C^1-umbilics with arbitrarily high indices
aa r X i v : . [ m a t h . DG ] A ug C -UMBILICS WITH ARBITRARILY HIGH INDICES NAOYA ANDO, TOSHIFUMI FUJIYAMA, AND MASAAKI UMEHARAA
BSTRACT . In this paper, the existence of C -umbilics with arbitrarily high indicesis shown. This implies that more than C -regularity is required to prove Loewner’sconjecture.
1. I
NTRODUCTION
The index of an isolated umbilic on a given regular surface is the index of thecurvature line flow of the surface at that point, which takes values in the set of half-integers.
Loewner’s conjecture asserts that any isolated umbilic on an immersed sur-face must have index at most . Carath´eodory’s conjecture asserts the existenceof at least two umbilics on an immersed sphere in R , which follows immediatelyfrom Loewner’s conjecture. Although this problem was investigated mainly on real-analytic surfaces, several geometers recently became interested in non-analytic cases(cf. [A, B, GH, GMS, SX]). In particular, Smyth-Xavier [SX] observed that En-neper’s minimal surface is inverted to a branched sphere such that the index of thecurvature line flow at the branch point is equal to two. Bates [B] found that the graphof the function(1.1) B ( x, y ) := 2 + xy √ x p y has no umbilics on R and inversion of it gives a genus zero surface without self-intersections, which is differentiable at the image of infinity under that inversion.Ghomi-Howard [GH] gave similar examples of genus zero surfaces using inversion.Moreover, they showed that Carath´eodory’s conjecture for closed convex surfaces canbe reduced to the problem of existence of umbilics of certain entire graphs over R . Abrief history of Carath´eodory’s conjecture and recent developments are written alsoin [GH]. Recently, Guilfoyle-Klingenberg [GK1] and [GK2] gave an approach toproving the Caratheodory and Loewner conjecture in the smooth case.Let P : U → R be a C -immersion defined on an open subset U of R suchthat P is C ∞ -differentiable on U \ { q } and not C -differentiable at q . Then the point q ∈ U is called a C -umbilic if the umbilics of P on U \ { q } do not accumulate to Date : September 18, 2018.2010
Mathematics Subject Classification.
Primary:53A05; 53C45; Secondary: 57R42; 37C10; 53A30. The first author was partly supported by the Grant-in-Aid for Young Scientists (B) 24740048, JapanSociety for the Promotion of Science. The third author was partly supported by the Grant-in-Aid forScientific Research (A) 262457005, Japan Society for the Promotion of Science. q . At that point q , we can compute the index of the curvature line flow of P . In thispaper, we prove the following assertion. Theorem 1.1.
Let U ( ⊂ R ) be the unit disk centered at the origin. For each positiveinteger m , there exists a C -function f : U → R satisfying the following properties : (1) f is real-analytic on U ∗ := U \ { (0 , } , (2) (0 , , f (0 , is a C -umbilic of the graph of f with index m/ . It should be remarked that the inversion of the graph of Bates’ function B ( x, y ) has a differentiable umbilic of index although not of class C (see Example 2.3). Itwas classically known that curvature line flows are closely related to the eigen-flowsof the Hessian matrices of functions (see Appendix A). As an application of the aboveresult, we can show the following: Corollary 1.2.
For each m ( ≥ , there exists a C -function λ : U → R satisfying(1) λ is real-analytic on U ∗ , and(2) the eigen-flow of the Hessian matrix of λ has an isolated singular point (0 , with index m/ . When we consider the eigen-flow of the Hessian matrix of f , it is well-known thatthe index of the flow at an isolated singular point is equal to half of the index of thevector field(1.2) d f := 2 f xy ∂∂x + ( f yy − f xx ) ∂∂y . In addition, if o := (0 , is an isolated singular point of the eigen-flow of the Hes-sian matrix of f , then its index is equal to o ( δ f ) / (see Appendix B), where ind o ( δ f ) is the index of the vector field(1.3) δ f := 2( rf rθ − f θ ) ∂∂x + ( − r f rr + rf r + f θθ ) ∂∂y , at o , and x = r cos θ , y = r sin θ . In order to prove the above theorem, we introducevector fields D f and ∆ f analogous to d f and δ f , respectively (cf. Propositions 3.3and 4.2), and prove the theorem by computing the index of ∆ f at infinity for each ofthe functions (cf. Section 5)(1.4) ( f =) f m ( r, θ ) := 1 + tanh ( r a cos mθ ) (0 < a < / , m = 1 , , . . . ) . In addition, we give an alternative proof of Theorem 1.1 without use of inversion, byan explicit example of λ (cf. (6.1)) satisfying (1) and (2) of Corollary 1.2 (see Section6). 2. T HE REGULARITY OF THE INVERSION
Let R be a positive number. Consider a function f : R \ Ω R → R , where(2.1) Ω R := { ( x, y ) ∈ R ; p x + y ≤ R } . Then F = ( x, y, f ( x, y )) gives a parametrization of the graph of f . The inversion of F is given by F/ ( F · F ) , where the dot denotes the inner product on R . We considerthe following coordinate change(2.2) x = uu + v , y = vu + v . Then(2.3) Ψ f := 1 ρ ˆ f + 1 ( u, v, ρ ˆ f ) , ˆ f ( u, v ) := f (cid:18) uρ , vρ (cid:19) gives a parametrization of the inversion, where ρ := √ u + v . The map Ψ f is de-fined on the domain(2.4) U ∗ /R := U /R \ { o } (cid:18) U /R := (cid:26) ( u, v ) ∈ R ; p u + v < R (cid:27)(cid:19) , where o := (0 , . If we set(2.5) x = r cos θ, y = r sin θ, where r > , then (2.2) yields(2.6) ρ = 1 r , u = ρ cos θ, v = ρ sin θ. In particular, the angular parameter is common in the xy -plane and the uv -plane. Proposition 2.1.
Let f : R \ Ω R → R be a C ∞ -function such that f /r is bounded.Then the inversion Ψ f : U ∗ /R → R can be continuously extended to (0 , . More-over, if (2.7) (cid:12)(cid:12)(cid:12)(cid:12) f − rf f r r (cid:12)(cid:12)(cid:12)(cid:12) < r > R ) , then the image of Ψ f = ( X, Y, Z ) can be locally expressed as the graph of a function Z = Z f ( X, Y ) on a neighborhood of (0 , in the XY -plane. Under the assumption (2.7) , the function Z f ( X, Y ) is differentiable if and only if lim r →∞ fr = 0 . Proof.
