The behavior of the principal distributions on a real-analytic surface

Naoya Ando

Abstract. The purposes of this paper are: (1) to study the behavior of the principal distributions around an isolated umbilical point on a real-analytic surface of a certain type, which contains all the real-analytic, special Weingarten surfaces; (2) to give one condition such that if a real-analytic surface satisfies the condition, then the index of an isolated umbilical point of the surface is less than or equal to one.

1 Introduction

Let S be a smooth surface in R^{3} and Umb(S) the set of the umbilical points of S and set Reg(S) := S \ Umb(S). If Reg(S) ≠ ∅, then there exists a principal distribution D_{S} on S, which is a one-dimensional continuous distribution on Reg(S) such that D_{S}(p) is one of the principal directions at p ∈ Reg(S). Let p_{0} be an isolated umbilical point of S. Then as a quantity in relation to the behavior of D_{S} around p_{0}, the index ind_{p_{0}}(S) of p_{0} is defined([3, pp. 137]).

For each l ∈ N ∪ {0}, let A_{o}^{(l)} be the set of the real-analytic functions defined on a neighborhood of (0, 0) in R^{2} such that for each F ∈ A_{o}^{(l)},

\frac{∂^{m+n}F}{∂x^{m}∂y^{n}} (0, 0) = 0 holds for non-negative integers m, n satisfying 0 ≦ m +

n < l. Let F be an element of A_{o}^{(2)} and G_{F} the graph of F. If the origin o of R^{3} is a non-interior point of Umb(G_{F}), then there exists an element f_{F} ∈ A_{o}^{(3)} satisfying Reg(G_{F −f_{F}}) = ∅, and there exists a nonzero homogeneous polynomial φ_{F} of degree k_{F} satisfying f_{F} −φ_{F} ∈ A_{o}^{(k_{F} +1)}. Let A_{o}^{l} be the set of the elements of A_{o}^{(l)} such that on their graphs, o is an isolated umbilical point, and P_{o} the set of the homogeneous polynomials contained in A_{o}^{2}, and A_{oo}^{l} the subset of A_{o}^{l} such that φ_{F} ∈ P_{o} holds for F ∈ A_{oo}^{l}. The purposes of this paper are

(1) to study the behavior of the principal distributions around o on the graph G_{F} of F ∈ A_{oo}^{2};

(2) to give one condition such that if F ∈ A_{o}^{2} satisfies the condition, then ind_{o}(G_{F}) ≦ 1 holds.

For F ∈ A_{o}^{2}, there exist two principal distributions D_{F}^{(1)}, D_{F}^{(2)} on G_{F} which give the principal directions to each point of Reg(G_{F}), and there exists a positive number ρ_{0} > 0 satisfying {0 < x^{2} + y^{2} < ρ_{0}^{2}} ⊂ Reg(G_{F}). Let ϕ_{F}^{(i)} be a continuous function defined on B_{0} := (0, ρ_{0}) × R satisfying

cos ϕ_{F}^{(i)}(ρ, θ)\frac{∂}{∂x} + sin ϕ_{F}^{(i)}(ρ, θ)\frac{∂}{∂y} ∈ D_{F}^{(i)}(ρ cos θ, ρ sin θ)

for (ρ, θ) ∈ B_{0}. In Section 3, we shall prove the following:

Proposition 1.1 For F ∈ A_{o}^{2} and for θ_{0} ∈R,

(1) there exists the number ϕ_{F,o}^{(i)}(θ_{0}) satisfying

lim_{ρ→+0} ϕ_{F}^{(i)}(ρ, θ_{0}) = ϕ_{F,o}^{(i)}(θ_{0});

(2) there exist the numbers ϕ_{F,o}^{(i)}(θ_{0} + 0), ϕ_{F,o}^{(i)}(θ_{0} − 0) satisfying

lim_{θ→θ_{0}±0} ϕ_{F,o}^{(i)}(θ) = ϕ_{F,o}^{(i)}(θ_{0} ± 0);

(3) there exists the element Γ_{F,o}(θ_{0}) of Z_{0} := {nπ/2}_{n∈Z} satisfying

Γ_{F,o}(θ_{0}) = ϕ_{F,o}^{(i)}(θ_{0} + 0) − ϕ_{F,o}^{(i)}(θ_{0} − 0)

for i = 1, 2.

For θ ∈R, let Hess_{φ_{F}} (θ) be the Hessian of φ_{F} at (cos θ, sin θ), and η_{φ_{F}} a continuous function on R such that^{t}(cos η_{φ_{F}} (θ), sin η_{φ_{F}} (θ)) is an eigenvector of Hess_{φ_{F}} (θ) for any θ ∈R, and S_{F,o} the set of the numbers at each of which Hess_{φ_{F}} is scalar. In Section 3, we shall prove the following:

Proposition 1.2 For F ∈ A_{o}^{2},

(1) if θ_{0} ∈ R satisfies Γ_{F,o}(θ_{0}) ≠ 0, then θ_{0} ∈ S_{F,o} holds;

(2) for θ ∈R, ind_{o}(G_{F}) is represented as

ind_{o}(G_{F}) = \frac{η_{φ_{F}} (θ + 2π) − η_{φ_{F}} (θ)}{2π} + \frac{1}{2π} ∑_{θ_{0}∈S_{F,o}∩[θ,θ+2π)} Γ_{F,o}(θ_{0}).

