An umbilical point on a non-real-analytic surface

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33 (2003), 1–14

An umbilical point on a non-real-analytic surface

Naoya Ando

(Received May 29, 2001) (Revised March 4, 2002)

Abstract. LetF be a smooth function of two variables which is zero at ð0;0Þand positive on a punctured neighborhood of ð0;0Þ. Then the function expð1=FÞ is smoothly extended to ð0;0Þand then the origin o of R³ is an umbilical point of its graph. In this paper, we shall study the behavior of the principal distributions around o on condition that the norm of the gradient vector ﬁeld of logF is bounded from below by a positive constant on a punctured neighborhood of ð0;0Þ.

1. Introduction

Let S be a surface in R³ and p₀ an isolated umbilical point of S. Then the index of p0 on S is deﬁned by the index of p0 with respect to a principal distribution.

It is known that if Sis a surface with constant mean curvature and if Sis connected and not totally umbilical, then each umbilical point of S is isolated and its index is negative ([Ho, p139]); ifS is a special Weingarten surface, then the same result is obtained ([HaW]).

It has been expected that the index of an isolated umbilical point on a surface is not more than one. We call this conjecture the index conjecture. In relation to the index conjecture, the following two conjectures are known:

Caratheódory’s conjecture and Loewner’s conjecture. Caratheódory’s conjecture asserts that there exist at least two umbilical points on a compact, strictly convex surface in R³. If the index conjecture is true, then we see from Hopf- Poincare´’s theorem that there exist at least two umbilical points on a compact, orientable surface of genus zero, and this immediately gives the a‰rmative answer to Caratheódory’s conjecture. Let Fbe a real-valued, smooth function of two real variablesx;y, and setq_z:¼ ðq=qxþ ffiffiffiffiffiffiffi

p1

q=qyÞ=2. ThenLoewner’s conjecture for a positive integer nAN asserts that if a vector ﬁeld

Reðq_zⁿFÞ q

qxþImðq_zⁿFÞ q qy

2000 Mathematics Subject Classiﬁcation. Primary 53A05; Secondary 53A99, 53B25.

Key words and phrases. principal distributions, the indices of isolated umbilical points.

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has an isolated zero point, then its index with respect to this vector ﬁeld is not more than n ([K], [T]). Loewner’s conjecture for n¼1 is a‰rmatively solved;

Loewner’s conjecture for n¼2 is equivalent to the index conjecture. We may ﬁnd [B], [GSaB], [SX1], [SX2] and [SX3] as recent papers in relation to Carathe´odory’s and Loewner’s conjectures.

Let gbe a homogeneous polynomial of degree kin two variables such that on its graph, the origin o:¼ ð0;0;0Þ of R³ is an isolated umbilical point. In [A1], we studied the behavior of the principal distributions around o on the graph of g. In particular, we showed that the index of o is an element of f1k=2þig^½k=2_i¼0 . In [A2], we studied the behavior of the principal distributions around each of an isolated umbilical point of the graph of g and the point added to the graph by the one-point compactiﬁcation for the graph.

In [A3], we have studied the behavior of the principal distributions around an isolated umbilical point on a real-analytic surface. Let F be a real-analytic function on a neighborhood of ð0;0Þ satisfying

Fð0;0Þ ¼qF

qxð0;0Þ ¼qF

qyð0;0Þ ¼0

and the condition thatois an isolated umbilical point of its graph. Then there exists a real number a_F AR satisfying

Fðx;yÞ ¼a_Fðx²þy²Þ=2þoðx²þy²Þ: ð1Þ Let sF be a function deﬁned by

sF :¼ 8>

>>

<

>>

>:

0 if a_F ¼0;

1 a_F jaFj

a_F

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

a_F² ðx²þy²Þ s

if a_F00:

Then we obtain FDsF and we see that there exist an integer kFf3 and a nonzero homogeneous polynomial g_F of degree k_F such that all the partial derivatives of FsFgF of order less thankF þ1 vanish at ð0;0Þ. In [A3], we have proved that if o is an isolated umbilical point of the graph ofg_F, then the index of o on the graph of F is not less than the index of o on the graph of g_F and not more than one. In addition, we have proved that if the graph of F is special Weingarten, then the index of o is equal to 1kF=2. In [A4], we have described a similar discussion on the graph of a smooth function with such coe‰cients as the above F has in Taylor’s formula.