We can write(2.8) Ψ f ( u, v ) = 11 + ϕ ( u, v ) (cid:16) u, v, ϕ ( u, v ) p u + v (cid:17) , where(2.9) ϕ ( u, v ) = p u + v ˆ f ( u, v ) = f ( x, y ) r . Since f /r is bounded, the function ϕ is bounded on U ∗ /R . Thus, one can prove lim ρ → Ψ f = (0 , , using (2.8), that is, Ψ f ( u, v ) can be continuously extended to (0 , . We denote by Π : R ∋ ( x, y, z ) ( x, y ) ∈ R the orthogonal projection.By setting ψ ( ρ, θ ) := ρ ϕ ( ρ cos θ, ρ sin θ ) , NAOYA ANDO, TOSHIFUMI FUJIYAMA, AND MASAAKI UMEHARA it holds that(2.10) Π ◦ Ψ f ( u, v ) = (cid:18) ψ ( ρ, θ ) cos θ, ψ ( ρ, θ ) sin θ (cid:19) . Since ˆ f ( ρ cos θ, ρ sin θ ) = f (cos θ/ρ, sin θ/ρ ) , we have ϕ ρ = f − rf r . In particular, it holds that ψ ρ = 1 − ( f − rf f r ) /r (1 + f /r ) . By (2.7), there exists ε > such that ρ ψ ( ρ, θ ) ( | ρ | ≤ ε ) is a monotone increasingfunction for each θ . Thus, by (2.10), we can conclude that Π ◦ Ψ f : U ε → R is aninjection. Since a continuous bijection from a compact space to a Hausdorff space isa homeomorphism, the inverse map G : Ω → U ε of Π ◦ Ψ f | U ε is continuous, where Ω is a neighborhood of the origin of the XY -plane in R . Then the graph of(2.11) Z f (cid:18) = ρϕ ϕ (cid:19) = ϕ ( G ( X, Y )) ρ ( G ( X, Y ))1 + ϕ ( G ( X, Y )) coincides with the image of Ψ f = ( X, Y, Z ) around (0 , , . Then X = u ϕ , Y = v ϕ , Z = ρϕ ϕ . Since ρ → as ( X, Y ) → (0 , , we obtain(2.12) lim ( X,Y ) → (0 , Z f ( X, Y ) √ X + Y = lim ( X,Y ) → (0 , ϕρ √ u + v = lim ρ → ϕ = lim r →∞ fr , proving the last assertion. (cid:3) Corollary 2.2.
Suppose that f : R \ Ω R → R is a bounded C ∞ -function satisfying (2.13) lim r →∞ f r r = 0 . Then the inversion Ψ f : U ∗ /R → R can be continuously extended to (0 , . More-over, the image of Ψ f is locally a graph which is differentiable at (0 , .Example . Bates’ example (cf. (1.1)) mentioned in the introduction is differen-tiable. In fact, B ( x, y ) is bounded and B r /r converges to zero as r → ∞ . However,the inversion of ( x, y, B ( x, y )) is not C . In fact, the unit normal vector field of thegraph of B is not continuously extended to the point at infinity. Since the inversionpreserves the angle, the unit normal vector field of its inversion cannot be continu-ously extended to (0 , , . Example . Ghomi-Howard [GH] gave an example(2.14) f GH = 1 + λ x + y p x + y ) ( λ > . The graph of f GH is an umbilic free (see Example 3.5 in Section 3). The function f GH is bounded. In addition, since ( f GH ) r is bounded, (2.13) is obvious. Therefore, as pointed out in [GH], the inversion of ( x, y, f GH ( x, y )) is differentiable. However,it is not a C -map. In fact, the limit of the unit normal vector field along y = 0 of thegraph of f GH is not equal to that along x + y = 0 at the point at infinity.Next, we give a condition for Ψ f to be extendable as a C -map to (0 , . Proposition 2.5.
Suppose that f : R \ Ω R → R is a bounded C ∞ -function satisfy-ing (a) lim r →∞ f r = 0 , (b) lim r →∞ f θ /r = 0 . Then Ψ f = ( X, Y, Z ) can be extended to (0 , as a C -map. Moreover, the map ( u, v ) ( X ( u, v ) , Y ( u, v )) is a C -diffeomorphism from a neighborhood of theorigin in the uv -plane onto a neighborhood of the origin in the XY -plane. To prove this, we prepare the following lemma.
Lemma 2.6.
The conditions ( a ) and ( b ) in Proposition 2.5 are equivalent to thefollowing two conditions, respectively : (1) lim ρ → ρ ˆ f ρ = 0 , (2) lim ρ → ρ ˆ f θ = 0 . Proof.
The equivalency of (2) and (b) is obvious. The equivalency of (1) and (a)follows from the identity ˆ f ρ = − f r /ρ . (cid:3) Proof of Proposition 2.5.
We see by Corollary 2.2 that Ψ f can be extended to (0 , asa differentiable map and the map ( u, v ) ( X ( u, v ) , Y ( u, v )) is a homeomorphismfrom a neighborhood of (0 , onto a neighborhood of (0 , . We set(2.15) h := ρ ˆ f (= ρϕ ) , k := ( ρ ˆ f ) (= ϕ ) . By (2.3), we can write(2.16) Ψ f = ( X, Y, Z ) = 1 k + 1 ( u, v, h ) . To show that Ψ f is a C -map at (0 , , it is sufficient to show that h, k are C -functions. Since h and k are C ∞ -functions on U ∗ /R , they satisfy h u = ρ (cid:18) (2 ˆ f + ρ ˆ f ρ ) cos θ − ˆ f θ sin θ (cid:19) , h v = ρ (cid:18) (2 ˆ f + ρ ˆ f ρ ) sin θ + ˆ f θ cos θ (cid:19) , (2.17) k u = 2 ˆ f ρ (cid:18) cos θ ( ˆ f + ρ ˆ f ρ ) − ˆ f θ sin θ (cid:19) , k v = 2 ˆ f ρ (cid:18) sin θ ( ˆ f + ρ ˆ f ρ ) + ˆ f θ cos θ (cid:19) (2.18)on U ∗ /R . Using (1), (2) in Lemma 2.6, (2.17) and (2.18), one can easily see that(2.19) lim ρ → h u = lim ρ → h v = lim ρ → k u = lim ρ → k v = 0 , which shows that Ψ f extends to (0 , as a C -map. By (2.16) and (2.19), we have X u (0 ,
0) = 1 , X v (0 ,
0) = 0 , Y u (0 ,
0) = 0 , Y v (0 ,
0) = 1 . Thus the second assertion follows from the inverse mapping theorem, because theJacobi matrix of the map ( u, v ) ( X ( u, v ) , Y ( u, v )) is regular at (0 , . (cid:3) NAOYA ANDO, TOSHIFUMI FUJIYAMA, AND MASAAKI UMEHARA
In Section 5, we need the following:
Proposition 2.7.
Let f : R \ Ω R → R be a bounded C ∞ function satisfying theconditions (a) and (b) of Proposition 2.5. If there exists a constant ≤ c < / suchthat r − c/ f r , r − c/ f θ , r − c f rr , r − c f rθ , r − c f θθ are bounded on R \ Ω R , then the map ( u, v ) ( X ( u, v ) , Y ( u, v )) is a C -map at (0 , , where Ψ f = ( X, Y, Z ) . We prepare the following lemmas:
Lemma 2.8.
