Solid angles and Seifert hypersurfaces

Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $\Phi\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of $\Phi$ is an explicit Seifert hypersurface for $M$.


Introduction
It has been known since Seifert [15] that every link L " š S 1 Ă R 3 possesses a Seifert surface, that is, a compact oriented surface Σ Ă R 3 such that BΣ " L. Seifert gave an explicit algorithm for finding a Seifert surface from a link diagram.
In 1969 Erle [8] proved that any codimension two embedding M n Ă R n`2 of a closed oriented connected manifold M has a trivial normal bundle and admits a Seifert hypersurface Σ n`1 Ă R n`2 with BΣ " M Ă R n`2 ; see also [1]. The proof of the existence of the latter fact is not constructive, it relies on the Pontryagin-Thom construction applied to any smooth map f : cl.pR n`2 zMD 2 q Ñ S 1 representing the generator 1 P rR n`2 zMˆD 2 , S 1 s " H 1 pR n`2 zM q " H n pM q " Z with Σ " f´1ptq for a regular value t P S 1 . To the best of our knowledge, there is no known algorithm for constructing Seifert hypersurfaces in higher dimensions.
In this paper we present a construction of a Seifert hypersurface Σ Ă R n`2 as Σ " Φ´1ptq for a concrete smooth map Φ : R n`2 zM Ñ R{Z " S 1 obtained geometrically. The construction is based on three-dimensional physical intuition. Namely, suppose M Ă R 3 is a loop with constant electric current. The scalar magnetic potential r Φ of M at a point x R M is the solid angle subtended by M , that is, the signed area of a spherical surface bounded by the image of M under the radial projection, as seen from x; see [12,Chapter III] or [9,Section 8.3]. As the complement R 3 zM is not simply connected, the potential r Φ is defined only modulo a constant, which we normalize to be 1. The potential induces a well-defined function Φ : R 3 zM Ñ R{Z. This physical interpretation suggests that there exists an open neighborhood N of M such that Φ| N zM is a locally trivial fibration. In particular, a level set Φ´1ptq should be a (possibly disconnected) Seifert surface for M . In [5, Chapter VII] the second author proved that this is indeed the case, although the proof is rather involved. In fact, even for a circle the exact formula for Φ is complicated; it was given by Maxwell in [12,Chapter XIV] in terms of power series and also by Paxton in [13]. These formulae for Φ for the circle show that the analytic behavior of Φ near M is quite intricate, although we can show that Φ is a locally trivial fibration in U zM for some small neighborhood U of M ; see Section 5. 3.
The construction can be generalized to higher dimensions, even though the physical interpretation seems to be a little less clear. For any closed oriented submanifold M n Ă R n`2 , by the result of Erle [8] there exists a Seifert hypersurface. For any such hypersurface Σ and a point x R Σ we define r Φpxq to be the high-dimensional solid angle of M , that is, the signed area of the image of the radial projection of Σ to the pn`1q-sphere of radius 1 and center x. The value of r Φpxq depends on the choice of the hypersurface Σ, but it turns out that under a suitable normalization, Φpxq :" r Φpxq mod 1 is independent of the choice of the Seifert hypersurface. Moreover, there is a formula for Φpxq in terms of integrals of some concrete differential forms over M , so the existence of Σ is needed only to show that Φ is well-defined.
As long as t ‰ 0 P R{Z, the preimage Φ´1ptq is a bounded hypersurface in R n`2 zM . If, additionally, t is a non-critical value, Φ´1ptq is smooth. To prove that Φ´1ptq is actually a Seifert hypersurface for M , we need to study the local behavior of Φ near M . It turns out that the closure of Φ´1ptq is smooth up to boundary. We obtain the following result, which we can state as follows.
Theorem 1.1. Let M Ă R n`2 be a smooth codimension 2 embedding. Let Φ : R n`2 zM Ñ R{Z be the solid angle map (or the scalar magnetic potential map).
‚ On the set of points tx P R n`2 zM : p0, 0, . . . , 0, 1q R Sec x pM qu, the map Φ is given by where Sec x is the secant map Sec x pyq " y´x }y´x} and λ is an explicit function depending on the dimension n described in (2.18). ‚ Let t ‰ 0 be a non-critical value of Φ. Then Φ´1ptq is a smooth (open) hypersurface whose closure is Φ´1ptq Y M . The closure of Φ´1ptq is a possibly disconnected Seifert hypersurface for M , which is a topological submanifold of R n`2 , smooth up to boundary. ‚ For t ‰ 0, the preimage Φ´1ptq has finite pn`1q-dimensional volume.
The structure of the paper is the following. Section 1 defines rigorously the solid angle map Φ. Then a formula (2.18) for Φ in terms of an integral of an n-form over M is given. In Section 3 we prove that for t ‰ 0 the inverse images of Φ´1ptq Ă R n`1 are bounded. It is also proved that Φ extends to a smooth map S n`2 zM Ñ R{Z. In Section 4 we calculate explicitly Φ for a linear subspace. The resulting simple formula is used later in the proof of the local behavior of Φ for general M . In Section 5 we derive the Maxwell-Paxton formula for Φ if M is a circle. These explicit calculations allow us to study the local behavior of Φ in detail and give insight for the general case. In Sections 6 and 7 we study the local behavior of Φ for general M . This is the most technical part of the paper. We prove Theorem 1.1 in Section 8.
Acknowledgements. The present article is an extended and generalized (to arbitrary dimension) version of the second part of the University of Edinburgh thesis of the second author [5] written under supervision of Andrew Ranicki. A significant part of the paper was written during a Ph.D. internship of the second author in Warsaw. He expresses his gratitude to WCMS for the financial support.
The authors would also like to thank Jae Choon Cha and Brendan Owens for stimulating discussions. The first author was supported by the National Science Center grant 2016/22/E/ST1/00040. The second author was supported by Chiang Mai University.
2. Definition of the map Φ Consider a point x P R n`2 and define the map Sec x : R n`2 ztxu Ñ S n`1 given by The map Sec x can be defined geometrically as the radial projection from x onto the sphere: for a point y ‰ x take a half-line l xy stemming from x and passing through y. We define Sec x pyq as the unique point of intersection of l xy and S x , where S x is the unit sphere with center x.
Let ω n`1 be the pn`1q-form that is, the volume of the unit pn`1q-dimensional sphere; for instance, σ 1 " 2π, σ 2 " 4π. Let M be a closed oriented connected and smooth manifold in R n`2 with dim M " n. Definition 2.1. A compact oriented pn`1q-dimensional submanifold Σ of R n`2 such that BΣ " M is called a Seifert hypersurface for M .
Remark 2.2. For simplicity, throughout the paper we drop the assumption that Σ be connected. By Erle [8] any closed oriented submanifold M Ă R n`2 admits a Seifert hypersurface. Given such a surface Σ, consider x P R n`2 zΣ. The map Sec x restricts to a map from Σ to S n`1 , which we will still denote by Sec x . Definition 2.3. The solid angle of Σ viewed from x is defined as

