Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 31

In paper [Hass and Scott, 1994] a simple construction of flow of curves is given which homotops a finite set of curves in a surface in such a way that:

(1)The number of self-intersections does not increase for each curve,

(2)The number of intersections does not increase for each pair of curves,

(3)Each curve either disappears in finite time or becomes close to a geodesic,

(4)This flow can be extended continuously to k-parameter families of curves.

This algorithm can be programmed easily and admits a generalisation to higher dimensions.

Let γ be a piecewise smooth immersed curve in a Riemannian surface F. Let us cover F by convex discs D₁, . . . , Dn whose radii are smaller than the injectivity radius. We choose discs to be in general position so that any point in F belongs to the boundary of no more than two discs, the boundaries of the discs intersect transversely the curve γ, and the discs of halved radii with the same centers cover still the surface F. Such a cover will be called well situated with respect to γ. Let us number the discs D₁, D₂, . . . cyclically so that D_n+i = Di for any i ≥ 1.

Roughly speaking, a disc flow will be defined as homotopy of each arc in γ ∩ D₁ to the unique geodesics with the same ends, then the process repeats for each disc D₂, D₃, . . . .

We investigate properties and convergence of the flow, then we show that the number of intersection points does not increase.

Fig. 3.21Disc flow

We start with several combinatorial lemmas.

We call an embedded loop of the curve γ any embedded subarc of γ with coinciding ends which bounds an embedded disc, see Fig. 3.22 left. An embedded bigon is a pair of subarcs of γ with common ends that bound an embedded disc, see Fig. 3.22 right. An embedded loop (bigon) is called innermost if the correspondent embedded disc does not contain any other arcs of γ.

Fig. 3.22An embedded loop and an embedded bigon

Lemma 3.1. 1. Given a triangle ABC and several embedded curves crossing it, such that any two curves have at most one intersection point in the triangle and neither curve intersects the side BC, then on each side AB and AC there is an innermost triangle adjacent to it.

2. Given an innermost bigon with embedded curves crossing it, for each edge of the bigon there is an innermost triangle adjacent to it.

Proof. 1. The proof is by induction on the number k of the intersecting curves. For k =1 the statement is evident. Assume that the statement holds for k = n. Consider the case of k = n + 1 intersecting curves. Let D be the intersection point on the side AB closest to A, and E be the intersection point on the side AC which belongs to the same intersecting curve as D. Then the triangle EAD contains at most n intersecting curves and none of them intersects the side AD. By induction assumption there is an innermost triangle adjacent to the side EA, thus, adjacent to AC. The reasonings for the side AB are analogous.

2. Since the bigon is innermost, any two curves inside intersect in at most one point. Draw a curve intersecting the bigon near to one of its vertices and not intersecting other curves. This curve splits the bigon into two triangles one of which is innermost. By the first statement, there are innermost triangles adjacent to edges of the triangle of the splitting. These innermost triangles cannot be adjacent to the splitting curve, thus, we can remove the supplementary splitting curve.

Remark 3.3.

(1)The lemma does not require the intersecting curves to be in general position. There can be multiple intersection points.

(2)The second statement of the lemma remains valid if one takes an embedded loop, which does not contains any bigons, instead of the bigon. That is, there is an innermost triangle adjacent to the edge of such loop.

Lemma 3.2. A finite set of piecewise smooth transversal curves in a convex disc can be homotoped (with respect to the boundary) to a set of geodesics such that the number of self-intersection and intersection points of the curves do not increase during the homotopy.

Proof. The proof is based on induction on double point number (with multiplicity). We count here an intersection point of multiplicity k as double points.

Assume that the number of double points is minimal. Then the curves do not have self-intersections and any two curves intersect in at most one point.

If there are closed curves, then take an innermost closed curve. This curve can be contracted to a point without intersecting other curves. Repeating this operation we homotop our set of curves to a set without closed curves. So we can suppose there are no closed curves in the set.

Let {ai}, i = 1, . . . , n, be the set of curves and let di be the geodesics connecting the ends of the curve ai in the disc for each i = 1, . . . , n. We can isotop slightly (with respect to the boundary) the curves {ai} so that they lie in general position with {di}. So we can suppose that all intersections of {ai} and {di} are transversal.

Assume that there are extra intersections of d₁ with the curves {ai}. Then there is a bigon formed by a subarc of d₁ and a subarc of some curve aj.

Take an innermost bigon among all bigons adjacent to d₁. Its edges belong to d₁ and aj for some j. Since each two curves in {ai} intersect in at most one point, this bigon is an innermost bigon among all the bigons in the disc. If some curves {ai} intersect the bigon, then by Lemma 3.1 there is an innermost triangle in the bigon adjacent to aj. So, by moving aj, we can either decrease the number of intersection points in the bigon (if the triangle is adjacent to one edge of the bigon, see Fig. 3.23 left) or decrease the number of arcs intersecting the bigon (if the triangle is adjacent to two edges of the bigon, see Fig. 3.23 right).

