The Total Curvature of Disks Extending Regular Closed Plane Curves

U.U.D.M. Project Report 2008:6

Examensarbete i matematik, 30 hp

Handledare och examinator: Tobias Ekholm Maj 2008

The Total Curvature of Disks Extending Regular Closed Plane Curves

Albin Eriksson Östman

The Total Curvature of Disks Extending Regular Closed Plane Curves

Albin Eriksson ¨Ostman Advisor: Tobias Ekholm

May 7, 2008

Abstract

An immersion of the circle into the plane extends to an immersion of the 2-dimensional disk into 3-space if and only if its Whitney index is odd. Given an immersion of the circle with odd Whitney index, we calculate the infimum of the total curvature over all immersed disks extending the immersed circle which are cylindrical near the boundary.

The infimum depends only on the Whitney index of the curve and it is attained only if the curvature function of the curve does not change sign.

Acknowledgments

I would like to thank Tobias Ekholm for suggesting the problem and for helping me during the work, as well as introducing me to interesting subjects in mathematics.

1 Introduction

If c : S¹ → R² is a regular closed plane curve, or for short an immersed circle, we define the Whitney index I(c) of c as the degree of the map S¹→ S¹, t 7→ ˙c(t)/| ˙c(t)|.

Let c : S¹→ R²× {0} ⊂ R³ be an immersed circle. We say an immersed disk f : D → R³ is an admissible extension of c if f |_S¹ = c and ^∂f_∂r = −e₃ in a neighborhood of S¹ ⊂ D, where D denotes the closed 2-dimensional unit disk, (r, θ) are polar coordinates on D, and (e₁, e₂, e₃) is the standard basis in R³. It is well known that such an extension exist if and only if I(c) is odd. We present a proof of this fact in Section 3.

Let R³× R³ → R³, (u, v) 7→ u × v denote the standard vector product, and let x = (x₁, x₂) be the standard coordinates on D. We denote the 2-dimensional sphere with S².

Definition 1.1. Let f : D → R³ be an immersed disk and let

N : D → S², x 7→

∂f

∂x1 ×_∂x^∂f₂

|_∂x^∂f

1 ×_∂x^∂f

2|

denote the Gauss map. The the total curvature K(f ) of f is K(f ) =

Z

D

|dN |.

Our main goal is the following theorem.

Theorem 1.2. Let c : S¹ → R² × {0} ⊂ R³ be an immersed circle with Whitney index I(c) = 2k + 1 for some k ∈ Z. Then

inf{K(f ) : f is an admissible extension of c} = 2π|2k + 1|.

Moreover, if the infimum is attained then the curvature function of c does not change sign.

2 Background

In this section we present background material we need in order to prove Theorem 1.2.

2.1 Regular homotopy of immersed circles

A regular plane curve is an immersion of the closed unit interval I into 2-space.

Definition 2.1. Let c : I → R² be a regular plane curve. The tangential change φ(c) of c is defined as

φ(c) = Z

˙c

ω₀, where

ω₀ = −y dx + x dy x²+ y²

is the winding form on R²\ {0} and ˙c is the curve defined by the derivative of c.

Remark 2.2. The tangential change is a counterpart of the Whitney index to non-closed curves. If c is an immersed circle, then I(c) = ^φ(c)_2π .

A regular homotopy between two immersed circles c₀, c₁ is a continuous 1-parameter family c : I × S¹ → R², (s, t) 7→ c_s(t) of immersed circles connecting c₀ and c₁. In particular ^∂c_∂t is continuous on I × S¹. If such a 1-parameter family exist, c₀ and c₁ are called regular homotopic. In [1]

Whitney proved the following theorem.

Theorem 2.3. Let c₀, c₁ : S¹ → R² be immersed circles. Then there is a regular homotopy between c₀ and c₁ if and only if I(c₀) = I(c₁).

From this we derive a slightly stronger result.