In other words, r
Φpxq is a signed area of a spherical surface spanned by Sec x pM q, that is, the radial projection of M from the point x.
Remark 2.5. One should not confuse the solid angle with the cone angle studied extensively by many authors, like [3,4,6]. To begin with, the cone angle is unsigned and takes values in R ě0 , whereas the solid angle is an element in R{Z. This indicates that there exist fundamental differences between the two notions.
We have the following fact.
Lemma 2.6. The value r Φpxq mod 1 does not depend on the choice of Σ. In particular r Φ induces a well-defined function Proof. Take another surface Σ 1 . For simplicity assume that the interiors of Σ and Σ 1 intersect transversally. Let Ξ " Σ Y Σ 1 . It is an exercise in Mayer-Vietoris sequence to see that H n`1 pΞ, Zq -Z r`r 1`s´2 , where r and r 1 are numbers of connected components of Σ and Σ 1 , while s is the number of connected components of Σ X Σ 1 . The union Σ Y Σ 1 is a cycle and it defines an element in H n`1 pΞ, Zq.
On the other hand, ω n`1 is a generator of H n`1 pS n`1 ; Zq. The pull-back 1 σ n`1 Secx ω n`1 belongs to H n`1 pΞ; Zq. With this point of view the integral ż Ξ Secx ω n`1 can be regarded as the evaluation of an integral cohomology class on an integral cycle of Ξ, so it is an integer. Therefore, ż Σ Secx ω n`1´ż Secx ω n`1 P Z.
This shows that Φ " r Φ mod 1 is well defined. To determine the domain of Φ, notice that for any point x R M there exists Σ as above that misses x. This means that Φ is defined on the whole complement of M .
From the definition of Φ, we recover its first important property.
Proposition 2.7. The map Φ is smooth away from the complement of M Ă R n`2 .
Proof. Take a point y R M . There exists a smooth compact surface Σ such that BΣ " M and y R Σ. Then, a small neighborhood U of y is disjoint from Σ. Thus, the map Sec x depends smoothly on the parameter x. It follows that r Φ is smooth in U .
2.1. Φ via integrals over M . The fact that the definition of Φpxq involves a choice of a Seifert hypersurface Σ is quite embarrassing. In fact, it might be hard to find estimates for Φ because we have little control over Σ. We want to define Φ via integrals over M itself. The key tool will be the Stokes' formula. We use the fact that while the volume form ω n`1 on S n`1 itself is not exact, its restriction ω 1 to the punctured sphere S n`1 ztzu is.
We need the following result.
Proposition 2.8. Let x P R n`2 zM and let z P S n`1 be such that z R Sec x pM q. Then, there exists a Seifert hypersurface Σ for M such that Sec x pΣq misses z.
Remark 2.9. The result is non-trivial in the sense that one can construct a Seifert surface Σ even for an unknot in R 3 such that the restriction Sec x | Σ is onto. However, notice that Sec x | M is never onto S n`1 in general because dim M ă dim S n`1 .
Proof. Let H be the half line tx`tz, t ą 0u. In other words H " Sec´1 x pzq. Choose any Seifert hypersurface Σ. We might assume that H is transverse to Σ. The set of intersection points of H and Σ is bounded and discrete, hence finite. Let tw 1 , . . . , w m u " H X Σ and assume these points are ordered in such a way that on H the point w 1 appears first (with smallest value of t), then w 2 and so on. Choose the last point w m of this intersection and a small disk D Ă Σ with center w m . We can make D small enough so that for any w 1 P D the intersection tx`tw 1 , t ą 1u X Σ is empty. Set now T " tx`tw 1 , t ě 1, w 1 P Du and B v T " tx`tw 1 , t ě 1, w 1 P BDu.
Consider a sphere S " Spx, rq, where r is large. Set S 1 " SzpS XT q. Increasing r if necessary we may and will assume that S 1 is disjoint from Σ. The new Seifert hypersurface is defined as With this construction we have H X Σ 1 " tw 1 , . . . , w m´1 u. Repeating this construction finitely many times we obtain a Seifert hypersurface disjoint from H.
Let η z be an n-form on S n`1 ztzu such that dη z " ω n`1 and suppose Σ is a Seifert hypersurface for M such that z R Sec x pΣq. By Stokes' formula ż Σ Secx ω n`1 " Therefore we obtain the following formula for Φ: The necessity of making the map modulo 1 comes now from different choices of the point z P S n`1 zM .
We shall need an explicit formula for η z . For simplicity, we consider the case when z " p0, 0, . . . , 1q P S n`1 Ă R n`2 and define η :" η z ; the general case can be obtained by rotating the coordinate system. We start with the following proposition.
To obtain a formula for η, it remains to solve (2.12). Rewriting (2.12), we have The integrating factor of this ordinary differential equation is p1´u 2 n`2 q n`1 2 , so the general solution of (2.12) can be written as The requirement that the solution be smooth at u n`2 "´1 translates into the following formula (2.14) λpu n`2 q " p´1q n p1´u 2 n`2 q´n`1 2 The integral in (2.14) can be explicitly calculated. If n is odd, the result is a polynomial. If n is even, successive integration by parts eventually reduces the integral to ş ? 1´s 2 ds. For small values of n, the function λ is as follows.
We see that λ is smooth for u n`2 P r´1, 1q and has a pole at u n`2 " 1. We shall work mostly in regions, where u n`2 is bounded away from 1, so that λ and its derivatives will be bounded.