On the other hand, there exists a nonconstant smooth function such that its coe‰cients are not helpful. For example, a smooth function

expð1=ðx²þy²ÞÞ

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deﬁned on R²nfð0;0Þg is smoothly extended to ð0;0Þ and then we see that all the coe‰cients at ð0;0Þ vanish and that o is an umbilical point of its graph.

Let l denote a positive integer or y and C_o^ðy;lÞ the set of the smooth functions on a connected neighborhood of ð0;0Þ in R² such that all the coef- ﬁcients of order less than l vanish at ð0;0Þ. The following hold:

C_o^ð^y;^lÞIC_o^ð^y;lþ1ÞIC_o^ð^y;y^ÞZf0g;

where lAN. ForlANUfyg, letC_o;^ðy;þ^lÞ be the subset ofC_o^ð^y;lÞ such that each element of C_o;þ^ð^y;lÞ is positive on a punctured neighborhood of ð0;0Þ. For a positive number a>0, we set

E0ðaÞ:¼a; E1ðaÞ:¼expð1=aÞ:

In addition, we set

EnðaÞ:¼E1ðEn1ðaÞÞ

for eachnANinductively. Then forF AC_o;^ð^y;lÞ_þ andnAN, the smooth function E_nðFÞ deﬁned on a punctured neighborhood of ð0;0Þ is smoothly extended to ð0;0Þ and then we see that EnðFÞ is an element of C_o;þ^ð^y;^y^Þ and that o is an umbilical point of its graph. The purpose of this paper is to study the behavior of the principal distributions around o on the graph of E_nðFÞ for FAC_o;þ^ð^y;^lÞ such that the norm of the gradient vector ﬁeld of logF is bounded from below by a positive constant on a punctured neighborhood of ð0;0Þ.

We set

jgrad_Fj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qF

qx

2

þ qF qy

2

s

:

We shall prove

Theorem1.1. Let F be an element of C_o;þ^ðy;lÞsuch that for each c>0, there exists a punctured neighborhood of ð0;0Þ on which jgrad_Fj=F >c holds. Then o is an isolated umbilical point with index one on the graph of E_nðaFÞ for any nAN and any a>0.

Example 1.2. For lAN, let P^l be the set of the homogeneous poly- nomials of degree l in two variables and set P_þ^l :¼P^lVC_o;^ðy;lÞ_þ . For example, x²þy² is an element ofP_þ². LetF be an element ofC_o;þ^ð^y^;lÞ forlANsuch that there exists an element gAP_þ^l satisfying FgAC_o^ð^y;^lþ1Þ. Then for each c>0, we may ﬁnd a punctured neighborhood ofð0;0Þon whichjgrad_Fj=F >c holds. Therefore we see from Theorem 1.1 that o is an isolated umbilical point with index one on the graph of EnðaFÞ for any nAN and any a>0.

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Example 1.3. Let F be an element of C_o;þ^ðy;lÞ such that its graph is locally strictly convex at each point, where a surface S is called locally strictly convex at a point pAS if there exists a punctured neighborhood of p in Swhich does not share any point with the tangent plane at p. For each yAR, we set

FðF;yÞðrÞ:¼Fðrcosy;rsinyÞ

for each rAðr0;r0Þ, where r0 is some positive number. Then dF_ðF_;yÞ=dr is increasing. Therefore for any rAð0;r₀Þ, the following hold:

F_ðF_;yÞðrÞ ¼ ðr

0

dF_ðF;yÞ

ds ðsÞds<rdF_ðF_;yÞ

dr ðrÞerjgrad_Fðrcosy;rsinyÞj:

Then we see that for each c>0, there exists a punctured neighborhood of ð0;0Þ on which jgrad_Fj=F>c holds. Therefore we see that o is an isolated umbilical point with index one on the graph of EnðaFÞ for any nAN and any a>0.

Remark 1.4. Our discussions in [A1]@[A4] crucially depend on coef- ﬁcients of functions; in order to show that the function in Example 1.2 satisﬁes the assumption in Theorem 1.1, we again depend on coe‰cients of the function. However, in Example 1.3, we do not depend on coe‰cients of the function.