In Section 4, we shall give one way of computing η_{φ_{F}} (θ + 2π) − η_{φ_{F}} (θ). In Section 5, we shall prove the following:

Theorem 1.3 For F ∈ A_{oo}^{2},

(1) −π/2 ≦ Γ_{F,o}(θ_{0}) ≦ π/2 hold for θ_{0} ∈ S_{F,o}; (2) ind_{o} (G_{φ_{F}} ) ≦ ind_{o}(G_{F}) ≦ 1 hold.

Remark For φ_{F} ∈ P_{o}, ind_{o} (G_{φ_{F}} ) ∈ {1 + i − k_{F} /2}_{i=0}^{[k_{F} /2]} holds([1]).

We know a conjecture which says that ind_{o}(G_{F}) ≦ 1 holds for any F ∈ A_{o}^{2}. This is part of Loewner’s conjecture([4]). In Section 5, we shall prove the following:

Theorem 1.4 Let F be an element of A_{o}^{2} satisfying Γ_{F,o}(θ_{0}) ≦ π for any θ_{0} ∈ S_{F,o}. Then ind_{o}(G_{F}) ≦ 1 holds.

Let S be a real-analytic, Weingarten surface and w_{S} a function of two variables satisfying w_{S}(K_{S}, H_{S}) ≡ 0 on S, where K_{S} and H_{S} are Gaussian curvature and the mean curvature of S, respectively. In this paper, we suppose that w_{S} is real-analytic, and according to [2], we call S special if w_{S} satisfies

H_{S} \frac{∂w_{S}}{∂X} (K_{S}, H_{S}) + \frac{1}{2} \frac{∂w_{S}}{∂Y} (K_{S}, H_{S}) ≠ 0  (1.1)

on Umb(S). In Section 6, we shall prove the following:

Theorem 1.5 Let F be an element of A_{o}^{(2)} whose graph is a non-totally umbilical, special Weingarten surface with an umbilical point o. Then the following hold:

(1) F ∈ A_{oo}^{2}, (2) ind_{o}(G_{F}) = ind_{o} (G_{φ_{F}} ) = 1 − k_{F} /2.

Remark Since k_{F} ≧ 3, we see that if F is as in Theorem 1.5, then ind_{o}(G_{F}) < 0 holds, which was also obtained in [2].

2 Preliminaries

Let f be a smooth function of two variables and G_{f} the graph of f. We set

p_{f} := \frac{∂f}{∂x}, q_{f} := \frac{∂f}{∂y}, and

E_{f} := 1 + p_{f}^{2},  F_{f} := p_{f}q_{f},  G_{f} := 1 + q_{f}^{2}.  (2.1)

The first fundamental form of G_{f} is the symmetric tensor field I_{f} on G_{f} of type (0, 2) represented in terms of the coordinates (x, y) as

I_{f} := E_{f}dx^{2} + 2F_{f}dxdy + G_{f}dy^{2},

where

dx^{2} := dx ⊗ dx, dxdy := \frac{1}{2}(dx ⊗ dy + dy ⊗ dx), dy^{2} := dy ⊗ dy.

We set r_{f} := \frac{∂^{2}f}{∂x^{2}} , s_{f} := \frac{∂^{2}f}{∂x∂y} , t_{f} := \frac{∂^{2}f}{∂y^{2}} , and

L_{f} := \frac{r_{f}}{\sqrt{det(I_{f})}}, M_{f} := \frac{s_{f}}{\sqrt{det(I_{f})}}, N_{f} := \frac{t_{f}}{\sqrt{det(I_{f})}},  (2.2)

where det(I_{f}) := E_{f}G_{f} − F_{f}^{2}. The second fundamental form of G_{f} is the symmetric tensor field II_{f} on G_{f} of type (0, 2) represented in terms of the coordinates (x, y) as

II_{f} := L_{f}dx^{2} + 2M_{f}dxdy + N_{f}dy^{2}.

Let T_{(x_{0},y_{0})}(G_{f}) be the tangent plane to G_{f} at (x_{0}, y_{0}) and U_{(x_{0},y_{0})}(G_{f}) the subset of T_{(x_{0},y_{0})}(G_{f}) defined by

U_{(x_{0},y_{0})}(G_{f}) := {u ∈ T_{(x_{0},y_{0})}(G_{f}) ; I_{f,(x_{0},y_{0})}(u, u) = 1},

and L_{(x_{0},y_{0})}(G_{f}) the set of the one-dimensional subspaces of T_{(x_{0},y_{0})}(G_{f}). Let ν_{f,(x_{0},y_{0})} be the function on U_{(x_{0},y_{0})}(G_{f}) defined by