The boundedness of the five functions in Proposition 2.7 is equivalentto the boundedness of the functions (2.20) ρ c/ ˆ f ρ , ρ c/ ˆ f θ , ρ c ˆ f ρρ , ρ c ˆ f ρθ , ρ c ˆ f θθ on U \ { (0 , } , where U is a sufficiently small neighborhood of (0 , .Proof. Differentiating ˆ f = ˆ f ( ρ cos θ, ρ sin θ ) by ρ , we have ρ ˆ f ρ = − rf r and ρ ˆ f ρρ =2 rf r + r f rr . Using these relations, the assertion can be easily checked. (cid:3) Lemma 2.9.
Suppose that the five functions in (2.20) are bounded on U \ { (0 , } .Then ρ c k uu , ρ c k uv and ρ c k vv are also bounded on U \ { (0 , } , where k is thefunction given in (2.15) .Proof. In fact, each of k uu , k uv , k vv is written as a linear combination of , ρ ˆ f ρ , ˆ f θ , ( ρ ˆ f ρ ) , ρ ˆ f ρ ˆ f θ , ˆ f θ , ρ ˆ f ρρ , ρ ˆ f ρθ , ˆ f θθ with coefficients that are bounded functions. For example, k uv = sin 2 θ (cid:16) ρ ˆ f ρ + ˆ f (cid:0) ρ ˆ f ρρ + 3 ρ ˆ f ρ − ˆ f θθ (cid:1) − ˆ f θ (cid:17) + 2 cos 2 θ (cid:16) ˆ f θ (cid:0) ρ ˆ f ρ + ˆ f (cid:1) + ρ ˆ f ˆ f ρθ (cid:17) . Thus, we get the assertion. (cid:3)
Proof of Proposition 2.7.
By Lemmas 2.8 and 2.9, the fact that c < yields that(2.21) lim ρ → ρk uu = lim ρ → ρk uv = lim ρ → ρk vv = 0 . Since X uu = 2 uk u − k + 1) k u − u ( k + 1) k uu ( k + 1) ,X uv = − k v ( − uk u + k + 1) + u ( k + 1) k uv ( k + 1) , X vv = − u (cid:0) ( k + 1) k vv − k v (cid:1) ( k + 1) , we have that X uu , X uv , X vv tend to as ρ → . This implies that X u , X v are C -functions. Similarly, Y u , Y v are also C -functions. (cid:3)
3. T
HE PAIR OF IDENTIFIERS FOR UMBILICS
Let U be a domain on R . Consider a flow (i.e. a -dimensional foliation) F de-fined on U \{ p , . . . , p n } , where p , . . . , p n are distinct points in U . We are interestedin the case that F is • the curvature line flow of an immersion P : U → R , • the eigen-flow of a matrix-valued function on U , or • the flow induced by a vector field on U .We fix a simple closed smooth curve γ : T → U \ { p , . . . , p n } , where T := R / π Z . We set ∂ x := ∂∂x , ∂ y := ∂∂y . Then one can take a smooth vector field V ( t ) := a ( t ) ∂ x + b ( t ) ∂ y along the curve γ ( t ) such that V ( t ) is a non-zero tangent vector of R at γ ( t ) whichpoints in the direction of the flow F . Then the map(3.1) ˇ V : T ∋ t ( a ( t ) , b ( t )) p a ( t ) + b ( t ) ∈ S := { x ∈ R ; | x | = 1 } is called the Gauss map of F with respect to the curve γ . The mapping degree of themap ˇ V is called the rotation index of F with respect to γ and denoted by ind( F , γ ) ,which is a half-integer, in general. If γ surrounds only p j , then ind( F , γ ) is inde-pendent of the choice of such a curve γ . So we call it the ( rotation ) index of theflow F at p j , and it is denoted by ind p j ( F ) . If the flow F is generated by a vectorfield V defined on U \ { p , . . . , p n } , then ind p j ( F ) is an integer, and we denote it by ind p j ( V ) .We denote by S ( R ) the set of real symmetric -matrices. Let U be a domain in R , and A = (cid:18) a ( x, y ) a ( x, y ) a ( x, y ) a ( x, y ) (cid:19) : U → S ( R ) a C ∞ -map. A point p ∈ U is called an equi-diagonal point of A if a = a and a = 0 at p . We now suppose that p is an isolated equi-diagonal point. Without lossof generality, we may assume that A has no equi-diagonal points on U \ { p } . Sincetwo eigen-flows of A are mutually orthogonal, the indices of the two eigen-flows ofthe S ( R ) -valued function A are the same half-integer at p . We denote it by ind p ( A ) .It is well-known that for an S ( R ) -valued function A , the formula(3.2) ind p ( A ) = 12 ind p ( v A ) holds, where v A is the vector field on U given by(3.3) v A := ( a − a ) ∂ x + a ∂ y . We shall apply these facts to the computation of the indices of isolated umbilics onregular surfaces in R as follows. Let f : U → R be a C ∞ -function. The symmetric NAOYA ANDO, TOSHIFUMI FUJIYAMA, AND MASAAKI UMEHARA matrices associated with the first and the second fundamental forms of the graph of f are given by(3.4) I := (cid:18) f x f x f y f x f y f y (cid:19) , II := (cid:18) f xx f xy f xy f yy (cid:19) . We consider a GL (2 , R ) -valued function(3.5) P := p f x − q (1 + f x + f y ) / (1 + f x ) f x f y / p f x ! , which satisfies the identity P P T = I , where P T is the transpose of P . Then A f := P − II ( P T ) − = P T ( I − II )( P T ) − is an S ( R ) -valued function. The umbilics of the graph of f correspond to the equi-diagonal points of A f . We show the following: Proposition 3.1.
The symmetric matrix A f ( p ) is proportional to the identity matrixat p ∈ U if and only if p gives an umbilic of the graph of f . Moreover, if p is anisolated umbilic, then ind p ( A f ) coincides with the index of the umbilic p .Proof. The first assertion follows from the definition of A f . Without loss of general-ity, we may assume that p coincides with the origin o := (0 , , and the graph of f has no umbilics other than o on U . Take a sufficiently small positive number ε > so that the circle γ ( t ) = ε (cid:0) cos t, sin t (cid:1) ( ≤ t ≤ π ) is null-homotopic in U .We denote by ( a ( t ) , b ( t )) T and ( a ( t ) , b ( t )) T eigen-vectors of I − II and A f at γ ( t ) , respectively. We may suppose ( a ( t ) , b ( t )) P ( γ ( t )) = ( a ( t ) , b ( t )) (0 ≤ t ≤ π ) . We set w i ( t ) := a i ( t ) ∂ x + b i ( t ) ∂ y ( i = 1 , . Then w points in one of the principal directions of the graph of f . The matrix P ( γ ( t )) takes values in the set(3.6) T := (cid:26)(cid:18) x − y z (cid:19) ; x, y > , z ∈ R (cid:27) . Since the set T is null-homotopic, the mapping degree of ˇ w ( t ) with respect to theorigin is equal to that of ˇ w ( t ) . Since the degree of ˇ w ( t ) with respect to o coincideswith ind o ( A f ) , we get the second assertion. (cid:3) By a straightforward calculation, one can get the following identity: ˜ A f := hk A f = (cid:18) f x f y ( f x f y f xx − hf xy ) + h f yy lklk k f xx (cid:19) , where h := 1 + f x , k := q f x + f y , l := − hf xy + f x f y f xx . Then the coefficients of the vector field v ˜ A f = v ∂ x + v ∂ y defined as in (3.3) for A = ˜ A f are given by v = ˜ a − ˜ a = ( − f x ) f y f xx − hf xx − hf x f xy f y + h f yy ,v = ˜ a = − k ( hf xy − f x f y f xx ) , where ˜ A f = (˜ a ij ) i,j =1 , . Hence, we get the following identity v = 2 f x f y k v + h (cid:18) − f xx (1 + f y ) + (1 + f x ) f yy (cid:19) . Consequently, we get the following fact (cf. Ghomi-Howard [GH, (10)]):
Fact 3.2.
