© Hindawi Publishing Corp. ON SOME NEW PROPERTIES OF THE SPHERICAL CURVATURE OF STEREOGRAPHICALLY PROJECTED ANALYTIC CURVES

We discover new information about the spherical curvature of stereographically projected analytic curves. To do so, we first state formulas for the spherical curvature and spherical torsion of the curves on S2 which result after stereographically projecting the image curves of analytic, univalent functions belonging to the class 𝒮. We then derive results concerning the location of the critical points of the spherical curvature, considered both as a function of one and two variables. Further analysis leads to a maximum principle for the spherical curvature functions.


Introduction.
We begin by defining the spherical curvature and spherical torsion of the stereographically projected images of circles and radii in U .
Let f ∈ and let Π denote the stereographic projection of the image plane of f onto the unit sphere S 2 .For each fixed r , 0 < r < 1, let For each fixed θ, −π < θ ≤ +π , let The local curvature at a specified point on each of the curves Ꮿ r and ᏸ θ has been the object of intense research [1, pages 126 and 262].Of course, the local torsion at any specified point on each of these curves is equal to zero.Our immediate objective is to determine formulas for the spherical curvature and spherical torsion at a specified point on the stereographically projected curves Ꮿ r and ᏸ θ on the unit sphere S 2 .A parametrization of the curves under consideration is easily prescribed.Indeed, if we write f (z) = f (r ,θ) = u(r ,θ),v(r ,θ) z = r e iθ ∈ U , (1.3) then Ꮿ r = X(r , θ) : −π < θ ≤ +π , ᏸ θ = X(r , θ) : 0 < r < 1 , (1.4) where In terms of the parametrization (1.5), the local spherical curvature κ(θ; r ,f ) and the local spherical torsion τ(θ; r ,f ) at the point X(r , θ) on the curve Ꮿ r are classically defined by the formulas where the subscripts on X denote the variable with respect to which the partial derivative is taken.The local spherical curvature κ(r ; θ, f ) and spherical torsion τ(r ; θ, f ) at the point X(r , θ) on the curve ᏸ θ are given by similar formulas.
In our first result below, we provide explicit formulas for the spherical curvatures and spherical torsions under study.In all results to follow, the quantity will denote the spherical derivative of f (z) and the quantity will denote the Schwarzian derivative of f (z).Note that (1.11) provides an explicit connection between these two quantities.In all formulas to follow, the subscripts on ᐆ denote the variable(s) with respect to which the partial derivative is taken.
(a) At the point X(r , θ) on the sphere curve Ꮿ r , the local spherical curvature κ(θ; r ,f ) is given by the formula and the local spherical torsion τ(θ; r ,f ) is given by the formula these quantities are related by (1.12) (b) At the point X(r , θ) on the sphere curve ᏸ θ , the local spherical curvature κ(r ; θ, f ) is given by the formula and the local spherical torsion τ(r ; θ, f ) is given by the formula Furthermore, these quantities are related by The proof of this result is lengthy, but straightforward, and is omitted.
Remark 1.2.In deriving formulas (1.9) and (1.13), it becomes clear that the partial derivatives −r ᐆ r (r , θ; f ) and ᐆ θ (r , θ; f ) are actually the real and imaginary parts of the same quantity.Indeed, if we set then, with z = r e iθ , we have Hence, the spherical curvatures are related to each other via Φ(z; f ).
shows that ᐆ(r , θ; f ) is subharmonic in the punctured disk U − {0}.Indeed, with the additional formulas (to be used in the proof of Theorem 3.1) we can explicitly compute the Laplacian of ᐆ(r , θ; f ) to be More generally, we see that ᐆ α (r , θ; f ) is also subharmonic for every α > 0 since (1.23) Also, we note that (1.24) In Section 2, for an arbitrary f ∈ , we are able to determine all critical points of κ(θ; r ,f ) and κ(r ; θ, f ) when they are considered as functions of one variable, necessarily introducing the level sets λ r 0 (f ), λ θ 0 (f ), and Λ 0 (f ).Using these data, a strategy for locating the extreme values of the spherical curvature κ(θ; r ,f ) on the curve Ꮿ r and the spherical curvature κ(r ; θ, f ) on the curve ᏸ θ is discussed.
In Section 3, for an arbitrary f ∈ , we are also able to determine the stationary points of κ(θ; r ,f ) and κ(r ; θ, f ) when they are considered as functions of two variables.
To complement this result, in Section 4, we show that on certain subdomains D, there exist real α and β for which κ α (θ; r ,f ) and κ β (r ; θ, f ) are subharmonic on D. Although it is not necessarily true that κ(θ; r ,f ) and κ(r ; θ, f ) are subharmonic, we will show nevertheless that maximum principles for these spherical curvature functions are still valid on certain subdomains of U.