Theorem 2.4. Let c₀, c₁ : S¹ → R² be immersed circles. Then there is a differentiable regular homotopy

H(s, t) : I × S¹→ R²

such that H|_[0,²]×S1 = c₀ and H|_{[1−²,1]×S}1 = c₁(t), for some ² > 0, if and only if I(c₀) = I(c₁).

Proof. Suppose I(c₀) = I(c₁). Let c : I × S¹ → R², (s, t) 7→ c_s(t) be a regular homotopy which exist by Whitney’s theorem. Then ^∂c_∂t : I ×S¹ → R² is continuous. Since I × S¹ is compact and |^∂c_∂t| 6= 0 for all (s, t) ∈ I × S¹, there is some δ > 0 such that |^∂c_∂t| > δ on I × S¹. Moreover ^∂c_∂t is uniformly continuous. Therefore there is some N ∈ N such that |s − s⁰| < _N¹ implies

|^∂c_∂t(s, t) −^∂c_∂t(s⁰, t)| < ^δ₂. We deform ck

N to ck+1

N for k = 0, ..., N − 1, keeping the regularity. Let v(t) = ck+1

N (t) − ck

N(t) and let ϕ : I → I be an increasing differentiable function with ϕ|_[0,1

3]= 0, ϕ|_[2

3,1] = 1. Define H on [_N^k,^k+1_N ] × S¹ by H(u, t) = ck

N(t) + ϕ((u − k

N)N )v(t).

Then H satisfies the required properties. The other implication follows immediately from the preceding theorem.

2.2 Morse theory

We also need some well-known results from Morse theory. Let M and N be differentiable manifolds and let f : M → N be a differentiable function. A point p ∈ M is called regular if (df )_p has maximal rank, and critical if not.

A point q ∈ N is called a regular value if f⁻¹(q) just contains regular points, and a critical value if not. In the special case N = R a critical point p ∈ M is called non-degenerate if the Hessian of f at p is a non-degenerate quadratic form. It can be shown that this definition is independent on coordinates, and that non-degenerate critical points are isolated. The index of a non- degenerate critical point p is defined as the number of negative eigenvalues of the Hessian of f at p. A function f : M → R, with only non-degenerate critical points, with the property that f⁻¹((−∞, a]) is compact for all a ∈ R is called a Morse function. For a reference, and a proof of the following theorem, see [2].

Theorem 2.5 (Lemma of Morse). Let f : M → R, be a Morse function on a n-dimensional differentiable manifold M and let p ∈ M be a critical point for f . Then there is a coordinate neighborhood U of p with x_i(p) = 0, i = 1, ..., n and f = −x²₁− ... − x²_λ+ x²_λ+1+ ... + x²_n holds in U , where λ is the index of p. In particular, the critical points of f are isolated.

Another result which one extracts from [2], using the gradient vector field, is the following.

Theorem 2.6. Let f : M → R, be a differentiable function form a dif- ferentiable manifold M to R. Suppose f⁻¹[a, b] is compact and does not contain any critical point of f . Then there is a differentiable function r : I × M → M, (s, p) 7→ r_s(p) such that r_s : f⁻¹(b) → f⁻¹(b − s(b − a)) is a diffeomorphism.

Moreover, even if f⁻¹[a, b] contains critical points, for a subset B ⊂ f⁻¹(b) which stays away from a neighborhood of the critical points under the above diffeomorphisms, we can still define r on I × B, with the above properties.

We also need the following theorem proved in [4].

Theorem 2.7. Let M ⊂ Rⁿ be a embedded differentiable manifold, and let f : M → R be a differentiable function. Then

f_a(x) := f (x) + a₁x₁+ ... + a_nx_n

have non-degenerate critical points for almost all a = (a₁, ..., a_n) ∈ Rⁿ, where x₁, ..., x_n are the standard coordinate functions on Rⁿ.

2.3 Integral geometric measure

In the proof of the following theorem we need Sard’s theorem. Let M and N be differentiable manifolds, and let f : M → N be a differentiable function.

Let F ⊂ N be the set of critical values of f . Then Sard’s theorem says E has measure zero. For a proof, see [3].