ν_{f,(x_{0},y_{0})}(u) := II_{f,(x_{0},y_{0})}(u, u)

for u∈ U_{(x_{0},y_{0})}(G_{f}). Suppose that ν_{f,(x_{0},y_{0})} attains an extremum at u_{0} ∈ U_{(x_{0},y_{0})}(G_{f}). Then the extremum ν_{f,(x_{0},y_{0})}(u_{0}) is called a principal curvature of G_{f} at (x_{0}, y_{0}) and the element of L_{(x_{0},y_{0})}(G_{f}) determined by u_{0} is called a principal direction of G_{f} at (x_{0}, y_{0}). The set of the principal directions at (x_{0}, y_{0}) is denoted by P D_{(x_{0},y_{0})}(G_{f}). The Weingarten map of G_{f} is the tensor field W_{f} on G_{f} of type (1, 1) satisfying

[W_{f}(\frac{∂}{∂x}), W_{f}(\frac{∂}{∂y} )] = [\frac{∂}{∂x}, \frac{∂}{∂y} ]W_{f},

where

W_{f} := (\begin{matrix}E_{f} & F_{f} \\ F_{f} & G_{f} \end{matrix})^{−1} (\begin{matrix}L_{f} & M_{f} \\ M_{f} & N_{f} \end{matrix}) .

By Lagrange’s method of indeterminate coefficients, we obtain

Proposition 2.1 The principal curvatures and the principal directions of G_{f} at (x_{0}, y_{0}) are given by the eigenvalues and by the one-dimensional eigenspaces of W_{f,(x_{0},y_{0})}, respectively.

Gaussian curvature K_{f}(x_{0}, y_{0}) and the mean curvature H_{f}(x_{0}, y_{0}) of G_{f} at (x_{0}, y_{0}) are given by the following:

K_{f}(x_{0}, y_{0}) := det (W_{f,(x_{0},y_{0})}), H_{f}(x_{0}, y_{0}) := tr (W_{f,(x_{0},y_{0})})/2.  (2.3)

Let PD_{f} be the symmetric tensor field on G_{f} of type (0, 2) represented in terms of the coordinates (x, y) as

PD_{f} := \frac{1}{\sqrt{det(I_{f})}}\{A_{f}dx^{2} + B_{f}dxdy + C_{f}dy^{2}\},

where

A_{f} := F_{f}L_{f} − E_{f}M_{f}, B_{f} := G_{f}L_{f} − E_{f}N_{f}, C_{f} := G_{f}M_{f} − F_{f}N_{f}.

For v_{1} , v_{2} ∈ T_{(x_{0},y_{0})}(G_{f}), the following holds:

\frac{1}{2} ∑_{\{i,j\}=\{1,2\}} W_{f,(x_{0},y_{0})}(v_{i}) ∧ v_{j} = \frac{PD_{f,(x_{0},y_{0})}(v_{1}, v_{2})}{\sqrt{det (I_{f,(x_{0},y_{0})})}}(\frac{∂}{∂x} ∧ \frac{∂}{∂y} )_{(x_{0},y_{0})}.

Then by Proposition 2.1, we obtain

Proposition 2.2 A tangent vector v_{0} ∈ T_{(x_{0},y_{0})}(G_{f}) belongs to an element of P D_{(x_{0},y_{0})}(G_{f}) if and only if PD_{f,(x_{0},y_{0})}(v_{0}, v_{0}) = 0 holds.

A point (x_{0}, y_{0}) is called an umbilical point of G_{f} if ν_{f,(x_{0},y_{0})} is constant.

Proposition 2.3 The following are mutually equivalent:

(1) (x_{0}, y_{0}) ∈ Umb(G_{f}), (2) P D_{(x_{0},y_{0})}(G_{f}) = L_{(x_{0},y_{0})}(G_{f}),

(3) A_{f}(x_{0}, y_{0}) = B_{f}(x_{0}, y_{0}) = C_{f}(x_{0}, y_{0}) = 0.

Let D_{f}, N_{f} be the symmetric tensor fields on G_{f} of type (0, 2) represented in terms of the coordinates (x, y) as

D_{f} := s_{f}dx^{2} + (t_{f} − r_{f})dxdy − s_{f}dy^{2},
N_{f} := (s_{f}p_{f}^{2} − p_{f}q_{f}r_{f})dx^{2} + (t_{f}p_{f}^{2} − r_{f}q_{f}^{2})dxdy + (p_{f}q_{f}t_{f} − s_{f}q_{f}^{2})dy^{2}.

By (2.1) and by (2.2), we obtain det(I_{f})PD_{f} = D_{f} + N_{f}. For a vector field V on G_{f}, we set

D̃_{f}(V) := D_{f}(V, V),  Ñ_{f}(V) := N_{f}(V, V),
P̃D_{f}(V) := PD_{f}(V, V).

We set

grad_{f} := ( \begin{matrix}p_{f} \\ q_{f} \end{matrix} ) , grad_{f}^{⊥} := ( \begin{matrix}−q_{f} \\ p_{f} \end{matrix} ) , Hess_{f} := (\begin{matrix}r_{f} & s_{f} \\ s_{f} & t_{f} \end{matrix}) .

For ϕ ∈R, we set

u_{ϕ} := ( \begin{matrix}cos ϕ \\ sin ϕ \end{matrix} ) , U_{ϕ} := cos ϕ\frac{∂}{∂x} + sin ϕ\frac{∂}{∂y} .