The graph of the function z = f ( x, y ) defined on U has an umbilic at p ∈ U if and only if the functions d ( x, y ) := (1 + f x ) f xy − f x f y f xx , d ( x, y ) := (1 + f x ) f yy − f xx (1 + f y ) both vanish at p . We consider the vector field D f := d ∂ x + d ∂ y defined on the domain U in the xy -plane. Suppose that p is a zero of D f . Thefollowing assertion holds: Proposition 3.3. If p gives an isolated umbilic of the graph of f , then half of the indexof the vector field D f at p coincides with the index of the umbilic p .Proof. The half of the index of the vector field X := − v ˆ A f = (2 f x f y d − hd ) ∂ x + kd ∂ y at p is equal to ind p ( ˜ A f ) . We now set X s := ( ∂ x , ∂ y ) sf x f y − − sf x q s ( f x + f y ) 0 !(cid:18) d d (cid:19) (0 ≤ s ≤ . Then X = X and X = − d ∂ x + d ∂ y , and the rotation index of X s at p does notdepend on s ∈ [0 , . Since the rotation index of D f = ( d , d ) at p coincides withthat of X , we can conclude that X has the same rotation index as D f at p . (cid:3) We call d , d the Cartesian umbilic identifiers of the function f . Example . For a function f ( x, y ) := Re( z ) = x − xy ( z = x + iy ), theCartesian umbilic identifiers are given by d = − yϕ , d = − xϕ , where ϕ := − x + 9 y + 1 , ϕ := 9 x + 18 x y + 9 y + 2 . Since ϕ i ( i = 1 , ) are positive at the origin (0 , , the vector field D f can be con-tinuously deformed into the vector field − y∂ x − x∂ y preserving the property that theorigin is an isolated zero. Thus D f is of index − , and the graph of the function f has an isolated umbilic of index − / at the origin. Example . Bates’ function B ( x, y ) has no umbilics since d > on R . On theother hand, the identifier d with respect to Ghomi-Howard’s function f GH ( x, y ) in(2.14) vanishes if and only if y = 0 or x = − y . Since d never vanishes on thesetwo sets, the graph of f GH also has no umbilics on R .4. T HE PAIR OF POLAR IDENTIFIERS FOR UMBILICS
Let U be a domain in the xy -plane, and f : U → R a C ∞ -function. Let ( r, θ ) bethe polar coordinate system associated to ( x, y ) as in (2.5). Then F ( r, θ ) := ( r cos θ, r sin θ, f ( r cos θ, r sin θ )) gives a parametrization of the graph of f with the unit normal vector ν := 1 p f θ + r (1 + f r ) (cid:18) f θ sin θ − rf r cos θ, − rf r sin θ − f θ cos θ, r (cid:19) . Then ˆ I := (cid:18) f r f r f θ f r f θ r + f θ (cid:19) is the symmetric matrix consisting of the coefficientsof the first fundamental form of F . If we set Q = (cid:18) p f r − p f θ + r (1 + f r ) / p f r f r f θ / p f r (cid:19) , then QQ T = ˆ I . The symmetric matrix consisting of the coefficients of the secondfundamental form is given by b II := 1 p f θ + r (1 + f r ) (cid:18) rf rr rf rθ − f θ rf rθ − f θ r ( f θθ + rf r ) (cid:19) . Then the symmetric matrix B f = Q − b II ( Q − ) T = Q T ( ˆ I − b II )( Q T ) − satisfies ˜ B f = ˆ h ˆ k B f = rf r f θ f rr + ˆ hf r (cid:16) − rf θ f rθ + 2 f θ + r ˆ h (cid:17) + r ˆ h f θθ ˆ l ˆ k ˆ l ˆ k r ˆ k f rr ! , where ˆ h := 1 + f r , ˆ k := q f θ + r (1 + f r ) , ˆ l := f θ (cid:16) ˆ h + rf r f rr (cid:17) − r ˆ hf rθ . The following holds.
Proposition 4.1.
The symmetric matrix ˜ B f ( p ) is proportional to the identity matrixat p ∈ U \ { o } if and only if p gives an umbilic of the graph of f . Moreover, if o isan isolated umbilic of the graph of f , then the index of the umbilic at o is equal to o ( ˜ B f ) .Proof. The first assertion follows from the above discussions. So we now provethe second assertion. Suppose o is an isolated umbilic. We take a simple closedsmooth curve γ ( t ) ( ≤ t ≤ π ) in the xy -plane which surrounds the origin o anti-clockwisely, and does not surround any other umbilics. Let w : [0 , π ] → R be a vector field along γ such that w ( t ) is an eigen-vector of the matrix I − II at γ ( t ) foreach t ∈ [0 , π ] . Since ∂ r = cos θ∂ x + sin θ∂ y , ∂ θ = − r sin θ∂ x + r cos θ∂ y , we have that ( ∂ r , ∂ θ ) = ( ∂ x , ∂ y ) T , T := (cid:18) cos θ − r sin θ sin θ r cos θ (cid:19) . Then, it holds that ˆ I − b II = ( T ) − ( I − II ) T . In particular, w ( t ) := T ( γ ( t )) − w ( t ) (0 ≤ t ≤ π ) gives an eigen-vector of the matrix ˆ I − b II at γ ( t ) . Let T s : U → GL (2 , R ) ( ≤ s ≤ ) be a map defined by T s := (cid:18) cos θ − ( r (1 − s ) + s ) sin θ sin θ ( r (1 − s ) + s ) cos θ (cid:19) (0 ≤ s ≤ . Then it gives a continuous deformation of T to the rotation matrix T . Since thewinding number of the curve γ ( t ) with respect to the origin o is equal to , the dif-ference between the rotation indices of w and w is equal to . Since the eigen-flowof the symmetric matrix ˜ B f is associated with that of the matrix ˆ I − b II by Q , the factthat Q takes values in the set T in Section 3 yields that the index of the umbilic o isequal to o ( ˜ B f ) . (cid:3) We now set δ := − ˜ b ˆ k = − f θ (cid:0) f r + rf r f rr (cid:1) + r (cid:0) f r (cid:1) f rθ , where ˜ B f = (˜ b ij ) i,j =1 , . Then we have ˜ b − ˜ b = − f r f θ δ + r (cid:0) f r (cid:1) δ , where δ := (cid:0) f r (cid:1) ( rf r + f θθ ) − f rr (cid:0) r + f θ (cid:1) . Thus, as in the proof of Proposition 3.3, we get the following assertion.
