Comparison Theorems for ODEs and Their Application to Geometry of Weingarten Hypersurfaces
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1 Comparison Theorems for ODEs and Their Application to Geometr of Weingarten Hpersurfaces Takashi Okaasu Ibaraki Universit Differential Geometr and Tanaka Theor Differential Sstem and Hpersurface Theor Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
2 Comparison Theorem A Assume F (X, Y ) satisfies F (0, 1/ 0 ) > 0 and F/ X < 0 for X>0. Solve the following equations with the initial conditions x(0) = x 0, (0) = 0, α(0) = α(0) = 0. dx (I) dα ds = F ( sin α x dx ds =cosα, d ds =sinα, (II) ds =cosα, d ds =sinα, dα ds, cos α ). = F (0, cos α ). = α() < α() for all satisfing 0 <α() < π 2 Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
3 Comparison Theorem B Assume F (0, 1/ 0 ) > 0 and F/ X, F/ Y < 0 for X, Y > 0. Solve the following equations with the initial conditions x(0) = x 0, (0) = 0, α(0) = α(0) = α 0 =0. (II) dx (I) dα ds = F ( sin α x dx ds =cosα, d ds =sinα, ds =cosα, d ds =sinα,, cos α ). dα ds = F ( sin α x 0, cos α 0 ). = α() > α() for all satisfing 0 <α() < π 2 Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
4 Example F (X, Y )=1 X Y Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
5 Our Aim We would like to construct new examples of complete Weingarten hpersurfaces in the Euclidean spaces. Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
6 Our Aim We would like to construct new examples of complete Weingarten hpersurfaces in the Euclidean spaces. Definition M R n+1 is called a Weingarten hpersurface λ 1,,λ n have a functional relation Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
7 Our Aim We would like to construct new examples of complete Weingarten hpersurfaces in the Euclidean spaces. Definition M R n+1 is called a Weingarten hpersurface λ 1,,λ n have a functional relation constant mean curvature hpersurfaces λ λ n =const constant scalar curvature hpersurfaces i j λ iλ j =const hpersurfaces whose second fundamental form h have constant length h 2 = i λ2 i =const Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
8 Delaune surfaces Figure: unduloid Figure: nodoid Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
9 A rotational surface with h = const Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
10 Noncompact Complete Hpersurfaces with Constant Scalar Curvature Known Examples flat generalized clinders clinders S p R n p 1-parameter famil of rotational hpersurfaces (Leite, 1990) a complete hpersurface with constant negative scalar curvature in E 4 (O, 1989) a complete hpersurface with 0 scalar curvature in E 4 (Palmas,2000) a complete hpersurface with 0 scalar curvature in E 2n (Sato, 2000) Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
11 Result & Method Result We can construct a new famil of complete hpersurfaces with constant positive scalar curvature. The are diffeomorphic to R S p S q. Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
12 Result & Method Result We can construct a new famil of complete hpersurfaces with constant positive scalar curvature. The are diffeomorphic to R S p S q. Method Method of equivariant geometr We use a subgroup O(p +1) O(q +1) O(p + q +2)=O(n +1)to construct Generalized rotational hpersurfaces. PDE ODE This method was used b W.Y. Hsiang et al. to construct man CMC hpersurfaces in the 80 s. Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
13 O(p +1) O(q +1)-invariant hpersurfaces O(p +1) O(q +1) R p+1 R q+1 The orbit space= the first quadrant R 2 + The orbit through (x, ) =S p (x) S q () Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
14 Generalized Rotational Hpersurfaces γ : a curve in the first quadrant R 2 + M γ : O(p +1) O(q +1)-invariant hpersurface generated b γ Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
15 Main Theorem 1 There exists a new famil of complete hpersurfaces M n E n+1 (n 5) with constant positive scalar curvature. 0 x x 0 x Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
16 Scalar Curvature Equation principal curvatures x x, x, x : multiplicities 1, p, q (the curve is parametrized b the arc length) The scalar curvature of M γ S = λ i λ j i j ) ( = 2(x x ) (p x q x + p(p 1) x ( x ) 2 +q(q 1) 2pq x x ) 2 Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
17 Constant Scalar Curvature Equation S = constant (I) dx ds =cosα, d ds =sinα, dα ds = p(p 1)( sin α x ) 2 ( ) 2pq sin α cos α x +q(q 1) cos α 2 S ( ). 2 q cos α p sin α x Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
18 Comparison Equation (I) dx ds =cosα, d ds =sinα, dα ds = p(p 1)( sin α x ) 2 ( ) 2pq sin α cos α x +q(q 1) cos α 2 S ( ). 2 q cos α p sin α x (II) dx ds =cosα, d ds =sinα, ( ) cos α 2 S dα ds = q(q 1) 2q cos α. Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
19 The solution of (II) 1 l 2 l 1 1 x Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
20 Comparison Theorem 0 < = q(q 1)/S Solve (I), (II) with the same initial conditions x(0) = x 0, (0) = 0, α(0) = 0. α() < α() where α is the solution of (II) Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
21 Comparison of Solution Curves x 0 x Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
22 Main Theorem 2 There exists a new famil of complete hpersurfaces M n E n+1 (n 5) with h =const. Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
23 = α() > α(), when 0 <α< π 2 Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21 Comparison Theorem for h = h 0 (const) Solve the following equations with the initial conditions x(0) = x 0, (0) = 0, α(0) = α(0) = α 0. (I) dx ds =cosα, d ds =sinα, dα ds = h 2 0 p ( sin α x ) 2 ( ) q cos α 2. (II) dx ds =cosα, d ds =sinα, dα ds = h 2 0 p ( sin α x 0 ) 2 ( ) q cos α0 2.
24 Theorem (Ye, 1991) Let (M n+1,g) be a Riemannian manifold. Suppose that p 0 is a nodegenerate critical point of the scalar curvature of M.Then there exists r 0 > 0, such that for all ρ (0,r 0 ), the geodesic sphere S ρ (p 0 ) ma be perturbed to a constant mean curvature hpersurface S ρ with H =1/ρ. Theorem Let (M n+1,g) be a Riemannian manifold. Suppose that p 0 is a nondegenerate critical point of the scalar curvature of M. Then there exists r 0 > 0, such that for all ρ (0,r 0 ), the geodesic sphere S ρ (p 0 ) ma be perturbed to a closed hpersurface S ρ whose second fundamental form are of constant length n/ρ. Takashi Okaasu (Ibaraki Universit) Comparison Theorems for ODEs Januar 25th, / 21
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