Theorem 2.8. Let X and Y be differentiable manifolds with dim X = dim Y , and X compact. Let f : X → Y be a differentiable map, then

Z

X

| det(df )| = Z

Y

µ,

where µ(v) := |f⁻¹(v)|, i.e. µ counts how many times each point in Y is hit.

Proof. Let F ⊂ Y be the set of all critical values of f . Then F has measure zero. Since X is compact each y ∈ Y \F have an open neighborhood V_y such that f⁻¹(V_y) = U_y is a finite union of disjoint open subsets U_y = U_y¹∪...∪U_y^ry and f : U_yⁱ → V_y is a diffeomorpism. Write Y \ F = ∪_y∈Y0V_y, where Y₀ is a countable subset of Y . Let {ϕ_y}_y∈Y0 be a partition of unity subordinate to {V_y}_y∈Y0.

The usual change of variable formula implies Z

Vy

µϕ_y = r_y Z

Vy

ϕ_y = Z

Uy

ϕ_y◦ f | det(df )|.

Let E := | det(df )|⁻¹(0). We now conclude:

Z

Y

µ = Z

Y \F

µ = Z

Y \F

X

y∈Y0

ϕ_yµ

= X

y∈Y0

Z

Vy

ϕ_yµ = X

y∈Y0

Z

Uy

ϕ_y◦ f | det(df )| = X

y∈Y0

Z

X\F

ϕ_y◦ f | det(df )|

= Z

X\F

X

y∈Y0

ϕ_y◦ f | det(df )| = Z

X\F

| det(df )| = Z

X

| det(df )|.

3 Immersed disks extending immersed circles

For some immersed circles it is easy to see there is an admissible extension.

See Figure 1. Since the curves are symmetric, we can construct disks f extending them by taking the surface of revolution of half of the curve, rotating trough an angle π around the y-axis, and then smoothing near the

Figure 1: Standard curves c₁, c₃ and c₅.

boundary so that f satisfies ^∂f_∂r = (0, 0, −1) in a neighborhood of S¹ and so that the Gaussian curvature of f does not change sign. In the same manner we construct admissible extensions f_2k+1 of similar curves c_2k+1 with I(c_2k+1) = 2k + 1. For an example, see Figure 2.

3.1 When is an extension possible

Theorem 3.1. Given an immersed circle c : S¹ → R²× {0} ⊂ R³, there is an admissible extension f of c if and only if I(c) = 2k + 1, k ∈ Z.

Proof. Suppose I(c) = 2k + 1, k ∈ Z. Let c_2n+1 denote our standard curve with Whitney index 2k + 1 and let f_2k+1denote the standard disk extending c_2k+1. We use a differentiable regular homotopy H between c and c_2k+1 as in 2.4. Define f : D → R³ by

f (r, θ) :=

(

H(2r − 1, θ) + (1 − r)e₃, ¹₂ ≤ r ≤ 1 f_2k+1¹ (2x)e₁+ f_2k+1² (2x)e₂+ (¹₂f_2k+1³ (2x) +¹₂)e₃, 0 ≤ |x| ≤ ¹₂ . Where f_2k+1ⁱ are the components of f_2k+1, (e₁, e₂, e₃) is the standard basis in R³, x = (x₁, x₂) are the standard coordinates in D, and (r, θ) are polar coordinates in D. Now f is an immersed disk, f |_S¹ = c and ^∂f_∂r = (0, 0, −1) in a neighborhood of S¹, so f is an admissible extension of c.

To prove the other direction, suppose there is an admissible extension f : D → R³ of c : S¹ → R²× {0}, with I(c) = 2k, k ∈ Z.