2.2.
The pull-back of the form η. We shall gather some formulae for evaluating the pull-back Secx η. This will allow us to estimate the derivative of Φ.
First notice that where we d y means that we take the exterior derivative with respect to the y variable. Consider the expression Secx du 1^¨¨¨^x du i^¨¨¨^d u n`1 .
To calculate the pull-back we replace du i by d y y i´xi }y´x} . Notice that if in the wedge product the term py i´xi qd y }y´x}´1 from (2.16) appears twice or more, this term will be zero. Therefore the pull-back takes the form Secx du 1^¨¨¨^x du i^¨¨¨^d u n`1 " 1 }y´x} n dy 1^¨¨¨^x dy i^¨¨¨d y n`11 }y´x} n´1 ÿ j‰i p´1q θpi,jq py j´xj qd y }y´x}´1^dy 1^¨¨¨^{ dy i , dy j^¨¨¨^d y n`1 , where θpi, jq is equal to j´1 if j ă i and j´2 if j ą i. Using the above expression together with d y }y´x}´1 "´1 }y´x} 3 ppy 1´x1 qdy 1`¨¨¨`p y n`1´xn`1 qdy n`1 q , we can calculate the pull-back of the form
Notice that we can change the order of the sums in the last term of the above expression to be ř n`1 j"1 ř i‰j . Since i`θpi, jq " j`θpj, iq˘1, the last two sums cancel out. Hence, we obtain (2.17) Secx ω n " 1 }y´x} n`1 n`1 ÿ i"1 p´1q i`1 py i´xi qdy 1^¨¨¨^x dy i^¨¨¨^d y n`1 .
In particular, using Definition 2.15 we get a proof of the first part of Theorem 1.1.
It is worth mentioning the formula for n " 1 and a general z (not necessarily p0, 0, 1q), which was given in [5,Theorem 5.3.7].
We conclude by remarking that if then analogous arguments as those that led to formula (2.17) imply that (2.20) Secx ω n`1 " 1 }y´x} n`2 n`2 ÿ i"1 p´1q i`1 py i´xi qdy 1^¨¨¨^x dy i^¨¨¨^d y n`2 .