Let ⟨ , ⟩ be the scalar product in R^{2}. Then we obtain Lemma 2.4 For any ϕ ∈R, the following hold:

D̃_{f}(U_{ϕ}) = ⟨ Hess_{f} u_{ϕ}, u_{ϕ+π/2}⟩,
Ñ_{f}(U_{ϕ}) = ⟨ grad_{f}, u_{ϕ}⟩⟨ grad_{f}^{⊥}, Hess_{f} u_{ϕ}⟩.

We set Grad_{f} := p_{f} \frac{∂}{∂x} + q_{f} \frac{∂}{∂y}. By Lemma 2.4, we obtain

Lemma 2.5 P̃D_{f}(Grad_{f}) = D̃_{f}(Grad_{f}).

3 The behavior of the principal distributions around an umbilical point

Let A_{o}^{⟨2⟩} be the set of the elements of A_{o}^{(2)} such that for F ∈ A_{o}^{⟨2⟩}, o is a non-interior point of Umb(G_{F}). For F ∈ A_{o}^{⟨2⟩}, let Φ_{F} be a real-analytic function on (−ρ_{0}, ρ_{0}) × R × R defined by

Φ_{F} (ρ, θ, ϕ) := det (I_{F,(ρ cos θ,ρ sin θ)})P̃D_{F,(ρ cos θ,ρ sin θ)}(U_{ϕ}).

Then by Lemma 2.4 and by Φ_{F −f_{F}} ≡ 0, we obtain

lim_{ρ→0} \frac{Φ_{F} (ρ, θ, ϕ)}{ρ^{k_{F} −2}} = ⟨ Hess_{φ_{F}} (θ)u_{ϕ}, u_{ϕ+π/2}⟩.  (3.1)

Therefore we obtain

Proposition 3.1 Let F be an element of A_{o}^{⟨2⟩} satisfying S_{F,o} = ∅. Then F ∈ A_{oo}^{2} holds.

Suppose θ_{0} ∈ S_{F,o}. Then we set

Φ_{F,θ_{0},ϕ}(ρ) := ρ^{2}Φ_{F} (ρ, θ_{0}, ϕ), X_{F,θ_{0}} := {ϕ_{0} ∈ R ; Φ_{F,θ_{0},ϕ_{0}} ≢ 0}.

For ϕ_{0} ∈ X_{F,θ_{0}}, let k_{F,θ_{0},ϕ_{0}} be the order of Φ_{F,θ_{0},ϕ_{0}} at 0, and set

k_{F,θ_{0}} := min_{ϕ_{0}∈X_{F,θ_{0}}} k_{F,θ_{0},ϕ_{0}}.

Then there exists a non-scalar matrix M(θ_{0}) satisfying

lim_{ρ→0} \frac{Φ_{F} (ρ, θ_{0}, ϕ)}{ρ^{k_{F,θ_{0}}−2}} = ⟨M(θ_{0})u_{ϕ}, u_{ϕ+π/2}⟩.  (3.2)

Proof of Proposition 1.1 By (3.1) and by (3.2), we obtain (1), and we see that for θ_{1}, θ_{2} ∈R satisfying θ_{1} < θ_{2} and S_{F,o} ∩ (θ_{1}, θ_{2}) = ∅, there exists the element z^{(i)}(θ_{1}, θ_{2}) of Z_{0} satisfying

ϕ_{F,o}^{(i)}(θ) = η_{φ_{F}} (θ) + z^{(i)}(θ_{1}, θ_{2})  (3.3)

for any θ ∈ (θ_{1}, θ_{2}). Therefore noticing that the set S_{F,o} is empty or discrete, we obtain (2). By (3.1) and by (3.2), we obtain ϕ_{F,o}^{(2)} − ϕ_{F,o}^{(1)} ≡ z_{0} for some z_{0} ∈ Z_{0}. Therefore by (3.3), we obtain (3). □ Proof of Proposition 1.2 By (3.3), we obtain (1). For θ ∈R, the following holds:

ind_{o}(G_{F}) = \frac{ϕ_{F,o}^{(i)}(θ + 2π) − ϕ_{F,o}^{(i)}(θ)}{2π}.  (3.4)

By (3.3), we obtain

\begin{aligned}ϕ_{F,o}^{(i)}(θ + 2π) − ϕ_{F,o}^{(i)}(θ) = η_{φ_{F}} & (θ + 2π) − η_{φ_{F}} (θ) \\ & + ∑_{θ_{0}∈S_{F,o}∩[θ,θ+2π)} Γ_{F,o}(θ_{0}). \end{aligned}  (3.5)

From (3.4) and from (3.5), we obtain (2). □

4 Homogeneous polynomials

Let P be the set of the homogeneous polynomials in two variables. For φ ∈ P, set φ̃(θ) := φ(cos θ, sin θ). A number θ_{0} ∈R is called a root of φ if

\frac{dφ̃}{dθ}(θ_{0}) = 0. The set of the roots of φ is denoted by R_{φ}. The straight line

L(θ_{0}) := {(ρ cos θ_{0}, ρ sin θ_{0})}_{ρ∈R} on R^{2} determined by θ_{0} ∈ R_{φ} is called a root line of φ.