Proposition 4.2.
Let U be a neighborhood of the origin o := (0 , . Let f : U → R be a C ∞ -function. Then the graph of f has an umbilic at p ∈ U \{ o } if and only if thetwo functions δ ( r, θ ) , δ ( r, θ ) both vanish at p , where x = r cos θ and y = r sin θ .Moreover, if o is an isolated umbilic, then half of the index of the vector field ∆ f := δ ∂ x + δ ∂ y at o equals − I f ( o ) , where I f ( o ) is the index of the umbilic o . We call δ , δ the polar umbilic identifiers of the function f . Example . Consider the function ( z = x + iy ) f ( x, y ) := Re( z ¯ z ) = x + xy = r cos θ. By straightforward calculations, we have δ = − r sin θ, δ = − r (cid:0) − r − r cos 2 θ (cid:1) cos θ. Since − r − r cos 2 θ is positive for sufficiently small r > , the vector field ∆ f can be continuously deformed into the vector field − sin θ∂ r − cos θ∂ θ preserving theproperty that the origin is an isolated zero. Thus the rotation index of ∆ f at o is equalto − , and I f ( o ) = 1 − / / . We give a modification of Proposition 4.2 for the computation of index of thecurvature line flow of a surface along an arbitrarily given simple closed curve sur-rounding the origin as follows. Let z = f ( x, y ) be a C ∞ -function defined on R admitting only isolated umbilics. Suppose that γ : R → R be a C ∞ -map satisfy-ing γ ( t + 2 π ) = γ ( t ) which gives a simple closed curve in the xy -plane such that itsurrounds a bounded domain containing the origin o anti-clockwisely. Moreover, weassume that γ ( t ) does not pass through any points corresponding to umbilics of thegraph of f . We denote by I f ( γ ) (resp. ind γ (∆ f ) ) the rotation index of the curvatureline flow (resp. of the vector field ∆ f ) along the simple closed curve γ . Then theformula(4.1) I f ( γ ) = 1 + ind γ (∆ f )2 can be proved by modifying the proof of Proposition 4.2. Suppose that there existat most finitely many points t = t , . . . , t k ∈ [0 , π ] such that δ ( γ ( t )) vanishes at t = t j . We now assume that δ ′ ( γ ( t )) := dδ ( γ ( t )) /dt does not vanish at t = t j ( j = 1 , . . . , k ). We set ε ( t j ) = δ ( γ ( t j )) < , δ ′ ( γ ( t j )) > and δ ( γ ( t j )) > , − δ ′ ( γ ( t j )) < and δ ( γ ( t j )) > . Then, it holds that(4.2) ind γ (∆ f ) = − k X j =1 ε ( t j ) .
5. P
ROOF OF THE MAIN THEOREM
In this section, using the function f = f m ( m = 1 , , , . . . ) given in (1.4), weprove Theorem 1.1 and Corollary 1.2 in the introduction. More generally, we considerthe function(5.1) ( g :=) g m ( r, θ ) := 1 + F ( r a cos mθ ) (0 < a < / , m = 1 , , , . . . ) , which is defined on { ( r, θ ) ; r > R } , where R is an arbitrarily fixed positive number,and F : R → R is a bounded C ∞ -function satisfying the following conditions:(i) F ( x ) is an odd function, that is, it satisfies F ( − x ) = − F ( x ) , Ψ f (0,0) F IGURE
1. The inversion of the graph f for a = 1 / (left) and itsenlarged view (right). In these two figures, the z -axis points towardthe downward direction.(ii) the derivative F ′ ( x ) of F is a positive-valued bounded function on R ,(iii) the second derivative F ′′ ( x ) is a bounded function on R such that F ′′ ( x ) < for x > ,(iv) there exist three constants α, β and γ ( β = 0 , γ > ) such that lim x →∞ e γx F ′ ( x ) = α, lim x →∞ e γx F ′′ ( x ) = β. One can easily construct a bounded C ∞ -function F ( x ) satisfying the properties (i-iv). For example, one can construct an odd C ∞ -function satisfying (ii) and (iii) sothat F ( x ) = 1 − e − x ( x ∈ [ M, ∞ )) . Then it satisfies also (iv). However, to prove Theorem 1.1, we must choose the func-tion F ( x ) to be real-analytic, and F ( x ) := tanh x satisfies all of the properties required. From now on, we shall prove Theorem 1.1 andCorollary 1.2 using only the above four properties of F ( x ) .The function g can be considered as a C ∞ -function on R \ Ω R in the xy -planefor any R > . The graph of g lies between two parallel planes orthogonal to the z -axis, and is symmetric under rotation by the angle π/m with respect to the z -axis(the entire figure of the inversion of the graph of f is given in the left-hand side of Figure 1). The partial derivatives of the function g are given by g r = ar a − c m F ′ ( r a c m ) , g θ = − mr a s m F ′ ( r a c m ) , (5.2) g rr = ar a − c m (cid:18) ar a c m F ′′ ( r a c m ) + ( a − F ′ ( r a c m ) (cid:19) ,g rθ = − amr a − s m (cid:18) r a c m F ′′ ( r a c m ) + F ′ ( r a c m ) (cid:19) ,g θθ = m r a (cid:18) r a s m F ′′ ( r a c m ) − c m F ′ ( r a c m ) (cid:19) , where(5.3) c m := cos mθ, s m := sin mθ. Since F ( x ) is a bounded function, g is bounded and satisfies (2.13), since a < .Therefore, the inversion Ψ g can be expressed as a graph near (0 , , . Since < a < , the function g satisfies (a) and (b) of Proposition 2.5. Then Z = Z f ( X, Y ) as in(2.11) with f := g is a C -function at (0 , . The graph of Z g for g = f near (0 , , is indicated in the right-hand side of Figure 1. To prove Theorem 1.1, it is sufficient toshow that (0 , , is a C -umbilic of the graph of Z g ( X, Y ) with index m/ . Inthe following discussions, we would like to show that there exists a positive number R such that the graph of g has no umbilics if r > R . We then compute the index I g (Γ) with respect to the