2.3.
Estimates for derivatives of Secx η. The following results are direct consequences of the pull-back formula for η, (2.18). We record them for future use in Sections 3 and 6. Recall from Section 2.2 that η was defined as a form on S n`1 zp0, . . . , 0, 1q. The form η z for general z P S n`1 is obtained by rotation of the coordinate system. Lemma 2.21. For any m ě 0, there exists a constant C # m,n such that for each non-negative integers k 1 , . . . , k n`2 such that ř k i " m, the (higher) differential of the pull-back Secx η has the form and H j i are smooth functions satisfying |H j i | ď C # m,n }y´x}´p n`m´jq . Proof. If m " 0, the proof is a direct consequence of (2.18). The general case follows by an easy induction.
As a consequence of Lemma 2.21 we prove the following fact.
Lemma 2.23. For any D ă 1 and for any integer m ą 0, there is a constant C D n,m such that if z P S n`1 , y, x satisfy x y´x }y´x} , zy ă D and Secx η z is a sum of forms of type H i 1 ,...,in dy i 1^. . .d y in , where all the coefficients H i 1 ,...,in are bounded by C D n,m }y´x}´n´m. Proof. Apply a linear orthogonal map of R n`2 that takes z to p0, 0, . . . , 0, 1q. Let x 1 and y 1 be the images of x and y, respectively, under this map. We have }y 1´x1 } " }y´x} and the condition The condition x y´x }y´x} , zy ă D becomes y 1 n`2´x 1 n`2 }y´x} ă D. We will use (2.22). As D ă 1, on the interval r´1, Ds the function λ and its derivatives up to m-th inclusive are bounded above by some constant C D,m depending on D and m. The constant C D n,m can be chosen as In other words, all fibers of Φ except Φ´1p0q are bounded.
Proof. Choose a Seifert hypersurface Σ for M . We may assume that it is contained in a ball Bp0, rq for some r ą 0. As Σ is compact and smooth, there exists a constant C Σ such that if an pn`1q-form ω n`1 on R n`2 has all the coefficients bounded from above by T , then | ş Σ ω n`1 | ă C Σ T . Now take R " 0 and suppose x R Bp0, R`rq. Then the distance of x to any point y P Σ is at least R. Then Secx ω n`1 has all the coefficients bounded by R´n´1, see (2.20), and therefore | ş Corollary 3.2. The map Φ extends to a C n`1 smooth map from S n`2 zM to S 1 .

Sketch of proof.
Smoothness of Φ at infinity is equivalent to the smoothness of w Þ Ñ Φp w }w} 2 q at w " 0. The proof of Theorem 3.1 generalizes to show that for any m ą 0 there exists C m with a property that |D α Φpxq| ď C m¨} x}´n´1´| α| , and }D α w }w} 2 } ď C m }w}´| α|´1 whenever |α| ď m. Here α is a multi-index. Now by the di Bruno's formula for higher derivatives of the composite function, we infer that |D α Φp w }w} 2 q| ď C}w} n`2´|α| (the worst case occurs when Φ is differentiated only once, while w }w} 2 is differentiated |α| times). Hence, the limit at w Ñ 0 of all derivatives of w Þ Ñ Φp w }w} 2 q of order up to n`1 is zero.
We can also strengthen the argument of Theorem 3.1 to obtain a more detailed information about the behavior of Φ at a large scale.
where C Σ depends solely on Σ and not on R and r.