Let k_{φ} be the degree of φ. If R_{φ} =R, then k_{φ} is even and φ is represented by (x^{2} + y^{2})^{k_{φ}/2} with constant multiplication([1]). Suppose R_{φ} ≠R. Then

for any θ_{0} ∈ R_{φ}, there exists a positive integer m satisfying \frac{d^{m+1}φ̃}{dθ^{m+1}}(θ_{0}) ≠ 0. The minimum of such integers as m is called the multiplicity of θ_{0} and denoted by μ_{φ}(θ_{0}). A root θ_{0} ∈ R_{φ} is said to be

(1) related if θ_{0} satisfies φ̃(θ_{0}) = 0 or if μ_{φ}(θ_{0}) is odd;

(2) non-related if θ_{0} satisfies φ̃(θ_{0}) ≠ 0 and if μ_{φ}(θ_{0}) is even.

Suppose that θ_{0} ∈ R_{φ} is related. Then it is said that the critical sign of θ_{0} is positive(resp. negative) if the following holds:

φ̃(θ_{0})\frac{d^{μ_{φ}(θ_{0})+1}φ̃}{dθ^{μ_{φ}(θ_{0})+1}}(θ_{0}) ≦ 0(resp. > 0).

The critical sign of θ_{0} is denoted by c-sign_{φ}(θ_{0}).

For any θ, ϕ ∈R, we set

Hess_{φ}(θ) := Hess_{φ}(cos θ, sin θ), d_{φ}(θ, ϕ) := D̃_{φ,(cos θ,sin θ)}(U_{ϕ}), and let η_{φ} be a continuous function on R satisfying d_{φ}(θ, η_{φ}(θ)) = 0 for any θ ∈R. A number θ_{0} ∈R is called a root of φ on Hessian if θ_{0} − η_{φ}(θ_{0}) ∈ Z_{0} holds. The set of the roots of φ on Hessian is denoted by R(Hess_{φ}). For any θ ∈R. the following holds:

d_{φ}(θ, θ) = (k_{φ} − 1)\frac{dφ̃}{dθ}(θ).  (4.1)

Therefore we obtain R(Hess_{φ}) ⊂ R_{φ}.

Lemma 4.1 The following hold:

(1) The set R_{φ} \ R(Hess_{φ}) consists of the numbers at each of which Hess_{φ} is nonzero scalar;

(2) An element θ_{0} ∈ R_{φ}\R(Hess_{φ}) is related and satisfies c-sign_{φ}(θ_{0}) = −.

Proof If Hess_{φ}(θ_{0}) is not scalar for θ_{0} ∈ R_{φ}, then by (4.1) we obtain θ_{0} ∈ R(Hess_{φ}). Suppose φ̃(θ_{0}) = 0 for θ_{0} ∈ R_{φ}. Then there exist an integer n_{0} ≧ 2 and an element φ_{0} ∈ P satisfying φ̃(θ) = sin^{n_{0}}(θ − θ_{0})φ̃_{0}(θ) and φ̃_{0}(θ_{0}) ≠ 0. Then we obtain

\frac{d_{φ}(θ, ϕ)}{sin^{n_{0}−2}(θ − θ_{0})} =_{n_{0}}C_{2}φ̃_{0}(θ) sin 2(ϕ − θ_{0})
+ a_{φ_{0}}(θ, ϕ) sin(θ − θ_{0}) + d_{φ_{0}}(θ, ϕ) sin^{2}(θ − θ_{0}),

(4.2)

where a_{φ_{0}} satisfies φ̃_{0}(θ_{0})a_{φ_{0}}(θ_{0}, θ_{0}) > 0. By (4.2), we obtain θ_{0} ∈ R(Hess_{φ}). Suppose that Hess_{φ}(θ_{0}) is the unit matrix for θ_{0} ∈ R_{φ}, and θ_{0} = 0. Then we may represent φ as

φ(x, y) = \frac{1}{k_{φ}(k_{φ} − 1)}x^{k_{φ}} + \frac{1}{2}x^{k_{φ}−2}y^{2} + ∑_{i=3}^{k_{φ}} a_{i}x^{k_{φ}−i}y^{i}.  (4.3)

Therefore Hess_{φ}(θ) = ∑_{i=0}^{k_{φ}−2} M_{i} cos^{k_{φ}−2−i} θ sin^{i} θ holds, where M_{1} is not diagonal. Then we obtain 0 ∉ R(Hess_{φ}). Hence we have proved (1) of

Lemma 4.1. For θ_{0} ∈ R_{φ} \ R(Hess_{φ}), we obtain \frac{d^{2}φ̃}{dθ^{2}}(θ_{0}) = k_{φ}(k_{φ} − 2)φ̃(θ_{0}). Therefore we obtain μ_{φ}(θ_{0}) = 1 and c-sign_{φ}(θ_{0}) = −. Hence we have proved (2) of Lemma 4.1. □

We set U_{θ}(ε) := (θ − ε, θ + ε) for θ ∈ R and for ε > 0. It is said that the sign of θ_{0} ∈ R(Hess_{φ}) is positive(resp. negative) if there exists a positive number ε_{0} > 0 satisfying

{θ − η_{φ}(θ) − (θ_{0} − η_{φ}(θ_{0}))}(θ − θ_{0}) > 0(resp. < 0)

for any θ ∈ U_{θ_{0}}(ε_{0}) \ {θ_{0}}. Let n_{φ,+}(resp. n_{φ,−}) be the number of the root lines determined by the elements of R(Hess_{φ}) with positive(resp. negative) sign. Then referring to [1], we obtain Proposition 4.2 For any θ ∈R, the following holds:

\frac{η_{φ}(θ + 2π) − η_{φ}(θ)}{2π} = 1 − \frac{n_{φ,+} − n_{φ,−}}{2} .