Example 1.5. Let F be an element of C_o;þ^ð^y^;lÞ such that jgrad_Fj=F>C0

holds for some C₀>0 on a punctured neighborhood of ð0;0Þ. Then noticing jgrad_E₁_ðaFÞj

E₁ðaFÞ ¼ 1

aF jgrad_Fj F

for any a>0, we see that for each c>0, there exists a punctured neighborhood of ð0;0Þon whichjgrad_E₁_ðaF_Þj=E1ðaFÞ>cholds. Therefore we see by Theorem 1.1 that o is an isolated umbilical point with index one on the graph of Enþ1ðaFÞ for any nAN and any a>0. We also see that Theorem 1.1 for n¼1 implies Theorem 1.1 for any nAN.

In addition, we shall prove

Theorem 1.6. Let F be an element of C_o;^ð^y;_þ^lÞ such that jgrad_Fj=F>C₀ holds for some C0>0 on a punctured neighborhood of ð0;0Þ.

(a) If lf3, then o is an isolated umbilical point with index one on the graph of E1ðaFÞ for any a>0;

(b) If there exists a nonzero gAP² satisfying gf0 on R², gBP_þ² and FgAC_o^ðy;3Þ, then there exists a positive number a0 >0 such that o is an isolated umbilical point with index one on the graph of E1ðaFÞfor any aAð0;a0Þ.

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Example 1.7. Let F be an element of C_o;þ^ðy;lÞ such that jgrad_Fj=F>C0

holds for some C₀ >0 on a punctured neighborhood of ð0;0Þ. If Fðx;yÞ ¼y⁴þoððx²þy²Þ²Þ;

then we see from (a) of Theorem 1.6 that o is an isolated umbilical point with index one on the graph of E1ðaFÞ for any a>0. If

Fðx;yÞ ¼y²þoðx²þy²Þ;

then we see from (b) of Theorem 1.6 that there exists a positive number a₀ >0 such that o is an isolated umbilical point with index one on the graph of E₁ðaFÞ for any aAð0;a₀Þ; we have not succeeded to grasp the behavior of the principal distributions around o on the graph of E1ðaFÞ for a large a>0 yet.

After Section 2 for preliminaries, we shall prove Theorems 1.1 and 1.6 in Section 3. In [A2] and [A3], we have studied the limit of each principal distribution toward an isolated umbilical point along the intersection of a surface with each normal plane at this point. In Section 4, we shall make a similar study on the graph of EnðaFÞ and ﬁnd another phenomenon than those which appear in [A2] and [A3].

2. Preliminaries

Let f be a smooth function of two variables x;y and Gf the graph of f. We set f_x:¼qf=qx, f_y:¼qf=qy and

E_f :¼1þf_x²; F_f :¼f_xf_y; G_f :¼1þf_y²:

The ﬁrst fundamental form of Gf is a symmetric tensor ﬁeld If on Gf of type ð0;2Þ represented in terms of the coordinates ðx;yÞ as

I_f :¼E_f dx²þ2F_f dxdyþG_f dy²; where

dx²:¼dxndx; dxdy:¼1

2ðdxndyþdyndxÞ; dy²:¼dyndy:

We set fxx:¼q²f=qx², fxy:¼q²f=qxqy, fyy:¼q²f=qy² and L_f :¼ f_xx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi detðIfÞ

p ; M_f :¼ fxy

p ; N_f :¼ fyy

p ;

where detðIfÞ:¼EfGf F_f². The Weingarten map of G_f is a tensor ﬁeld Wf

on Gf of type ð1;1Þ satisfying

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W_f q

qx ;W_f q qy

¼ q qx; q

qy

W_f;

where

Wf :¼ E_f F_f Ff Gf

1

L_f M_f Mf Nf

:

A principal direction of G_f is a one-dimensional eigenspace of W_f. A point of Gf is called umbilical if at the point, Wf is represented by the identity transformation up to a constant. By the symmetry of W_f with respect to I_f, we see that at each non-umbilical point, there exist just two principal directions, which are perpendicular to each other with respect to If.

Let PD_f be a symmetric tensor ﬁeld on G_f of type ð0;2Þ represented in terms of the coordinates ðx;yÞ as

PDf :¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi detðIfÞ

p fAf dx²þ2Bf dxdyþCf dy²g;

where

A_f :¼E_fM_f F_fL_f; 2B_f :¼E_fN_f G_fL_f; C_f :¼F_fN_f G_fM_f: Then for tangent vectors v₁;v₂,

1 2

X

fi;jg¼f1;2g

v_i5W_fðvjÞ ¼PD_fðv1;v₂Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi detðIfÞ

p q

qx5 q qy

:

Therefore we obtain

Proposition 2.1 ([A3]). A tangent vector v₀ to G_f is in a principal direction if and only if PDfðv0;v₀Þ ¼0 holds.