Critical points of the spherical curvature functions considered as functions of one variable.
We now consider the problem of determining the critical points of the spherical curvature functions on the curves Ꮿ r and ᏸ θ for an arbitrary function f ∈ .This is an important task since extreme values of the spherical curvature functions will occur at critical points.
(b) By differentiating (1.13), we obtain Thus, for a critical point of the spherical curvature κ(r ; θ, f ) to occur at a point z = r e iθ ∈ ᏸ θ corresponding to a point X(r , θ) ∈ ᏸ θ , it is necessary and sufficient that either ᐆ θ (r , θ; f ) = 0 or that ᐆ r θ (r , θ; f ) = 0. (It is possible for both of these quantities to equal zero simultaneously.)If ᐆ θ (r , θ; f ) = 0, then κ(r ; θ, f ) = 1, an absolute minimum value, and the corresponding z value belongs to the level set λ θ 0 (f ), and conversely.If ᐆ r θ (r , θ; f ) = 0, then Im[z 2 {f ,z}] = 0, where z = r e iθ ∈ Λ 0 (f ), and conversely.Remark 2.2.As a consequence of Theorem 2.1, the procedure of determining the critical values of both spherical curvature functions, and subsequently the location of local and absolute minimum and maximum values of either of the spherical curvature functions, is now clear.We first determine the level sets λ r 0 (f ), λ θ 0 (f ), and Λ 0 (f ).We then compare the values of κ(θ; r ,f ) on the set (λ r 0 (f ) ∪ Λ 0 (f )) ∩ Ꮿ r ⊂ U to each other and the values of κ(r ; θ, f ) on the set (λ θ 0 (f ) ∪ Λ 0 (f )) ∩ ᏸ θ ⊂ U to each other.Alternately, after all of the critical points have been located, the standard second derivative test may be employed as follows in an attempt to classify them.
For the curves Ꮿ r , another differentiation of (2.2) with respect to θ yields (2.4) If the point X(r , θ) ∈ Ꮿ r corresponds to a critical point for which ᐆ r (r , θ; f ) = 0, that is, for which κ(θ; r ,f ) = 1, then the form of the second derivative reduces to which is clearly nonnegative at an absolute minimum as expected.On the other hand, if ᐆ r (r , θ; f ) ≠ 0 and ᐆ r θ (r , θ; f ) = 0, then the form of the second derivative reduces to By differentiating (1.11) with respect to θ, we obtain and the second derivative further reduces to the form clearly indicating the two quantities upon which the classification of the critical point depends.
For the curves ᏸ θ , another differentiation of (2.3) with respect to r yields (2.9) If X(r , θ) ∈ ᏸ θ corresponds to a critical point for which ᐆ θ (r , θ; f ) = 0, that is, for which κ(r ; θ, f ) = 1, then the form of the second derivative reduces to which is clearly nonnegative at an absolute minimum as expected.Now, X(r , θ) ∈ ᏸ θ corresponds to a critical point for which ᐆ θ (r , θ; f ) ≠ 0 and ᐆ r θ (r , θ; f ) = 0, then the form of the second derivative reduces to Theoretically, the second derivatives κ θθ (θ; r ,f ) and κ r r (r ; θ, f ) may also be used to determine the candidates for inflection points of the spherical curvatures κ(θ; r ,f ) and κ(r ; θ, f ) of the curves Ꮿ r and ᏸ θ , respectively, although in practice, this task is computationally formidable.

Stationary points of the spherical curvature functions considered as functions of two variables.
Although the prescribed spherical curvature functions κ(θ; r ,f ) and κ(r ; θ, f ) have been defined as a function of one variable (as a function of θ with r fixed, and vice versa), it is clear that each of them may be considered as a function of two variables on the punctured disk U − {0}.Upon adopting this point of view, it is natural to ask whether the spherical curvature functions have stationary points in U − {0} and how to locate them if they exist.The answer is provided in the following result.(a) Any stationary points of the spherical curvature κ(θ; r ,f ), when considered as a function of two variables, belong to the set λ r 0 (f ) ∪ (Λ 0 (f ) ∩ Ξ + 0 (f )), where Ξ + 0 (f ) is defined within the proof.Furthermore, if the stationary point z ∈ Λ 0 (f ) ∩ Ξ + 0 (f ), then it is necessary that z 2 {f ,z} be real and greater than 1/2 at that point.
(b) Any stationary points of the spherical curvature κ(r ; θ, f ), when considered as a function of two variables, belong to the set then it is necessary that z 2 {f ,z} be real and less than 1/2 at that point.
We conclude from Theorem 3.1 that no point z ∈ Λ 0 (f ) can simultaneously be a stationary point for both spherical curvature functions κ(θ; r ,f ) and κ(r ; θ, f ).
Having located the stationary points of both spherical curvature functions, the standard Hessian test may be used to classify them when the test is not inconclusive.