Fig. 3.23Removing intersecting curves from the bigon

Repeating this operation, we obtain an empty bigon. Then we eliminate the bigon by moving the curve aj.

In this manner we can homotop our curves to a position where the curves aj, j > 1, intersect d₁ in at most one point and the curves d₁ and a₁ form a bigon. This bigon is innermost so a₁ can be homotop to d₁ without increasing the number of intersection points.

Applying these reasonings consequently to a₂ and d₂, a₃ and d₃ etc., we homotop the set of curves to the geodesics.

Assume now that the number of double points is not minimal. Then there is either a self-intersection of some curve or two curves have two or more intersection points. Then there is an embedded loop or a bigon in the disc. Take an innermost loop or a bigon among all embedded loops and bigons.

An innermost loop is empty and can be contracted, so the number of intersection points will reduce, see Fig. 3.24.

Fig. 3.24Removing intersecting curves from the bigon

An innermost bigon may contain only arcs without self-intersections, and any two of these arcs can have at most one intersection point. Then there is an innermost triangle adjacent to an edge of the bigon. Applying the triangle move, we can reduce either the number of intersection points inside the bigon or the number of arcs intersecting the bigon. After all, we shall have an empty bigon which can be removed with eliminating two intersection points in the disc.

Thus, the configuration can be reduced to a case with the minimal number of intersection points and lemma is proved.

Note that the constructed homotopy will be regular everywhere except the moment when a loop is contracted to a point.

We can define the disc flow as follows. Consider a curve s = s₀. It can have several components and self-intersections. We define a family of curves st, t ≥ 0 inductively. Given a curve s_i−1, i ∈ , we reduce the radius of the disc Di by a factor so that s_i−1 is in general position with the boundary ∂Di. It means that s_i−1 and ∂Di intersect transversely and ∂Di contains no self-intersection points of s_i−1. Then we define st, t ∈ [i − 1, i] as the image of s_i−1 under the straightening homotopy of Lemma 3.2 in the (shrinked) disc Di. Note that the length of curves st is not monotonous but the length sequence of the curves si, i ∈ , is non increasing. The family of curves st, t ≥ 0, is called the disk flow of the curve s = s₀.

The disc flow is not canonical since the choice of homotopy in Lemma 3.2 is not unique. Nonetheless, it possesses several useful properties whose formulation requires some additional notation. Let Δ be the map defined on the set of curves F which transforms a curve s = s₀ to the curve sn obtained from s by consecutive straightening in the discs D₁, . . . , Dn. When we talk about convergence of curves, we use the topology in the curve space such that the ε-neighbourhood of a curve γ consists of curves γ′ that admits a parametrisation for which the curves γ and γ′ are ε-close in the Frechet topology. This topology is induced by the Frechet metric in the space of curves.

Theorem 3.5. Let γ, γ′ be transversally intersecting curve in a surface F. Let D₁, . . . , Dn be a disc covering of the surface F which is good with respect to γ ∪ γ′. Let γt be the images of the curves by the disc flow.

(1)The number of self-intersection points of the curve γt does not increase with the grow of t ∈ [0, +∞).

(2)The number of intersection points between the curves γt and does not increase with the grow of t, t ∈ [0, +∞).

(3)Either γt disappears in a finite time or a subsequence of the curves {γt} converges to a geodesic with t → ∞. In the second case, if U is an open neighbourhood of the set of geodesics which are homotopic to γt, then there exists T > 0 such that γt belongs to U with t > T.

(4)If the sequence {γi} converges to a geodesic γ_∞ with i → ∞, then length (Δ(γ)) converges to length (γ_∞).

(5)Length(Δ(γ)) ≤ length(γ), and the equality takes place only if γ is a geodesic or a point.

Proof. The first two properties follows from Lemma 3.2.

Assume γi do not disappear, then the sequence length(γi) is bounded from below. The sequence γi is a sequence of curves of bounded lengths in a compact manifold. Then by Ascoli theorem there is a subsequence {γj} which converges uniformly to a curve δ with length(δ) ≤ lim_j→∞ length(γj). Let us show δ is a geodesic.

If δ is not a geodesic, then there is a small subarc δ′ ∈ δ such that δ′ is not a geodesic and δ′ lies in some disc Di with the halved radius.

Let ε > 0 be the value by which the straightening process decreases the length of δ. Since γj converges to δ we have length(Δ(γj)) < length(γj) – ε/2 for sufficiently large j. On the other hand, length(γj) < length(δ)+ε/2. Then length(Δ(γj)) < length(γj) that gives a contradiction. Thus, any convergent subsequence of γj must converge to a geodesic.

Assume that we can form a subsequence {γk} with all γk lying outside the neighbourhood U of the set of geodesics. The reasonings above show that there is a subsequence in {γk