Define ˜F , ˜G : S¹ → GL⁺₃ by:

θ 7→ F (θ) :˜







e₁ 7→ ˜v₁ := df_θ(_∂x^∂₁) e₂ 7→ ˜v₂ := df_θ(_∂x^∂

2) e₃ 7→ ˜v₃ := ˜v₁× ˜v₂

θ 7→ G(θ) :˜



e₁ 7→ ˜w₁ := df_θ(_∂r^∂) e 7→ ˜w := df ( ^∂)

Figure 2: The surface of revolution extending c₃.

where GL⁺₃ is the general linear group of invertible linear maps R³ → R³ of positive determinant. Define maps F, G, R : S¹ → SO₃ by:

θ 7→ F (θ) :









e₁ 7→ v₁:= _{| ˜}^v_v^˜1_1|

e₂ 7→ v₂:= _{| ˜}^v_v^˜2₂^−hv_−hv¹₁^{, ˜}_{, ˜}^v_v²₂^iv_iv¹_1| e₃ 7→ v₃:= v₁× v₂

θ 7→ G(θ) :







e₁ 7→ w₁ := ˜w₁ e₂ 7→ w₂ := _{| ˜}^w_w^˜2

2|

e₃ 7→ w₃ := w₁× w₂

θ 7→ R(θ) :







e₁ 7→ cos(θ)e₁+ sin(θ)e₂ e₂ 7→ − sin(θ)e₁+ cos(θ)e₂ e₃ 7→ e₃

where SO₃ is the group of orthogonal linear transformations R³ → R³ of determinant 1.

The closed curve in SO₃defined by F is contractible, since we can extend F to D. G is contractible as well, since I(f |_S¹) = 2k and therefore G lifts to a closed curve in the cover space S³ of SO₃. Moreover F ' ˜F and G ' ˜G, where X ' Y stands for X and Y are homotopic, since we can perform the Gram-Schmidt orthogonalization continuously. Now R ' F ◦ R ' ˜F ◦ R = G ' G in GL˜ ⁺₃. But then R ' G already in SO₃, since we have a retract r : GL⁺₃ → SO₃ defined by performing Gram-Schmit orthogonalization.

This is a contradiction, since R is not contractible.

4 The total curvature of an admissible extension

The main goal of this section is to prove Theorem 1.2. We begin by ex- pressing the total curvature of an immersed disk in terms of critical points to Morse functions.

Definition 4.1. Let f : D → R³ be an immersed disk. For v ∈ S₊², let h_v : D → R, x 7→ hv, f (x)i denote the height function in direction v.

(Where R³ × R³ → R, (u, v) 7→ hu, vi denotes the usual inner product on R³, and S²₊:= {(x₁, x₂, x₃) ∈ R³ : x₃≥ 0}.) Moreover, let

µ : S₊² → N, v 7→ |{x ∈ D : dh_v(x) = 0}|

i.e. µ(v) is the number of critical points of the height function in direction v.

Lemma 4.2. h_v : D → R, x 7→ hv, f (x)i is a Morse function for almost all

Proof. Cover D with neighborhoods U₁, ..., U_n such that f |_Ui is an embed- ding. Then Theorem 2.7 imply there is a set E_i ⊂ S₊² of measure zero, such that h_v is a Morse function on U_i for v ∈ S₊² \ E_i. (Since h_v have non-degenerate critical points if and only if h_λv have, λ ∈ R \ {0}, a set of positive measure on S₊² inducing h_v’s with degenerate critical points gives a cone C ⊂ R³ with positive measure such that 0 + c₁x₁+ ... + c_nx_n have degenerate critical points for all c = (c₁, ..., c_n) ∈ C, contradicting Theorem 2.7.) Now take E = E₁∪ ... ∪ E_n.

Lemma 4.3. Let f : D → R³ be an immersed disk. Then K(f ) =R

S₊² µ.

Proof. K(f ) =R

D|dN | =R

S²|N⁻¹(v)|, by Theorem 2.8, now Z

S²

|N⁻¹(v)| = Z

S₊²

|N⁻¹(v)| + |N⁻¹(−v)| = Z

S₊²

µ, since x is a critical point of h_v if and only if N (x) = ±v.

Lemma 4.4. Let f_2k+1 be our standard disk extending our standard curve c_2k+1 with I(c_2k+1) = 2k + 1, then µ(v) = |2k + 1| for almost all v ∈ S₊². Proof. Straightforward.