This implies that
Write also Now suppose }x} ą R and }y} ă r. Then´ξ 1´ξ3 has all the coefficients bounded by rR n`2 pR´rq n`2 }x}´n´2. Likewise notice thaťˇˇˇˇn`2 ÿ where we used Schwarz' inequality in the last estimate. Therefore´ξ 2´ξ3 has all the coefficients bounded by }y} }y´x} n`2 , and by assumptions on }x} and }y} we have that }y} }y´x} n`2 ď rR n`2 pR´rq n`2 }x}´n´2. We conclude that (3.4)ˇˇˇˇż Σ´ξ 1`p n`2qξ 2`p n`1qξ 3ˇď C Σ pn`2q rR n`2 pR´rq n`2 }x}´p n`2q .
As Φpxq " ş Σ ξ 3 , we obtain the statement. The statement of Theorem 3.3, in theory, can be used to obtain information about C Σ from the behavior of Φ at infinity. The left hand side of (3.4) is n`1qΦˇˇˇˇ, and does not depend on Σ. Therefore if we know Φ and its derivatives, we can find a lower bound for C Σ , which roughly tells, how complicated Σ might be. Unfortunately we do not know of any examples where this can be used effectively.

Φ for an n-dimensional linear surface
Let M Ă R n`2 be given by tw P R n`2 : w 1 " 0, w 2 " 0u, the set of points having the first two coordinates zero. We wish to calculate the map Φ for M . We encounter some technical problems. Firstly, as M is not compact, we have no reason to expect that Φ has bounded fibers and indeed, the statement of Theorem 3.1 does not hold. Secondly, there is a more serious problem. The map Φ will depend on the choice of the "Seifert hypersurface". We used quotation marks in the previous sentence because M , as it is not compact, does not admit a compact Seifert hypersurface. However, if we choose a Seifert hypersurface for M to be an pn`1q-dimensional half-space, it turns out that the derivative of Φ does not depend on the choice of the half-space. This feature and calculations for BΦ Bx j will be important in Section 6.
For any point x R Σ, the value of the map Φpxq is (up to a sign) the area of the image Sec x pΣq. This image can be calculated explicitly.
Choose a point y " py 1 , . . . , y n`2 q P S n`1 . The half-line from x R Σ through x`y is given by t Þ Ñ x`ty, t ě 0; see Figure 2. By definition, y P Sec x pΣq if and only if this half-line intersects Σ, that is, for some t 0 ą 0 we have (4.1) x 2`t0 y 2 " 0 and x 1`t0 y 1 ď 0.
Note that if x 2 " 0, then the half-line through x and any point in Σ will meet M , which results in an n-dimensional image Sec x pΣq in S n`1 . Suppose x 2 ‰ 0. The condition t 0 ą 0 together with (4.1) implies that the signs of x 2 and y 2 must be opposite. Plugging t 0 from the first equation of (4.1) into the second one, we obtain The calculation of Φ boils down to the study of the set of x 1 , x 2 satisfying (4.2). Write x 1 " r cos 2πβ and x 2 " r sin 2πβ. Multiply (4.2) by y 2 x 2 (which is negative) to obtain the inequality There are four cases depending on in which quadrant of the plane contains px 1 , x 2 q, see Figure 3. We next calculate the area of the image Sec x pΣq of each of those four cases. To do so, we first deal with the calculations and then discuss the choice of the sign. For the moment, we choose a sign for the area as ǫ P t´1,`1u; refer to Section 4.1 for the discussion of the sign convention.
Notice that the area of the two-dimensional circular sector in Figure 3 is (up to normalization) equal to the pn`1q-dimensional area of the image Sec x pΣq. This is because the defining equations are homogeneous, and other variables y 3 , . . . , y n`2 do not enter in the definition of the region. Case 1 x 1 ě 0 and x 2 ą 0. The region Sec x pΣq is given by y 2 ă 0 (because the sign of y 2 is opposite to the sign of x 2 ), y 1 ď y 2 { tan 2πβ and tan 2πβ P p0, 8q. The area of the sector corresponding to Case 1 is equal to πβ, hence Φpxq " ǫβ, where ǫ is a sign. Case 2 x 1 ď 0 and x 2 ą 0. The region Sec x pΣq is given by y 2 ă 0, y 1 ď y 2 { tan 2πβ, where tan 2πβ P p´8, 0q. The area of the sector is equal to πβ and so Φpxq " ǫβ. Case 3 x 1 ď 0 and x 2 ă 0. Then y 2 ą 0 and tan 2πβ P p0, 8q. The area of the sector is π´πβ, but now the hypersurface Σ is seen from the other side, hence the signed area is ǫpπβ´πq. After normalizing and taking modulo 1, we obtain that Φpxq " ǫβ. Case 4 x 1 ě 0, x 2 ă 0. Then y 2 ą 0 and tan 2πβ P p´8, 0q. As in Case 3, we deduce that the area is π´πβ and we obtain Φpxq " ǫβ.
Putting all the cases together, we see that Φpxq " ǫβ. Suppose we take another 'Seifert surface' for M , denoted Σ 1 , given by u 1 " 0, u 2 ď 0. Let Φ 1 be the map Φ defined relatively to Σ 1 . To calculate Φ 1 , we could repeat the above procedure, yet we present a quicker argument. A counterclockwise rotation A in the pu 1 , u 2 q-plane by angle π 2 fixes M and takes Σ to Σ 1 . In particular Φ 1 pxq " ΦpAxq. Hence Φ 1 pxq " ǫpβ´1 4 q We notice that Φ 1 ‰ Φ, but on the other hand Φ 1´Φ is a constant. This approach shows that if we take a linear hypersurface (a half-space) for the 'Seifert surface' of Φ, then it is well defined up to a constant, and so the derivatives are well defined.
4.1. The sign convention. Given that Φ is defined as an integral of a differential form, changing the orientation of M induces a reversal of the sign of Φ. We use the example of a linear surface to show how the sign is computed.
Choose an orientation of M in such a way that B Bu 3 , . . . , B Bu n`2 is a positive basis of T M . Stokes' theorem is applicable if Σ is oriented by the rule "normal outwards first", see [16,Chapter 5], so that B Bu 1 , B Bu 3 , . . . , B Bu n`2 is an oriented basis of T Σ.

