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 → R 3 be a compact surface, with two principal curvatures k 1 and k 2 . Then The Gauss curvature and mean curvature are defined by K = k 1 k 2 H = 1 2 ( k 1 + k 2 )
Embedded constant mean curvature tori in the three-sphere Backgrounds Gauss-Bonnet Theorem Let M be a compact surface in R 3 , then � K dA = 2 πχ ( M ) , 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 R 3 , then � K dA = 2 πχ ( M ) , 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 R 3 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 R 3 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 R 3 , H 3 or a hemisphere S 3 + , 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 R 3 , S 3 and H 3 .
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 S 3 is the Clifford torus S 1 ( 1 2 ) × S 1 ( 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 S 3 is the Clifford torus S 1 ( 1 2 ) × S 1 ( 1 2 ) . √ √ In March 2012, Simon Brendle of Stanford University solved this conjecture. See his paper: “Embedded minimal tori in S 3 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 S 3 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 S 3 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 S 3 is a surface of rotation. 1 3 , − 1 (2) If H ∈ { 0 , 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 S 1 × Z m if | H | lies m 2 − 2 strictly between cot π m and 2 √ m 2 − 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. √ 1 3 , − 1 (2) For H � = { 0 , 3 } and is not Clifford torus, CMC embedded √ √ tori are the analogues of Delaunay Surface in R 3 . 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 S 3 .
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 S 3 are: Totally umbilic 2-spheres, √ Clifford torus T r ≡ S 1 ( r ) × S 1 ( 1 − r 2 ) , 0 < r < 1: � ( x 1 , x 2 , x 3 , x 4 ) ∈ S 3 : x 2 4 = 1 − r 2 � 1 + x 2 2 = r 2 , x 2 3 + x 2 T r ≡ .
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 R n + 1 moving under the mean curvature flow: Non-collapsing result For any embedded compact mean-convex hypersurface M ⊂ R n + 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 ) � F ( y ) − F ( x ) � 2 + � F ( y ) − F ( x ) , ν ( x ) � ≥ 0 (5.1) δ 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 M n = F (Σ n ) be an embedded hypersurface in S n + 1 ⊂ R n + 2 given by an embedding F , and bounding a region Ω ⊂ S n + 1 . The ball in S n + 1 with boundary curvature Φ which is tangent to F (Σ) at the point F ( x ) is B = B Φ − 1 ( p ) , where p = F ( x ) − Φ − 1 ν ( x ) , and ν is the unit normal to F (Σ) at F ( x ) in S n + 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 ) − p � 2 ≥ Φ − 2 , which can be written as follows: � F ( y ) − ( F ( x ) − Φ − 1 ν ( x )) � 2 − Φ − 2 ≥ 0 . This is equivalent to Z (Φ , x , y ) := Φ( x ) � F ( y ) − F ( x ) � 2 + � F ( y ) − F ( x ) , ν ( x ) � ≥ 0 . (5.2) 2
Embedded constant mean curvature tori in the three-sphere Outline of proof of Theroem Since F ( x ) , F ( y ) ∈ S n + 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 of the surface at x , and denote it by ¯ Φ( x ) .
Recommend
More recommend