circle(5.4) Γ( θ ) := ( r cos θ, r sin θ ) (0 ≤ θ ≤ π, r > R ) , using (4.1) and (4.2), which does not depend on the choice of r ( > R ) , as follows. Weset(5.5) ˇ δ j ( θ ) := δ j (Γ( θ )) ( j = 1 , . The first polar identifier is given by(5.6) δ = − mr a s m (cid:18) ar a c m F ′′ ( r a c m ) + ( a − F ′ ( r a c m ) (cid:19) . Since < a < , the condition (ii) yields that(5.7) ( a − F ′ ( r a c m ) < . On the other hand, by (i) and (iii), it holds that(5.8) xF ′′ ( x ) ≤ x := r a c m ) . By (5.7) and (5.8), we can conclude that ˇ δ ( θ ) changes sign only at the zeros of thefunction sin mθ . Since the function g is symmetric with respect to rotation by angle π/m , to compute the rotation index of ∆ g along Γ , it is sufficient to check the signchanges of ˇ δ i ( θ ) ( i = 1 , ) for θ = 0 and θ = π/m . By (5.6), (5.7) and (5.8), we getthe following:(5.9) d ˇ δ dθ (cid:12)(cid:12)(cid:12)(cid:12) θ =0 > , d ˇ δ dθ (cid:12)(cid:12)(cid:12)(cid:12) θ = π/m < . The second polar identifier δ is given by r − a δ = − r − a (cid:0) a c m − m s m (cid:1) F ′′ ( c m r a )+ ac m (cid:0) a c m − am + m s m (cid:1) F ′ ( c m r a ) − c m r − a (cid:0) a − a + m (cid:1) F ′ ( c m r a ) . We need the sign of ˇ δ ( θ ) at θ ∈ ( π/m ) Z . In this case, s m = 0 and c m = ± .Substituting these relations and using the fact that F ′ (resp. F ′′ ) is an even function(resp. an odd function), we have r − a δ = ∓ r − a a F ′′ ( r a ) ± a (cid:0) a − m (cid:1) F ′ ( r a ) ∓ r − a (cid:0) a − a + m (cid:1) F ′ ( r a ) . Since F ′ is bounded, the middle term is bounded. Hence, by (iv) and by the fact that < a < , there exists a positive number R such that the sign of δ is determined bythe sign of the first term ∓ r − a a F ′′ ( r a ) whenever r > R . Then, we have(5.10) − ˇ δ ( π/m ) = ˇ δ (0) > . In particular, the image of the graph of g has no umbilics when r > R . By the π/m -symmetry of g , (4.2), (5.9), and (5.10), the index ind Γ (∆ g ) is equal to − m . Thenthe index of the curvature line flow along Γ is equal to I g (Γ) = 1 − m/ by (4.1).Then after inversion, the Poincar´e-Hopf index formula yields that the index I of theumbilic of Ψ g at the origin is I = 2 − I g (Γ) = 1 + m/ . If we choose F ( x ) := tanh x , then the function Z g ( X, Y ) satisfies the properties ofTheorem 1.1.We next prove the corollary. We set(5.11) λ := Z p Z X + Z Y p Z X + Z Y , where Z := Z g is the function given in (2.11). Suppose that λ and λν are a C -function and a C -vector field defined on a sufficiently small neighborhood of ( X, Y ) =(0 , , respectively, where ν is a unit normal vector field of the graph of Z g . Then themap Φ : (
X, Y ) ( ξ ( X, Y ) , η ( X, Y )) given by (A.4) for f = Z f m is a local C -diffeomorphism, and is real-analytic on U \ { (0 , } . Then the proof of Fact A.1 in the appendix is valid in our situation,and we can conclude that the eigen-flow of the Hessian matrix of λ ( ξ, η ) is equal tothe curvature line flow of the map P ( ξ, η ) given by (A.8). Since the image of P ( ξ, η ) coincides with that of Ψ f m ( u, v ) , we get the proof of the corollary in the introduction.Thus, it is sufficient to show that λ and λν are C at ( X, Y ) = (0 , . By (5.11),we have the following expression(5.12) λν = ( ZZ X , ZZ Y , − Z )1 + p Z X + Z Y . By (5.11) and (5.12), we can conclude that λ ( X, Y ) and λ ( X, Y ) ν ( X, Y ) are C at (0 , if(5.13) lim ( X,Y ) → (0 , ZZ XX = lim ( X,Y ) → (0 , ZZ XY = lim ( X,Y ) → (0 , ZZ Y Y = 0 hold. So to prove the corollary, it is sufficient to show (5.13). It can be easily seenthat all of r − a g r , r − a g θ , r − a g rr , r − a g rθ and r − a g θθ are bounded functions on R \ Ω R . Since < a < / , Proposition 2.7 yields that the map ( u, v ) ( X, Y ) =Π ◦ Ψ g ( u, v ) is a C -map. Then (5.13) is equivalent to(5.14) lim ( u,v ) → (0 , ZZ uu = lim ( u,v ) → (0 , ZZ uv = lim ( u,v ) → (0 , ZZ vv = 0 . Since Z = h/ ( k + 1) , (5.14) follows from (2.19), (2.21) and the fact that lim ρ → ρh uu = lim ρ → ρh uv = lim ρ → ρh vv = 0 .
6. A
N ALTERNATIVE PROOF OF THE MAIN THEOREM
In the previous section, we have proved Corollary 1.2. However, it is natural toexpect that one can give an explicit description of the function with the desired prop-erties. The function λ given in (5.11) does not have a simple expression. On the otherhand, we will see that functions(6.1) (Λ =) = Λ m := r tanh( r − a cos mθ ) ( m = 1 , , , . . . ) satisfy (1) and (2) of Corollary 1.2 if < a < . We set ( λ :=) λ m := r F ( r − a cos mθ ) , where ξ = r cos θ, η = r sin θ , and F : R → R is a function satisfying the proper-ties (i–iv) given in the beginning of Section 5. Then Λ m is a special case of λ m for F ( x ) := tanh x . It holds that λ r = r (cid:18) F ( r − a c m ) − ac m r − a F ′ ( r − a c m ) (cid:19) , λ θ := − mr − a s m F ′ (cid:0) r − a c m (cid:1) ,λ rr = 2 F ( r − a c m ) + ar − a c m (cid:18) ( a − r a F ′ ( r − a c m ) + ac m F ′′ ( r − a c m ) (cid:19) ,λ rθ = ms m r − a (cid:18) ( a − r a F ′ ( r − a c m ) + ac m F ′′ ( r − a c m ) (cid:19) ,λ θθ = − m r − a (cid:18) r a c m F ′ ( r − a c m ) − s m F ′′ ( r − a c m ) (cid:19) , where c m and s m are defined in (5.3). We set ζ := 2( rλ rθ − λ θ ) , ζ := − r λ rr + rλ r + λ θθ . Then each component of the