We shall prove

Proposition 4.3 For θ_{0} ∈ R(Hess_{φ}), θ_{0} is related if and only if the sign of θ_{0} is positive or negative. In addition, for a related root θ_{0} ∈ R(Hess_{φ}),

(1) if φ̃(θ_{0}) ≠ 0, then the number

δ_{φ}(θ_{0}) := \frac{d^{μ_{φ}(θ_{0})+1}φ̃}{dθ^{μ_{φ}(θ_{0})+1}}(θ_{0})\frac{∂d_{φ}}{∂ϕ} (θ_{0}, θ_{0})

is nonzero and the sign of θ_{0} is given by the sign of δ_{φ}(θ_{0});

(2) if φ̃(θ_{0}) = 0, then the sign of θ_{0} is positive.

Proof Suppose that for θ_{0} ∈ R(Hess_{φ}), Hess_{φ}(θ_{0}) is not scalar. Then

\frac{∂d_{φ}}{∂ϕ} (θ_{0}, θ_{0}) ≠ 0 holds. Therefore by the implicit function theorem, we see

that η_{φ} is infinitely differentiable and satisfies

\frac{d^{m}}{dθ^{m}} (θ − η_{φ})|_{θ=θ_{0}} = (k_{φ} − 1)\frac{d^{m+1}φ̃}{dθ^{m+1}}(θ_{0})/\frac{∂d_{φ}}{∂ϕ} (θ_{0}, θ_{0})  (4.4)

for m = 1, …, μ_{φ}(θ_{0}). Therefore θ_{0} is related if and only if the sign of θ_{0} is positive or negative. By (4.4), we obtain (1) of Proposition 4.3.

Suppose φ̃(θ_{0}) = 0 for θ_{0} ∈ R(Hess_{φ}). Then by (4.2), we obtain (2) of Proposition 4.3.

Hence we have proved Proposition 4.3. □ If θ_{0} ∈ R(Hess_{φ}) is related, then the sign of θ_{0} is denoted by sign_{φ}(θ_{0}).

Proposition 4.4 Let θ_{0} be a related root of φ satisfying c-sign_{φ}(θ_{0}) = +. Then θ_{0} ∈ R(Hess_{φ}) and sign_{φ}(θ_{0}) = + hold.

Proof Suppose that a related root θ_{0} satisfies c-sign_{φ}(θ_{0}) = +. Then from Lemma 4.1, we obtain θ_{0} ∈ R(Hess_{φ}). If φ̃(θ_{0}) ≠ 0, then δ_{φ}(θ_{0}) > 0 holds([1]). Therefore we obtain sign_{φ}(θ_{0}) = +. If φ̃(θ_{0}) = 0, then Proposition 4.3 says sign_{φ}(θ_{0}) = +. Hence we obtain Proposition 4.4. □

An element φ ∈ P is called harmonic if tr(Hess_{φ}) ≡ 0 holds. If φ is harmonic, then we obtain φ̃(θ) = c cos k_{φ}(θ − α), where c, α ∈ R \ {0}. Then we immediately obtain

Proposition 4.5 For a nonzero harmonic element φ ∈ P,

(1) the number of the root lines of φ is equal to k_{φ};

(2) any root θ_{0} ∈ R_{φ} is related and satisfies c-sign_{φ}(θ_{0}) = +;

(3) for any θ ∈ R, Hess_{φ}(θ) is not scalar.

5 Proof of Theorem 1.3 and Theorem 1.4

Proof of Theorem 1.3 Let F be an element of A_{oo}^{2}. Then we obtain S_{F,o} = R_{φ_{F}} \ R(Hess_{φ_{F}}). We set

Ψ_{F} (ρ, θ) := D̃_{F,(ρ cos θ,ρ sin θ)}(Grad_{F} (ρ cos θ, ρ sin θ))/ρ^{k},

where

k := \{ \begin{matrix}3k_{F} − 4, & if F − f_{F} ≡ 0, \\ k_{F} , & if F − f_{F} ≢ 0. \end{matrix}

Then by (4.3) and by the implicit function theorem, we obtain

Lemma 5.1 For θ_{0} ∈ S_{F,o}, there exist a neighborhood V_{θ_{0}} of (0, θ_{0}) in R^{2} and a real-analytic curve C_{θ_{0}} in V_{θ_{0}} through (0, θ_{0}) satisfying

(1) C_{θ_{0}} = {(ρ, θ) ∈ V_{θ_{0}} ; Ψ_{F} (ρ, θ) = 0};

(2) C_{θ_{0}} is not tangent to the θ-axis at (0, θ_{0}).