Subharmonicity and maximum principles for spherical curvature.
It is natural to ask whether there exist maximum principles for the spherical curvature functions on subdomains of U −{0}.It is well known that subharmonic functions satisfy principles of this type.Hence, we attempt to discover conditions under which we may conclude that powers of the spherical curvature functions are subharmonic.
Let D be a simply connected domain with piecewise smooth boundary ∂D contained in U − {0}.(Of course, more general domains may be considered.)Note that both spherical curvature functions given by (1.9) and (1.13) have the algebraic form w = (1 + u 2 ) 1/2 , where either u = −(1/2)r ᐆ r (r , θ; f ) or u = +(1/2)ᐆ θ (r , θ; f ).A short computation shows that Since the function u∆u is continuous on D, it must have an absolute minimum there, which may be negative.For w α to be subharmonic, the other quantity within the square brackets must compensate for the possible negativity.This will occur provided that and that α is sufficiently large.On the other hand, if u∆u is positive, then the choice α = 1 implies that w is subharmonic.
Theorem 4.1.Let f ∈ and let D be a simply connected domain with piecewise smooth boundary ∂D contained in U −{0}.
Consequently, under the given conditions, the maximum values of κ α (θ; r ,f ) and κ β (r ; θ, f ), when considered as functions of two variables, must occur on the boundary of the domain D.

Topological properties of the level sets λ r
0 (f ), λ θ 0 (f ), and Λ 0 (f ).The importance of the level sets λ r 0 (f ), λ θ 0 (f ), and Λ 0 (f ) has been firmly established in Theorems 2.1, 3.1, and 4.1.Therefore, in view of our central purpose, it is important to discuss these sets in further detail.
In view of the representations (1.17) and (1.18), it is clear that the sets λ r 0 (f ) and λ θ 0 (f ) are the level sets of infinitely smooth surfaces for every f ∈ .Some additional topological properties of these level sets are established here.(c) The level sets λ r 0 (f ) and λ θ 0 (f ) do not contain Jordan curves (i.e., simple closed curves) which bound simply connected subdomains of U −{0}.
Proof.(a) Recall (1.17).The numerator of Re Φ(z; f ) is the difference of two functions: the first N 1 (z) of which is harmonic and the other N 2 (z) which is not.They can be equal on an open set if and only if they are both constant.The function N 1 (z) is constant on an open set if and only if f (z) = z, but in this eventuality, N 2 (z) is a nonconstant function of r .
The argument for λ θ 0 (f ) proceeds in a similar manner.
Remark 5.3.It is equally important to discuss the level set Λ 0 (f ).Since Im[z 2 {f ,z}] is a harmonic function, some of the properties of the level set Λ 0 (f ) follow from well-known properties of harmonic functions.For instance, Λ 0 (f ) will contain neither an isolated point nor an arc which terminates at an interior point of U since either of these occurrences would violate the mean value property for harmonic functions.Also, if the level set Λ 0 (f ) is to contain either an open subset of U or a closed Jordan curve, then Im[z 2 {f ,z}] would be identically zero on U , due to an application of the maximum principle for harmonic functions.This possibility could occur if, for instance, f (z) = z/(1− cz) for some constant c.

Corollary 4. 2 (
maximum principles for the spherical curvature functions).Let f ∈ and let D be a simply connected domain with smooth boundary ∂D contained in U − {0}.Then, under the conditions stated in Theorem 4.1, the maximum values of κ(θ; r ,f ) and κ(r ; θ, f ), considered as functions of two variables, must occur on the boundary of D.

Proposition 5. 1 .
Let f ∈ .(a) The level set λ r 0 (f ) does not contain an open subset of U. (b) The level set λ θ 0 (f ) contains an open subset of U if and only if f (z) = z, in which case λ θ 0 (f ) = U .