(4.3)
Notice that on the left hand side we have an integral of a differential form, whereas on the right hand side the integral is with respect to the pn`1qdimensional Lebesgue measure on a subset of R n`1 .

Φ for a circle
We now use the formula for Φ via the integrals of the pull-back of η, see (2.18), to give an explicit formula for Φ in the case when M is a circle. The output is given in terms of elliptic integrals. Detailed calculations can be found e.g. in [5], therefore we omit some tedious computations. We focus on the analysis of the behavior of Φ near the circle.

Elliptic Integrals.
For the reader's convenience we give a quick review of elliptic integrals and their properties. We shall use these definitions in future calculations. This section is based on [2].
1´k 2 sin 2 t is called an elliptic integral of the first kind. If ϕ " π{2, it is called a complete elliptic integral of the first kind, denoted by Kpkq :" Fpπ{2, kq.
(2) The integral is called an elliptic integral of the second kind. If ϕ " π{2, it is called a complete elliptic integral of the second kind, denoted by Epkq :" Epπ{2, kq.
is called an elliptic integral of the third kind. If ϕ " π{2, it is called a complete elliptic integral of the third kind, denoted by Πpα 2 , kq :" Πpπ{2, α 2 , kq. (4) Heuman's Lambda function Λ 0 pβ, kq can be defined by the formula Although Kpkq blows up at k " 1, we know how fast it goes to infinity as k approaches 1 from below.
In particular The differentials of Kpkq and Epkq are calculated e.g. in [2, page 282].
The derivative of the Heuman's Lambda function Λ 0 pβ, kq is given by the following formula, see [2, formulae 710.11 and 730.04].
Computation of Φ for the circle. In this section, we follow closely [5, Sections 6.2 and 6.3]. The circle U has the parametrization γ : r´π, πs Ñ R 3 given by γptq " pcos t, sin t, 0q. Suppose x P R 3 is such that x R tu 2 1`u 2 2 " 1, u 3 ď 0u. Then Sec x pU q does not contain p0, 0, 1q and (2.19) implies: where Q " 1`}x} 2´2 x 1 cos t´2x 2 sin t. Write x 1 " r cos θ, x 2 " r sin θ for r ě 0. Substituting this into (5.9), we observe that Φ does not depend on θ, hence we can write Φ " Φpr, x 3 q, that is, We have some special cases where we can compute the integral explicitly. If x 3 " 0, we use the identity cos t " 1´tan 2 pt{2q 1`tan 2 pt{2q and deal with improper integrals; there are two situations: ‚ r ă 1: we have Φpr, 0q " 1 4π "´t 2´a rctanˆ1`r 1´r tan t 2˙ π π "´1 2 ; x Figure 4. A Seifert surface for the circle with the property that its image under Sec x is a hemisphere.
‚ r ą 1: we have This agrees with the geometric interpretation. If we choose the disk D " tr ď 1, x 3 " 0u as a Seifert surface for U , then for x " pr cos θ, r sin θ, 0q with r ą 1, the image Sec x pDq is one-dimensional, so Φpxq " 0. Conversely, for x " pr cos θ, r sin θ, 0q with r ă 1 we choose a Seifert surface Σ to be the disk D with a smaller disk centered at x replaced by a hemisphere with center at x. In this way, the image Sec x pΣq is a hemisphere; see Figure 4.
Remark 5.11. The inverse image Φ´1p0q contains (and actually it is equal) to the set tx 2 1`x 2 2 ą 1, x 3 " 0u. This shows that the assumption that t ‰ 0 in Theorem 3.1 is necessary.