Suppose θ_{0} ∈ S_{F,o}. Let ψ_{F} be a continuous function on B_{0} such that for (ρ, θ) ∈ B_{0}, grad_{F} (ρ cos θ, ρ sin θ) is represented by u_{ψ_{F} (ρ,θ)} with constant multiplication. Noticing Lemma 2.5, we suppose ϕ_{F}^{(1)} = ψ_{F} on B_{0} ∩ C_{θ_{0}}. Since θ_{0} ∈ R_{φ_{F}} \ R(Hess_{φ_{F}}), ψ_{F} may be extended to \overline{B_{0} ∩ V_{θ_{0}}}. Let ε_{0} be a positive number satisfying {0} × U_{θ_{0}}(ε_{0}) ⊂ V_{θ_{0}}. We set ψ_{F,o}(θ) := ψ_{F} (0, θ) and χ_{F,o}(θ) := ϕ_{F,o}^{(1)}(θ) − ψ_{F,o}(θ) for θ ∈ \overline{U_{θ_{0}}(ε_{0})}. Let α be a continuous function on U_{θ_{0}}(ε_{0}) satisfying α(θ)(θ − θ_{0}) > 0 for any θ ∈ U_{θ_{0}}(ε_{0}) \ {θ_{0}}. Then we obtain

Γ_{F,o}(θ_{0}) = \{ \begin{matrix}π/2, & if αχ_{F,o} > 0 on U_{θ_{0}}(ε_{0}) \ \{θ_{0}\}, \\ −π/2, & if αχ_{F,o} < 0 on U_{θ_{0}}(ε_{0}) \ \{θ_{0}\}, \\ 0, & if χ_{F,o}(θ_{0} + ε_{0})χ_{F,o}(θ_{0} − ε_{0}) > 0. \end{matrix}

Therefore we obtain (1) of Theorem 1.3.

By Lemma 2.4, we obtain

Ñ_{φ_{F} ,(ρ cos θ,ρ sin θ)}(U_{ϕ})

= ρ^{3k_{φ_{F}} −4} \frac{det(Hess_{φ_{F}} (θ))}{k_{φ_{F}} − 1}⟨grad_{φ_{F}} (cos θ, sin θ)u_{θ}, u_{ϕ}⟩ sin(ϕ − θ). (5.1)

Therefore we may suppose ϕ_{φ_{F} ,ρ}^{(1)}(θ_{0}) = ψ_{φ_{F} ,o}(θ_{0}) = θ_{0}, where ϕ_{φ_{F} ,ρ}^{(1)}(θ) := ϕ_{φ_{F}}^{(1)}(ρ, θ). By (2) of Lemma 4.1, we obtain {ψ_{φ_{F} ,o}(θ) − θ}(θ − θ_{0}) > 0. By

(5.1), we obtain \frac{∂P̃D_{φ_{F}} (U_{ϕ})}{∂ϕ} ≠ 0 at (ρ cos θ_{0}, ρ sin θ_{0}). Therefore we see

that ϕ_{φ_{F} ,ρ}^{(1)} is differentiable at θ_{0}, and we obtain

\frac{d}{dθ} (θ − ϕ_{φ_{F} ,ρ}^{(1)})|_{θ=θ_{0}} = \frac{d^{2}φ̃_{F}}{dθ^{2}} (θ_{0})/k_{φ_{F}}^{3} (k_{φ_{F}} − 1)ρ^{2k_{φ_{F}} −2}φ̃_{F} (θ_{0})^{3}.

Then by (2) of Lemma 4.1, we obtain {θ − ϕ_{φ_{F} ,ρ}^{(1)}(θ)}(θ − θ_{0}) > 0. Therefore we obtain αχ_{F,o} < 0 on U_{θ_{0}}(ε_{0}) \ {θ_{0}}, and Γ_{φ_{F} ,o}(θ_{0}) = −π/2. By Proposition 1.2 and by (1) of Theorem 1.3, we obtain ind_{o} (G_{φ_{F}} ) ≦ ind_{o}(G_{F}).

For θ ∈ R, set n_{s} := ♯(S_{F,o} ∩ [θ, θ + 2π)), and let {θ_{n}}_{n=0}^{n_{s}} be the subset of S_{F,o} satisfying θ_{n−1} < θ_{n} and (θ_{n−1}, θ_{n}) ∩ S_{F,o} = ∅ for n = 1, …, n_{s}. Then by Lemma 4.1, we see that for each n ∈ {1, …, n_{s}}, the number of the related roots in R_{φ_{F}} ∩ (θ_{n−1}, θ_{n}) with positive critical sign is more than the number of the related roots in R_{φ_{F}} ∩ (θ_{n−1}, θ_{n}) with negative critical sign. Therefore by Proposition 4.2 and by Proposition 4.4, we obtain

\frac{η_{φ_{F}} (θ + 2π) − η_{φ_{F}} (θ)}{2π} ≦ 1 − n_{s}/4.