Let Df;Nf be symmetric tensor ﬁelds on Gf of type ð0;2Þ represented in terms of the coordinates ðx;yÞ as

D_f :¼f_xydx²þ ðf_yyf_xxÞdxdyf_xydy²; N_f :¼ ðf_xyf_x²f_xf_yf_xxÞdx²

þ ðfyyf_x²fxxf_y²Þdxdyþ ðfxfyfyyfxyf_y²Þdy²: Then detðIfÞPDf ¼Df þNf. For a tangent vector v, we set

D~

DfðvÞ:¼Dfðv;vÞ; NN~fðvÞ:¼Nfðv;vÞ;

PDf

PDfðvÞ:¼PDfðv;vÞ:

We set

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grad_f :¼ f_x

f_y ; grad^?_f :¼ fy

fx

; Hess_f :¼ f_xx f_xy

f_xy f_yy

:

For fAR, we set

uf:¼ cosf sinf

; U_f:¼cosf q

qxþsinf q qy: Let h;i be the scalar product in R². Then we obtain

Lemma 2.2 ([A3]). For any fAR, D~

DfðUfÞ ¼hHessf uf;ufþp=2i; N~

N_fðUfÞ ¼hgrad_f;u_fihgrad^?_f;Hess_f u_fi:

3. Proof of Theorems 1.1 and 1.6

Let F be an element of C_o;^ðy;lÞ_þ . We set

w_nðFÞ:¼ 1 if n¼0;

Qn

i¼1E_iðFÞ if nAN:

Then by induction with respect to nAN, we obtain Lemma 3.1. For any nAN,

grad_E_n_ðFÞ¼ EnðFÞ

F²w_n1ðFÞ grad_F; Hess_E_nðFÞ¼ EnðFÞ

F²w_n1ðFÞ HessF

þ E_nðFÞ F⁴w_n1ðFÞ

Xⁿ¹

i¼0

12E_iðFÞ w_iðFÞ

! F_x² F_xF_y F_xF_y F_y²

! :

By Lemma 2.2 together with Lemma 3.1, we obtain Lemma 3.2. For any nAN,

D~

D_E_nðFÞðUfÞ ¼ EnðFÞ

F²w_n1ðFÞDD~FðUfÞ þ E_nðFÞ

F⁴w_n1ðFÞ Xⁿ¹

i¼0

2E_iðFÞ 1 w_iðFÞ

!

hgrad_F;u_fihgrad^?_F;u_fi;

N~

N_E_n_ðF_ÞðU_fÞ ¼ EnðFÞ F²w_n1ðFÞ

3

N~ N_FðU_fÞ:

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Suppose that there exist positive numbers C0, r0>0 satisfying jgrad_Fj>C₀F >0

on f0<x²þy²<r₀²g. Let c_F be a continuous function on ð0;r0Þ R such that grad_Fðrcosy;rsinyÞis represented byu_c_F_ðr;yÞ up to a positive constant for any ðr;yÞAð0;r0Þ R. By Lemma 3.2, we see that for any ðr;yÞAð0;r0Þ R,

detðI_E_nðaFÞÞPDPDf_E_nðaFÞðUfÞð¼DD~_E_nðaFÞðUfÞ þNN~_E_nðaFÞðUfÞÞ

¼ E_nðaFÞ a²F²w_n1ðaFÞ

aDD~_FðUfÞ þ1 a

E_nðaFÞ F²w_n1ðaFÞ

2

N~ N_FðUfÞ

þ jgrad_Fj²

2F²w_n1ðaFÞYn1ðaFÞsin 2ðfc_Fðr;yÞÞ

holds at ðrcosy;rsinyÞ, where nAN, a>0 and Yn1ðaFÞ:¼Xⁿ¹

i¼0

w_n1ðaFÞ

w_iðaFÞ ð2EiðaFÞ 1Þ:

Therefore we see that at ðrcosy;rsinyÞ, U_f is in a principal direction of GEnðaFÞ if and only if the following holds:

aw_n1ðaFÞDD~FðUfÞ þw_n1ðaFÞ a

E_nðaFÞ F²w_n1ðaFÞ

2

N~ NFðUfÞ

þ1 2

jgrad_Fj²

F² Yn1ðaFÞsin 2ðfc_Fðr;yÞÞ ¼0: ð2Þ We notice the following:

(a) There exists a positive constant C1>0 satisfying w_n1ðaFÞjDD~_FðUfÞj<C₁

on a neighborhood of ð0;0Þ and for any fAR;

(b) for each positive number e>0, there exists a neighborhood of ð0;0Þ on which the following hold:

EnðaFÞ F²w_n1ðaFÞ<e;

jYn1ðaFÞ þ1j<e:

Proof of Theorem1.1. Suppose that for each c>0, there exists a punctured neighborhood of ð0;0Þ on which jgrad_Fj=F >c holds. Then noticing

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(2) and the above (a), (b), we see that for each positive number e0>0, there exists a number r₀Að0;r₀Þ such that for each ðr;yÞAð0;r₀Þ R, there exists the only one number f_E_n_ðaF_Þðr;yÞ satisfying

jf_E_n_ðaFÞðr;yÞ c_Fðr;yÞj<e₀ ð3Þ and the condition that atðrcosy;rsinyÞ, U_f

EnðaFÞðr;yÞ is in a principal direction of G_E_n_ðaF_Þ. Therefore we see that o is an isolated umbilical point on G_E_n_ðaF_Þ and that f_E_n_ðaF_Þ is continuous on ð0;r₀Þ R. Roughly speaking, around o, a principal distribution is approximated by the gradient vector ﬁeld of F. The index indoðG_E_nðaFÞÞ of o on G_E_nðaFÞ is represented as follows:

ind_oðGEnðaFÞÞ ¼f_E_n_ðaF_Þðr;yþ2pÞ f_E_n_ðaF_Þðr;yÞ

2p :

Noticing (3) and that the index is represented as the half of an integer, we obtain

ind_oðGEnðaFÞÞ ¼c_Fðr;yþ2pÞ c_Fðr;yÞ

2p : ð4Þ

We set

mðr₀Þ:¼ min

fx²þy²¼r₀²gFðx;yÞ:

Then mðr₀Þ>0. Noticing that grad_F does not vanish on f0<x²þy²<r₀²g, we see that the set

C_m:¼ fðx;yÞAR²;x²þy²<r₀²;Fðx;yÞ ¼mg

formAð0;mðr₀ÞÞis a connected and simple closed curve such that the bounded domain contains ð0;0Þ. Therefore by (4), we obtain

indoðG_E_nðaFÞÞ ¼1:

Hence we obtain Theorem 1.1. r

Proof of Theorem 1.6. Suppose lf3. Then for each positive number e>0, we obtain

jDD~FðUfÞj<e

on a neighborhood of ð0;0Þ and for any fAR. Therefore noticing (2) and the above-mentioned (b), we obtain the same result as in Theorem 1.1. Sup- pose that there exists a nonzero gAP² satisfying gf0 on R², gBP_þ² and FgAC_o^ðy;3Þ. Then HessF is not represented by the unit matrix up to any constant at ð0;0Þ. Therefore noticing (2) and the above-mentioned (a), (b), we

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may ﬁnd a positive number a0>0 such that for anyaAð0;a0Þ,o is an isolated umbilical point on G_E₁_ðaF_Þ. In addition, we may obtain ind_oðGE1ðaFÞÞ ¼1 for any aAð0;a0Þ in the same way as in the proof of Theorem 1.1. Hence we

obtain Theorem 1.6. r

4. On representation of the index

The purpose of this section is to study the limit of each principal distribution towardo along the intersection of the graph ofE_nðaFÞwith each normal plane at o. In order to do this, we shall ﬁrstly introduce some terms.

Let r₀ be a positive number and x a continuous function on ð0;r₀Þ R.

Then x is called admissible if there exists a number indðxÞ satisfying indðxÞ ¼xðr;yþ2pÞ xðr;yÞ

2p

for any ðr;yÞAð0;r₀Þ Rand if there exists a discrete subsetS ofRsatisfying the following:

(a) For each y₀ARnS, there exists a number x_oðy0Þ satisfying

r!0lim xðr;y0Þ ¼x_oðy0Þ;

(b) For each y0AR, there exist numbers x_oðy0þ0Þ, x_oðy00Þ satisfying

y!ylim0G0 x_oðyÞ ¼x_oðy0G0Þ and in addition, if y0ARnS, then the following hold:

x_oðy0þ0Þ ¼x_oðy00Þ ¼x_oðy0Þ: ð5Þ Suppose that x is admissible. Then the number indðxÞis called the indexofx.