Then
Cprq :" Using (5.12) and cos 2θ " 1´2 sin 2 θ we write We may write the formula in terms of Heuman's Lambda function Λ 0 using the formula relating Π and Λ 0 ; see [2, page 228] or [13]. After straightforward but tedious calculations, we obtain the following explicit formula.
‚ If x 2 1`x 2 2 " 1 and x 3 ă 0, then Another approach in computing the solid angle for an unknot was given by F. Paxton, see [13]. He showed that the solid angle subtended at a point P with height L from the unknot and with distance r 0 from the axis of the unknot is equal to where R max " a p1`r 0 q 2`L2 , ξ " arctan L |1´r 0 | and k is given by (5.14). It can be shown that the Paxton formula agrees with the result of Proposition 5.13.
Finally we remark that the computation of the solid angle of the unknot was already studied by Maxwell. He gave the formulae in terms of infinite series, see [12, Chapter XIV].

5.3.
Behavior of Φ near U . We shall now investigate the behavior of Φ and its partial derivatives near U .
Remark 5.16. The sign of the limit is´λ and not`λ. It is not hard to see that the orientation convention for the circle, that is, such that t Þ Ñ pcos t, sin t, 0q is an oriented parametrization of U is opposite to the convention adopted in Section 4.1.
Next we compute the derivatives of Φ near U . It is clear that the map Φ for the circle is invariant with respect to the rotational symmetry around the z-axis. Hence, if α is the longitudinal coordinate near U , then B Bα Φ " 0. The two coordinates we have to deal with are the meridional and radial coordinates λ and ε. The first result is the following. Φp1`ε cos 2πλ, 0, ε sin 2πλq " 1 4π 2 sin 2πλpKpkq´Epkqq p1`ε cos 2πλq ? 4`4ε cos 2πλ`ε 2 .
Remark 5.19. This extension of Φ through tε " 0u will be generalized in the Continuous Extension Lemma 7.10.
We have estimated the derivatives of Φ with respect to ε and λ. We can now give the following corollary, which is a straightforward consequence of (5.18).  Bx j Φpx 1 , x 2 , x 3 q, j " 1, 2, 3 have at most a logarithmic pole at points px 1 , x 2 , x 3 q close to U . More precisely, there exists a constant C circ such thaťˇˇˇB We remark that from (2.18), we get much weaker estimates on the derivative. We do not know if these weaker estimates can be improved for general manifolds M . To conclude, we show level sets of the function pr, x 3 q Þ Ñ Φpr, x 3 q for the circle in Figure 5. Notice that in the Figure the half-lines stemming from point p1, 0q (and not parallel to the x 3 " 0 line) intersect infinitely many level sets near the point p1, 0q. This suggests that the radial derivative B Bε Φp1ὲ cos 2πλ, ε sin 2πλq is unbounded as ε Ñ 0`. We proved this fact rigorously in Proposition 5.17.