Now we calculate an upper bound for the infimum of the total curvature.

Theorem 4.5. Let c : S¹ → R² × {0} ⊂ R³ be an immersed circle with Whitney index I(c) = 2k + 1 for some k ∈ Z. Then

inf{K(f ) : f is an admissible extension of c} ≤ 2π|2k + 1|.

Proof. Let f : D → R³, x 7→ (f₁(x), f₂(x), f₃(x)) be the disk extending c we constructed in 3.1. Since for ¹₂ ≤ r ≤ 1, ^∂f_∂r¹ = −e₃, the normal map N_f : D \ ¹₂D → S² misses a neighborhood V ⊂ S² of {e₃, −e₃}. Let V = V ∩ S˜ ₊². Now all critical points of h_v with v ∈ ˜V comes from our standard disk f_2k+1, i.e. µ(v) = |2k + 1| for v ∈ ˜V

Define for each λ in R a new disk

f^λ : D → R³, x 7→ (f₁(x), f₂(x), λf₃(x)).

Now K(f_λ) →R

S₊² |2k + 1| = 2π|2k + 1| as λ → ∞.

4.1 The upper bound is the best possible

The idea is to show that each admissible extension f : D → R³ of a curve c with Whitney index I(c) = 2k + 1 has to have at least |2k + 1| critical points for each height function h_v, v ∈ S₁⁺. To do so we investigate what happens with the Whitney index of f (D) ∩ P , where P is a plane orthogonal to v, when we let P pass critical points. To do so we need to attach a cylinder to our immersed disk f : D → R³. This can be done since the disk is an admissible extension of c.

Definition 4.6. Given an admissible extension f : D → R³ of an immersed circle c : S¹ → R²× 0 define the associated cylinder as ˜f : R² → R³

f (x) :=˜

(f (x), x ∈ D

f (_|x|^x) + (1 − |x|)e₃, x ∈ R²\ D

Definition 4.7. Let π : R³ → v^⊥ denote the orthogonal projection on the plane v^⊥ of all vectors in R³ orthogonal to v. If a is a regular value of h_v, then h⁻¹_v (a) ⊂ R² is a finite family of non-intersecting embedded circles.

Parametrize them with γ₁, ..., γ_n : S¹ → D, such that {^∂γ_∂tⁱ, gradh_v(γ_i)} is positively oriented in D, where grad(h_v) is the gradient of h_v. Define the total index at height a as

I_a:=

Xn i=1

I(π ◦ f ◦ γ_i).

Remark 4.8. If our plane P with normal vector v ∈ S₊² \ S¹ only intersect the “cylinder part” of ˜f the Whitney index of the intersection is the Whit- ney index of ±c, since the intersection and ±c are regular homotopic via a rotation of P and a projection on R² × {0}. This eventually occurs if we translate P downwards along the x₃-axis, since ˜f (D) is compact. In terms of the total index this means I_a= ±I(c) for a ¿ 0.

We now show that the total index change with one when we pass a critical point.

Lemma 4.9. If h⁻¹_v ([a, b]) does not contain any critical point of h_v, then I_a= I_b.

Proof. Flow the parameterizations with the funtion r_s defined in Theorem 2.6. Each component in f (h⁻¹_v (a)) is in this way regular homotopic to exactly one component in f (h⁻¹_v (b)).

Lemma 4.10. If h⁻¹_v ([a, b]) contains a critical point p of index 0 of h_v, but no other critical point, then I_a= I_b± 1.

Proof. This follows directly from Theorem 2.5. We just add an immersed circle with positive or negative orientation. The other components of f⁻¹(a) respectively f⁻¹(b) are regular homotopic like in the perseeding lemma.

Lemma 4.11. If h⁻¹_v ([a, b]) contains a critical point p of index 2 of h_v, but no other critical point, then I_a= I_b± 1.

Proof. Like the preceding lemma, but instead of add we remove an immersed circle with positive or negative orientation.