vector field δ λ := ζ ∂ x + ζ ∂ y is an identifier for theeigen-flow of the Hessian matrix of λ at the origin given in the introduction (cf. (1.3)). By a direct calculation, we have ζ = 2 mr − a s m (cid:18) ac m F ′′ ( r − a c m ) + ( a − r a F ′ ( r − a c m ) (cid:19) ,ζ = − r − a ( a c m − m s m ) F ′′ ( r − a c m ) − ( a − a + m ) r − a c m F ′ ( r − a c m ) . By the property (ii) of F , ( a − r a F ′ ( r − a c m ) is negative, and by (ii) and (iii), c m F ′′ ( r − a c m ) is also negative. So ζ is positively proportional to − s m (= − sin mθ ) .In particular, ζ vanishes only when s m = 0 . Moreover, for fixed r , it holds that dζ /dθ < (resp. dζ /dθ > ) if c m = 1 (resp. c m = − ).On the other hand, if s m = 0 and r tends to zero, then c m = ± and F ′ ( ± r − a ) and F ′′ ( ± r − a ) tend to zero with exponential order (cf. the condition (iv) for F ( x ) ).Therefore, the leading term of ζ for small r is − r − a ( a c m − m s m ) F ′′ ( r − a c m ) .Hence, for a fixed sufficiently small r , the function ζ is positive (resp. negative) if c m = 1 (resp. c m = − ). Summarizing these facts, one can easily show that theindex of the vector field δ λ at o := (0 , is equal to m . So the index of the eigen-flowof the Hessian matrix of λ at o is equal to m/ (cf. Appendix B). On can easilycheck that λ is a C -function at o and the function λ satisfies (1) and (2) of Corollary1.2. Since Λ is a special case of λ , we proved that Λ satisfies the desired properties.F IGURE
2. The image of P ( r ≤ / ) for m = 2 and a = 1 / .To give an alternative proof of Theorem 1.1, we consider the real analytic map P : R \ { o } → R defined by (cf. (A.8)) P ( ξ, η ) := ( ξ, η, Λ( ξ, η )) − Λ( ξ, η ) ν ( ξ, η ) , where(6.2) ν := 1Λ ξ + Λ η + 1 (2Λ ξ , η , Λ ξ + Λ η − . One can easily verify that Λ ξ = r − a (cid:18) ( ms s m − ac c m ) sech (cid:0) r − a c m (cid:1) + 2 r a c tanh (cid:0) r − a c m (cid:1)(cid:19) , Λ η = r − a (cid:18) r a s tanh (cid:0) r − a c m (cid:1) − ( as c m + mc s m ) sech (cid:0) r − a c m (cid:1)(cid:19) , where c = cos θ and s = sin θ . Using them, one can get the following expressions(6.3) Λ ξξ = 1 r a h ( r, θ ) , Λ ξη = 1 r a h ( r, θ ) , Λ ηη = 1 r a h ( r, θ ) , where h i ( r, θ ) ( i = 1 , , ) are continuous functions defined on R . Using (6.2), (6.3)and the fact lim r → Λ /r a = 0 , we have(6.4) lim r → Λ ν ξ = lim r → Λ r a ( r a ν ξ ) = 0 , and also(6.5) lim r → Λ ν η = 0 . Using (6.4), (6.5) and the fact d (Λ ν ) = ( d Λ) ν + Λ dν, we can conclude that Λ ν can be extended as a C -function at o . Thus P ( ξ, η ) can bealso extended as a C -differentiable map at o . One can also easily check that P ξ (0 ,
0) = (1 , , , P η (0 ,
0) = (0 , , . Hence P is an immersion at o , and Φ : ( ξ, η ) ( X ( ξ, η ) , Y ( ξ, η )) is a local C -diffeomorphism, where P = ( X, Y, Z ) . In particular, Z Λ := Z (Φ − ( X, Y )) gives a function defined on a neighborhood of ( X, Y ) = (0 , . By Fact A.1 in theappendix, the index of the curvature line flow at (0 , of the graph of Z Λ is equal tothe index of the eigen-flow of the Hessian matrix of Λ , which implies Theorem 1.1.The image of P for m = 3 and a = 1 / is given in Figure 2.7. T HE DUALITY OF INDICES
At the end of this paper, we consider the index at infinity for eigen-flows of Hessianmatrices. Let f : R \ Ω R → R , g : U /R \ { o } → R be C -functions, where Ω R and U /R are disks defined in Section 2. Let H f (resp. H g ) be the eigen-flow of the Hessian matrix of f (resp. g ). If the Hessian matrixof f has no equi-diagonal points, then we can consider the index ind ( H f , Γ) withrespect to the circle Γ given in (5.4) and it is independent of the choice of r > R .So we denote it by ind ∞ ( H f ) . Similarly, if the Hessian matrix of g has no equi-diagonal points, then we can consider the index ind ( H g , Γ ′ ) with respect to the circle Γ ′ ( θ ) := ( ρ cos θ, ρ sin θ ) ( ≤ θ ≤ π , ρ < /R ). Since it is independent of thechoice of ρ < /R , we denote it by ind o ( H g ) . Consider the plane-inversion ι : R ∈ ( u, v ) u + v ( u, v ) ∈ R . Then the following assertion holds. Proposition 7.1 (The duality of indices) . Let f : R \ Ω R → R be a C -functionwhose Hessian matrix has no equi-diagonal points. Then the function g : Ω R → R defined by g ( x, y ) := ( u + v ) f ◦ ι ( u, v ) (called the dual of f ) satisfies ind o ( H g ) + ind ∞ ( H f ) = 2 . Proof.
Using the identification of ( u, v ) and z = u + iv , it holds that u = ( z + ¯ z ) / and v = ( z − ¯ z ) / (2 i ) . In particular, f can be considered as a function of variables z and ¯ z , and can be denoted by f = f ( z, ¯ z ) . Since ι ( z ) = 1 / ¯ z , we can write g ( z, ¯ z ) := z ¯ zf (1 / ¯ z, /z ) . Then g zz ( z, ¯ z ) = ¯ zf ¯ z ¯ z (1 / ¯ z, /z ) z holds, where ∂∂z := 12 (cid:18) ∂∂u − i ∂∂v (cid:19) , ∂∂ ¯ z := 12 (cid:18) ∂∂u + i ∂∂v (cid:19) . Since Γ( θ ) = re iθ , we have that g zz (Γ( θ )) = f ¯ z ¯ z ( ι ◦ Γ( θ )) r e iθ . Thus, it holds that ind o ( g zz , Γ) = − o ( f ¯ z ¯ z , ι ◦ Γ) . By (B.1), we have ind o ( g zz , Γ) = − o ( H g ) , ind o ( f ¯ z ¯ z , ι ◦ Γ) = − ind o ( f zz , ι ◦ Γ) = 2 ind ∞ ( H f ) . Thus we get the assertion. (cid:3)
Applying Proposition 7.1 for the function g = Λ m (cf.(6.1)), we get the following: Corollary 7.2.