By Proposition 1.2 and by (1) of Theorem 1.3, we obtain ind_{o}(G_{F}) ≦ 1. Hence we have proved (2) of Theorem 1.3. □

Proof of Theorem 1.4 Let S_{F,o}^{(0)} be the set of the elements of S_{F,o} at each

of which Hess_{φ_{F}} is zero, and set S_{F,o}^{(1)} := S_{F,o} \ S_{F,o}^{(0)}. For θ ∈ R and for

i ∈ {0, 1}, set n_{s}^{(i)} := ♯(S_{F,o}^{(i)} ∩ [θ, θ + 2π)). Then by Proposition 4.2 and by Proposition 4.4, we obtain

\frac{η_{φ}(θ + 2π) − η_{φ}(θ)}{2π} ≦ 1 − n_{s}^{(0)}/2 − n_{s}^{(1)}/4.  (5.2)

If Γ_{F,o}(θ_{0}) ≦ π holds for θ_{0} ∈ S_{F,o}^{(0)}, then by Proposition 1.2, (1) of Theorem 1.3 and by (5.2), we obtain ind_{o}(G_{F}) ≦ 1. Hence we have proved Theorem 1.4. □

6 Special Weingarten surfaces

We shall prove

Proposition 6.1 Let F be an element of A_{o}^{⟨2⟩} whose graph is a special Weingarten surface. Then φ_{F} is harmonic.

To prove Proposition 6.1, we need lemmas.

For F ∈ A_{o}^{⟨2⟩}, we set c_{F} := H_{F} (0, 0). Then K_{F} (0, 0) = c_{F}^{2} holds. We represent K_{F} − c_{F}^{2} and H_{F} − c_{F} as

K_{F} − c_{F}^{2} := ∑_{i≧1}K_{F}^{(i)}, H_{F} − c_{F} := ∑_{i≧1}H_{F}^{(i)},

where K_{F}^{(i)} and H_{F}^{(i)} are elements of P of degree i. Noticing (2.3) and that K_{F −f_{F}} and H_{F −f_{F}} are constant, we obtain

Lemma 6.2 For F ∈ A_{o}^{⟨2⟩}, (1) if c_{F} = 0, then the following holds:

K_{F}^{(i)} = \{ \begin{matrix}0, & if i ∈ \{1, …, 2k_{F} − 5\}, \\ det(Hess_{φ_{F}}), & if i = 2k_{F} − 4, \end{matrix}

(2) if c_{F} ≠ 0, then the following holds:

K_{F}^{(i)} = \{ \begin{matrix}0, & if i ∈ \{1, …, k_{F} − 3\}, \\ c_{F} tr(Hess_{φ_{F}}), & if i = k_{F} − 2, \end{matrix}

(3) the following holds:

H_{F}^{(i)} = \{ \begin{matrix}0, & if i ∈ \{1, …, k_{F} − 3\}, \\ tr(Hess_{φ_{F}})/2, & if i = k_{F} − 2. \end{matrix}

Let w be an element of A_{o}^{(1)} satisfying

c_{F} \frac{∂w}{∂X} (0, 0) + \frac{1}{2} \frac{∂w}{∂Y} (0, 0) ≠ 0,

and set

Δ_{F,w}(x, y) := w(K_{F} (x, y) − c_{F}^{2} , H_{F} (x, y) − c_{F}).

We represent Δ_{F,w} as Δ_{F,w} := ∑_{i≧1} Δ_{F,w}^{(i)}, where Δ_{F,w}^{(i)} is an element of P of degree i. By Lemma 6.2, we obtain

Lemma 6.3 The following holds:

Δ_{F,w}^{(i)}

= \{ \begin{matrix}0, & if i ∈ \{1, …, k_{F} − 3\}, \\ (c_{F} \frac{∂w}{∂X} (0, 0) + \frac{1}{2} \frac{∂w}{∂Y} (0, 0)) tr(Hess_{φ_{F}}), & if i = k_{F} − 2. \end{matrix}

Proof of Proposition 6.1 If the graph of F ∈ A_{o}^{⟨2⟩} is a special Weingarten surface, then by (1.1) and by Lemma 6.3, we see that φ_{F} is harmonic. □ Proof of Theorem 1.5 Since φ_{F} is harmonic, from Proposition 3.1 and from (3) of Proposition 4.5, we obtain (1) of Theorem 1.5. By Proposition 1.2, Proposition 4.2, Proposition 4.4, and by Proposition 4.5, we obtain (2) of Theorem 1.5. □

Acknowledgement The author is a research fellow of the Japan Society for the Promotion of Science.

References

[1] N.Ando, An isolated umbilical point of the graph of a homogeneous polynomial, to appear in Geom.Dedicata.

[2] P.Hartman and A.Wintner, Umbilical points and W-surfaces, Amer. J. Math., 76(1954) 502-508.

[3] H.Hopf, Lectures on differential geometry in the large, Lecture Notes in Math.,vol.1000, Springer-Verlag.

[4] T.Klotz, On Bol’s proof of Carathéodory’s conjecture, Comm. Pure Appl. Math., 12(1959) 277-311.

Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. E-mail address: naoya@mpim-bonn.mpg.de Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguroku, Tokyo 153-8914, Japan. E-mail address: andou@ms.u-tokyo.ac.jp