Proof. Let c := h_v(p) and let v₁, v₂ ∈ v^⊥be an orthonormal base in v^⊥. In a neighborhood of p, we can parametrize the disk with the coordinates given by {v₁, v₂}, via h_v i.e. f (y₁, y₂) = (y₁, y₂, h_v(y₁, y₂)) and whitout loss of generality f (0, 0) = p If we Taylor expand h_v around p in these coordinates we see h_v(y₁, y₂) = a₁y₁²+ a₂y₁y₂+ a₃y₂²+ O(3) + c, since p is a critical point of h_v. Moreover, the quadratic form is non-degenerate since the critical point p is. Therefore, h⁻¹_v (c + ²) and h⁻¹_v (c − ²), looks locally like hyperbolas when ² is small. Fix four point P₁, ..., P₄ in h⁻¹_v (c + ²) as indicated in Figure 4.1. Let γ₁ be the curve joining P₄ and P₁ in the indicated direction, and let γ₂ join P₂ and P₄ in the same manner. If ² is small, we can flow γ₁ and γ₂ with the function r_s defined in Theorem 2.6 from h⁻¹_v (c + ²) to h⁻¹_v (c − ²) without altering the total tangent change much. Let ˜γ₁ and ˜γ₂ denote the resulting curves. Then

φ(γ₁) = φ( ˜γ₁) + δ₁ φ(γ₂) = φ( ˜γ₂) + δ₂,

for small error terms δ_i. Let γ₃ be the curve joining P₃ and P₂ and γ₄ the curve joining P₁ and P₄. Let ˜γ₃ be the curve joining Q₃ and Q₄ and ˜γ₄ the curve joining Q₁ and Q₂. We have

φ(γ₃) − φ( ˜γ₄) + φ(γ₄) − φ( ˜γ₃) + 4π = 2π + δ₃, for a small error term δ₃ if ² is small. This implies:

φ( ˜γ₃) + φ( ˜γ₄) = φ(γ₃) + φ(γ₄) + 2π − δ₃. Now,

φ( ˜γ₁)+φ( ˜γ₂)+φ( ˜γ₃)+φ( ˜γ₄) = φ(γ₁)+φ(γ₂)+φ(γ₃)+φ(γ₄)+2π−δ₁−δ₂−δ₃. Choose ² so small that |δ_i| < ^π₃. Since the total change is a multiple of 2π the lemma follows. (If the orientation is opposite, the argument is analogous.)

Remark 4.13. Lemma 4.9 to 4.12 also shows that if h⁻¹_v [a, b] contains n critical points then |I_a− I_b| ≤ n, since the calculations are local.

We can now prove our main result.

Proof of Theorem 1.2. The upper bound is already established. For the lower bound, Remark 4.8 and Remark 4.13 imply that each height function h_v has at least |2k + 1| critical points. The lower bound now follows from Lemma 4.3.

If the curvature function of a curve c of Whitney index of absolute value 2k + 1 changes sign then the total curvature of c is larger than 2π(2k + 1).

This implies that there exists some direction v in the plane such that the

Figure 3: The level curves h⁻¹_v (c + ²) and h⁻¹_v (c − ²).

height function h_v in direction of v has at least 2(2k + 1) + 2 non-degenerate critical points. Since the extension is cylindrical near the boundary it follows that there is some direction in space near v in which the number of critical points for the height function is at least 2(2k + 1) + 2. It follows that the infimum can not be attained unless the curvature function of c does not change sign.

References

[1] H. Whitney On regular closed curves in the plane, Compositio Mathe- matica, tome 4, p.276-284, 1937.

[2] J. Milnor: Morse Theory, Princeton University Press, Annals of Math- ematics Studies, Princeton, 1973.

[3] M. Golubitsky and V. Guillemin: Stable Mappings and Their Singu- larities, Springer-Verlag, Graduate Texts in Mathematics, New York, 1973.

[4] V. Guillemin and A. Pollack: Differential Topology, Prentice-Hall, New Jersey, 1974.