For each m ( ≥ , there exists a C -function f : R \ Ω R → R satisfying(1) f is real-analytic on R \ Ω R ,(2) the eigen-flow of the Hessian matrix of f has no singular points, and(3) the index at infinity of the eigen-flow of H f is equal to − m/ . The function Λ m used in the second proof of Theorem 1.1 coincides with the dualof the function f m − given in (1.4). A PPENDIX
A. T
HE CLASSICAL REDUCTION
In this appendix we show the existence of a special coordinate system ( ξ, η ) of thegraph of a function f ( x, y ) which reduces the curvature line flow to the Hessian ofa certain function, called Ribaucour’s parametrization (the third author learned thisfrom Konrad Voss at the conference of Thessaloniki 1997). Although, the existenceof such a coordinate system was classically known, and a proof is in the appendix of[S], the authors will give the proof here for the sake of convenience. We set P =( x, y, f ( x, y )) , and suppose that f (0 ,
0) = f x (0 ,
0) = f y (0 ,
0) = 0 . Consider asphere which is tangent to the graph of f at P and also tangent to the xy -plane at apoint Q . Then, it holds that(A.1) Q + λ e = P + λν, where e = (0 , , and ν = ( f x , f y , − / q f x + f y . Taking the third compo-nent of (A.1), we get(A.2) λ = f q f x + f y q f x + f y . In particular, λ (0 ,
0) = 0 . Since f x (0 ,
0) = f y (0 ,
0) = 0 , we have that(A.3) dλ (0 ,
0) = df (0 ,
0) = 0 . Taking the exterior derivative of (A.1), and using (A.3) and λ (0 ,
0) = 0 , we have dP (0 ,
0) = dQ (0 , . So, if we set Q = ( ξ ( x, y ) , η ( x, y ) , , then it holds that ( ξ x (0 , dx + ξ y (0 , dy, η x (0 , dx + η y (0 , dy,
0) = dQ = dP = ( dx, dy, f x (0 , dx + f y (0 , dy ) = ( dx, dy, , which implies that the Jacobi matrix of the map(A.4) Φ : ( x, y ) ( ξ ( x, y ) , η ( x, y )) is the identity matrix at (0 , . So we can take ( ξ, η ) as a new local coordinate system.Differentiating (A.1) by ξ and η , we get the following two identities: Q ξ + λ ξ e = P ξ + λ ξ ν + λν ξ , Q η + λ η e = P η + λ η ν + λν η . Taking the inner products of them and ν , these two equations yield(A.5) Q ξ · ν + λ ξ ν = λ ξ , Q η · ν + λ η ν = λ η , where we set ν = ( ν , ν , ν ) . Since Q = ( ξ, η, , we have that Q ξ = (1 , , and Q η = (0 , , . So it holds that Q ξ · ν = ν and Q η · ν = ν . Substituting this into(A.5), we have(A.6) λ ξ = ν − ν , λ η = ν − ν . This implies that ( λ ξ , λ η ) is the image of ν via the stereographic projection, and wecan write(A.7) ν = 11 + λ ξ + λ η (2 λ ξ , λ η , λ ξ + λ η − . By (A.1), we have(A.8) P = ( ξ, η, − λν + (0 , , λ ) . We prove the following
Fact A.1.
The curvature line flow of the graph z = f ( x, y ) coincides with the eigen-flow of the Hessian of the function λ ( ξ, η ) given by (A.2) .Proof. Noticing (A.8), we set ∆ ( ξ,η ) := det νdPdν = det νdξ, dη, dλdν . Then this gives a map ∆ ( ξ,η ) : T ( ξ,η ) R → R such that ∆ ( ξ,η ) (cid:18) a ∂∂ξ + b ∂∂η (cid:19) = det (cid:18) ν, aP ξ ( ξ, η )+ bP η ( ξ, η ) , aν ξ ( ξ, η )+ bν η ( ξ, η ) (cid:19) ∈ R . It is well-known that w ∈ T ( ξ,η ) R points in a principal direction of P at ( ξ, η ) ifand only if ∆ ( ξ,η ) ( w ) = 0 . Since ( ν ) + ( ν ) + ( ν ) = 1 , (A.6) yields that λ ξ ν + λ η ν = ( ν ) + ( ν ) − ν = 1 − ( ν ) − ν = 1 + ν , which implies ν = λ ξ ν + λ η ν − . We now set µ = 2 / (1+ λ ξ + λ η ) . Differentiating(A.7), we have dν = dµµ ν + µ ( dλ ξ , dλ η , λ ξ dλ ξ + λ η dλ η ) . The first term of the right hand-side of the above equation is proportional to ν anddoes not affect the computation of ∆ ( ξ,η ) . So we have that ∆ ( ξ,η ) = µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν ν λ ξ ν + λ η ν − dξ dη λ ξ dξ + λ η dηdλ ξ dλ η λ ξ dλ ξ + λ η dλ η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν ν − dξ dη dλ ξ dλ η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − µ (cid:12)(cid:12)(cid:12)(cid:12) dξ dηdλ ξ dλ η (cid:12)(cid:12)(cid:12)(cid:12) = µ (cid:18) ( λ ξξ − λ ηη ) dξdη − λ ξη ( dξ − dη ) (cid:19) . Fact A.1 follows from this representation of ∆ ( ξ,η ) . (cid:3) A PPENDIX
B. I
NDICES OF EIGEN - FLOWS OF H ESSIAN MATRICES
Let g : Ω R \ { o } → R be a C -function, where Ω R is the closed disk of radius R centered at the origin o := (0 , (cf. (2.1)). The Hessian matrix of g is given by H g := (cid:18) g xx g xy g yx g yy (cid:19) . We denote by H g the eigen-flow of H g . A point p ∈ Ω R \ { o } is called an equi-diagonal point of H g if H g ( p ) is proportional to the identity matrix. Consider thecircle Γ( θ ) := r (cos θ, sin θ ) (0 ≤ θ < π, r < R ) . If there are no equi-diagonal points on Ω R \ { o } , then we can define the index ind( H g , Γ) of the eigen-flow H g with respect to Γ , which does not depend on thechoice of r . We call it the index of H g at the origin and denote it by ind o ( H g ) .Consider the vector field d g := 2 g xy ∂∂x + ( g yy − g xx ) ∂∂y . It is well-known that the mapping degree of the Gauss map (cf. (3.1)) ˇ d g : T := R / π Z ∋ θ d g (Γ( θ )) | d g (Γ( θ )) | ∈ S := { ( x, y ) ∈ R ; x + y = 1 } is equal to o ( H g ) . Using the correspondence ( x, y ) x + iy , we identify R with C , where i = √− . Then g z = 12 ( g x − ig y ) , g zz = 14 (( g xx − g yy ) − ig xy ) , where g z := ∂g/∂z , g zz := ∂ g/∂z and ∂∂z := 12 (cid:18) ∂∂x − i ∂∂y (cid:19) . Thus, d g can be identified with the right-angle rotation of g zz . In particular, we have(B.1) ind o ( H g ) = −
12 ind o ( g zz ) . Here g zz is considered as a vector field and ind o ( g zz ) is its index at the origin. Let ( r, θ ) be as in (2.5). Then z = re iθ and g z = e − iθ r ( rg r − ig θ ) , g zz = e − iθ r (cid:18) ( r g rr − rg r − g θθ ) + 2 i ( g θ − rg rθ ) (cid:19) . We consider the vector field defined by(B.2) δ g := 2( rg rθ − g θ ) ∂∂x + ( − r g rr + rg r − g θθ ) ∂∂y . Since ind o ( g zz ) = 2 + ind o ( δ g ) , we obtain the following: Lemma B.1.
The identity ind o ( H g ) = 1 + ind o ( δ g ) / holds.Acknowledgements. The third author thanks Udo Hertrich-Jeromin for fruitful con-versations. The authors thank Wayne Rossman for valuable comments. R EFERENCES[A] N. Ando,
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