The minimum of such sets as S is denoted by Sx and each element of Sx is called a singular argument ofx. A singular argumenty₀ of x is just a number for which (5) does not hold. Noticing the deﬁnition of indðxÞ, we see that if y₀AS_x, then y₀þ2npAS_x for any nAZ. From x, we may obtain a continuous function b_x on R such that on each connected component of RnSx, b_xx_o is constant. Such a function as b_x is called a base function of x.

There exists a number indðb_xÞ satisfying

indðb_xÞ ¼b_xðyþ2pÞ b_xðyÞ 2p

for any yAR. The number indðb_xÞ is called the index of b_x. For each y0ASx,

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Gx;oðy0Þ:¼x_oðy0þ0Þ x_oðy00Þ

is called the gap of x in y0. The index indðxÞ is represented as follows:

indðxÞ ¼indðb_xÞ þ 1 2p

X

y0ASxV½y;yþ2pÞ

G_x;oðy0Þ: ð6Þ

Example 4.1. Let g be an element of P^l and h_g a continuous function on R such that u_h_gðyÞ is an eigenvector of Hessgðcosy;sinyÞ for any yAR.

Suppose that on G_g, o is an isolated umbilical point. Let r₀ be a positive number such that on f0<x²þy²<r₀²g, there exists no umbilical point of G_g and f_g a continuous function on ð0;r₀Þ R such that for any ðr;yÞA ð0;r₀Þ R, U_f_g_ðr;yÞ is in a principal direction of Gg at ðrcosy;rsinyÞ.

Suppose that g depends only on x²þy². Then each of two vector ﬁelds xq=qxþyq=qy, yq=qxþxq=qy is in a principal direction at any point of G_g. Therefore f_g is admissible; a base function of f_g is given by y and there exists no singular argument of f_g. Suppose thatg does not depend only onx²þy². Thenlf3, andf_g is admissible; a base function off_g is given byh_g; a number y₀ is a singular argument off_g if and only if Hess_gðcosy₀;siny₀Þis represented by the unit matrix up to a constant; the gapGf_g;oðy0Þfor eachy0 ASf_g is equal to p=2 ([A2]), and the index of f_g is just the index of o on Gg.

Example 4.2. Let F be a real-analytic function on a neighborhood of ð0;0Þ satisfying

Fð0;0Þ ¼qF

qxð0;0Þ ¼qF

qyð0;0Þ ¼0

and the condition that on G_F, o is an isolated umbilical point. Let r₀ be a positive number such that on f0<x²þy²<r₀²g, there exists no umbilical point of G_F and f_F a continuous function on ð0;r₀Þ R such that for any ðr;yÞAð0;r₀Þ R,U_f_F_ðr;_yÞ is in a principal direction ofGF atðrcosy;rsinyÞ.

Then f_F is admissible; a base function off_F is given by h_g_F, where g_F is as in Section 1; at any singular argument y0 of f_F, HessgFðcosy0;siny0Þ is represented by the unit matrix up to a constant; if ois an isolated umbilical point of G_g_F, then the gap G_f_F;oðy0Þfor eachy₀ AS_f_F is equal top=2, 0 orp=2 ([A3]), and the index of f_F is just the index of o on GF. Suppose that o is not any isolated umbilical point of G_g_F. Then it is in general very di‰cult to grasp Gf_F;oðy0Þfory0 ASf_F. The author considers that this di‰culty is just the thing which prevents us from solving the index conjecture (see Section 1) on a real- analytic surface. In relation to the index conjecture, the following holds: if Gf_F;oðy0Þep for any y0ASf_F, then indðf_FÞe1 ([A3]).

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Example 4.3. Let F be an element of C_o^ð^y;^2Þ such that on GF, o is an isolated umbilical point. Then there exists a real number a_F ARsatisfying (1).

We obtain FDsF. If FsF BC_o^ð^y;y^Þ, then F has properties as in Example 4.2 ([A4]).

Remark 4.4. In [A3], we presented one way of computing the index indðh_gÞ of h_g for each gAP^l and lf3.

Remark 4.5. Noticing that at any non-umbilical point of a surface, the two principal directions are perpendicular to each other with respect to the ﬁrst fundamental form, we see that if a principal distribution around an isolated umbilical point on a surface is given by an admissible function, then the other principal distribution is also given by another admissible function.

