, C) = 2π m (5.15) for some positive integer m.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

The following result is due to Perdomo Proposition 7 If H = 0, ± 1

√ 3, there exist compact embedded tori in S3 with constant

mean curvature H, which are not Clifford tori. In fact, for any integer m ≥ 2, if H satisfies cot π m < H < m2 − 2 2 √ m2 − 1 , (5.16) then there exists a compact embedded torus in S3 with constant mean curvature H whose isometry group contains O(2) × Zm which is not a Clifford torus.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

We can prove the following result Proposition 8 For any nonnegative real number H, K(H, C) is monotone decreasing in 2(H + √ 1 + H2) < C < ∞. When H = 0, proposition 8 was proved by T. Otsuki.

Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem

By Proposition 2, every embedded CMC torus is a surface of rotation, and we can the other statements (see our paper in details). So we can complete the proof of Main Theorem.

Embedded constant mean curvature tori in the three-sphere Reference

1

Backgrounds

2

Lawson Conjecture

3

Pinkal-Sterling Conjecture and our Theorem

4

Outline of proof of Theroem

5

Reference

Embedded constant mean curvature tori in the three-sphere Reference

References I

[Br]Simon Brendle, Embedded minimal tori in S3 and the Lawson conjecture. arXiv: 1203.6596 [AL]Ben Andrews and Haizhong Li Embedded constant mean curvature tori in the three-sphere. arXiv: 1204.5007 [Al]A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. I, Vestnik Leningrad. Univ. in Russian, 11(1956), no.19, 5-17. [An] Ben Andrews, Non-collapsing in mean-convex mean curvature flow, arxiv:1108.0247

Embedded constant mean curvature tori in the three-sphere Reference

References II

[Ho1]Heinz Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen,

• Math. Nachr.,4(1951), 232-249

[Ot]Tominosuke Otsuki, On a differential equation related with differential geometry,

• Mem. Fac. Sci. Kyushu Univ. Ser. A, 47(1993), no.2, 245-281.

[Per]Oscar M. Perdomo, Embedded constant mean curvature hypersurfaces on spheres, Asian J. Math., 14(2010), no.1,73-108. [PS]U. Pinkall and I. Sterling, On the classification of constant mean curvature tori,

• Ann. of Math. (2), 130(1989),no.2,407-451.

Embedded constant mean curvature tori in the three-sphere Reference

References III

[We]Henry C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math., 121(1986), no.1,193-243. [Ma]Dana Mackenzie, What’s Happening in the Mathematical Sciences, Volume 9 American Mathematical Society, 2012.

Embedded constant mean curvature tori in the three-sphere Reference

Thank you for your attention!