Example 4.6. Let F be an element of Co;þ^ðy;lÞ such that grad_F does not vanish on f0<x²þy²<r₀²g for some r₀>0 and c_F a continuous function on ð0;r₀Þ Rsuch that grad_Fðrcosy;rsinyÞis represented by u_c_Fðr;yÞ up to a positive constant for any ðr;yÞAð0;r₀Þ R. Suppose that there exists a nonzero gAP^l satisfying FgAC_o^ðy;lþ1Þ. Then gf0. Let c_g be a continuous function on R such that grad_gðcosy;sinyÞ is represented by u_c_gðyÞ up to a constant for anyyARandZgthe set of the numbers such thatgðcosy0;siny0Þ ¼0 holds for each y0AZg. We see the following: c_F is admissible; a base function of c_F is given by c_g; the index of c_g is represented as

indðc_gÞ ¼1#fZgV½y;yþpÞg; ð7Þ and any singular argument of c_F is an element of Zg. Let y0 be an element of Z_g. We set y₀¼0. Then c_gð0ÞAfð2iþ1Þp=2g_iAZ. Therefore noticing gf0, we see that there exist integers nþ;nAZ satisfying

c_F_;oðG0Þ ¼ ð4nGG1Þp=2:

Let e₀ be a positive number satisfying

Z_gV½e0;e₀ ¼ f0g and dþ;d elements of ð0;r₀Þ satisfying

Fðdþcose0;dþsine0Þ ¼Fðdcose0;dsine0Þð¼:m0Þ:

As we have seen in the proof of Theorem 1.1, if m0<mðr₀Þ, then the con- tour line C_m₀ is a connected, simple closed curve such that the bounded domain contains ð0;0Þ. Therefore noticing that Cm0 contains the two points ðdþcose₀;dþsine₀Þ, ðdcose₀;dsine₀Þ and that f_xq=qxþf_yq=qy is normal to Cm0 at any point of Cm0, we obtain nþ¼n and 0ASc_F (see Figure 1).

Therefore we obtain Sc_F ¼Zg and

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Gc_F;oðy0Þ ¼p ð8Þ for any y0ASc_F. By (6), (7) and (8), we obtain indðc_FÞ ¼1. This means that the index ofð0;0Þwith respect to f_xq=qxþf_yq=qy is equal to one and does not contradict the proof of Theorem 1.1.

Let F be an element of C_o;^ðy;_þ^lÞ satisfying jgrad_Fj=F >C0 for some C0 >0 on a punctured neighborhood of ð0;0Þ. Suppose lf3. Then Theorem 1.6 says that o is an isolated umbilical point of G_E₁ðaFÞ for any a>0. Let r₀ be a positive number such that on f0<x²þy² <r₀²g, there exists no umbilical point of G_E₁ðaFÞ and f_E₁_ðaF_Þ a continuous function on ð0;r₀Þ R such that for any ðr;yÞAð0;r₀Þ R, U_f_E

1ðaFÞðr;yÞ is in a principal direction of G_E₁_ðaF_Þ at ðrcosy;rsinyÞ. Referring to the proof of Theorem 1.6 and Example 4.6, we obtain

Proposition 4.7. Suppose that there exists a nonzero gAP^l satisfying FgAC_o^ðy;lþ1Þ. Then the following hold:

(a) f_E₁_ðaF_Þ is admissible;

(b) a base function of f_E₁_ðaF_Þ is given by c_g; (c) Sf_E₁ðaFÞ¼Zg;

(d) the gap Gf_E

1ðaFÞ;oðy0Þ is equal to p for any y0 ASf_E

1ðaFÞ.

Remark 4.8. Supposel¼2. Then for a suitable a>0,f_E₁_ðaF_Þ is admissible; S_f_E

1ðaFÞ ¼Z_g, and G_f_E

1ðaFÞ;oðy0Þ ¼p for anyy₀AS_f_E

1ðaFÞ. However, a base function of f_E₁_ðaF_Þ may not be given by c_g for any a>0.

Remark 4.9. For any lf2, any nAN and any a>0, f_E_nþ1_ðaF_Þ has properties as (a)@(d) in Proposition 4.7.

Fig. 1. Example 4.6

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Acknowledgement

(1) The author is grateful to the referee for helpful comments and sugges- tions; (2) the author is a research fellow of the Japan Society for the Promotion of Science.

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Department of Mathematics Faculty of Science Kumamoto University

2-39-1 Kurokami Kumamoto 860-8555

Japan

E-mail: [email protected]