Derivatives of Φ near M
We begin by recalling a well-known fact in differential geometry. Proposition 6.1. Let X Ă R n`2 be a k-dimensional, smooth, compact submanifold with smooth boundary. Then, there exists a constant C X such that for every x P R n`2 and for any r ą 0 we have vol k pX X Bpx, rqq ď C X r k .
Moreover, increasing C X if necessary, we may assume that if ω is a k-form on R n`2 whose coefficients are bounded by T , then | ş XXBpx,rq ω| ď C X T r k . The result is well-known to the experts, therefore we present only a sketchy proof.
Sketch of proof. Let δ k be the volume of unit k-dimensional ball. By smoothness of X we infer that lim rÑ0 vol k pX X Bpx, rqq r k is 0, 1 2 δ k or δ k depending on whether x R X, x P BX or x P XzBX. Using Vitali's theorem, one shows that there exists r 0 independent of x such that vol k pX X Bpx, rqq r k ď 2δ k for all r ă r 0 . We take C X to be the maximum of 2δ k and vol k pXq{r k 0 . The second part is standard and left to the reader. 6.1. The Separation Lemma. The form η z used in Section 2.2 has a pole at z P S n`1 . In the applications for given x P R n`2 zM we choose a point z such that z R Sec x pM q. Such a point exists, see Remark 2.9 above. However, in order to obtain a meaningful bound for Secx η z , we need to know that z is separated from Sec x pM q, in the sense that there exists a constant D such that xy, zy ď D for any y P Sec x pM q. In this section we show that the constant D ă 1 can be chosen independently of x. Lemma 6.2 (Separation Lemma). There exist ε 0 ą 0 and D ă 1 such that the set N 0 " pM`Bp0, ε 0 qqzM of points at distance less than ε 0 from M (and not lying in M ) can be covered by a finite number of open sets U 1 , . . . , U l with the following property: for each i there exists a point z i P S n`1 such that for any x P U i we have Sec x pM q Ă tu P S n`1 : xu, z i y ď Du. Remark 6.3. In general it is impossible for a given point x P M to find an element z P S n`1 and a neighborhood U Ă R n`2 of x, such that for every x 1 P U we have z R Sec x 1 pM q. In fact, the opposite holds. For any z P S n`1 the sequence x n " x´z n has the property that z P Sec xn pM q and x n Ñ x. This is the main reason why the proof of an apparently obvious lemma is not trivial.
Put differently, the subtlety of the proof of Lemma 6.2 lies in the fact that the image Sec x pM q can be defined for x P M as a closure of Sec x pM ztxuq, but we cannot argue that Sec x pM q depends continuously on x, if x P M .
Proof of Lemma 6.2. Take a point x P M . Let V be the affine subspace tangent to M at x, that is V " x`T x M . The image Sec x pV ztxuq is the intersection S x :" T x M X S n`1 .

Lemma 6.4.
For any open subset U Ă S n`1 containing S x , there exists r ą 0 such that the Sec x pM X Bpx, rqztxuq is contained in U .
Proof. Suppose the contrary, that is, for any n there exists a point y n P M such that }x´y n } ă 1 n and Sec x py n q R U . In particular y n Ñ x. As M is a smooth submanifold of R n`2 , the tangent space T x M is the linear space of limits of secant lines through x. This means that if y n Ñ x and y n P M , then, up to passing to a subsequence, Sec x py n q converges to a point in S x . But then, starting with some n 0 ą 0, we must have Sec x py n q P U for all n ą n 0 . Contradiction.
Choose now a neighborhood U of S x and r from Lemma 6.4. As S x is invariant with respect to the symmetry y Þ Ñ´y, we may and will assume that U also is. We assume that U is small neighborhood of S x , but in fact we will only need that U is not dense in S n`1 . We will need the following technical result.
Proof of Lemma 6.5. We act by contradiction. Assume the statement of the lemma does not hold. That is, there is a sequence x n converging to x such that both v and´v belong to Sec xn pM X Bpx, rqq. This means that for any Figure 6. Proof of the Lemma 6.5. To the left: a sequence of lines l x 1 , l x 2 converges to a line that is tangent to M at x, so that v P T x M . To the right: a sequence of lines l x 1 , l x 2 converges to a line that passes through x and intersects M at some point. Then v P Sec x pM q.
n the line l xn :" tx n`t v, t P Ru intersects M X Bpx, rq in at least one point for t ą 0 and at least one point for t ă 0. For each n, choose a point yǹ in M X Bpx, rq X l xn X tt ą 0u and a point yń in M X Bpx, rq X l xn X tt ă 0u. In particular Sec xn pyǹ q " v and Sec xn pyń q "´v. By taking subsequences of tyǹ u and tyń u we can assume that yǹ Ñ y`and yń Ñ y´for some y`, y´P M X Bpx, rq.
If y`‰ x, then the line l x " tx`tvu passes through y`, but this means that v P Sec x pM X Bpx, rqq, but the assumption was that v R U , so we obtain a contradiction. So y`" x. Analogously we prove that y´" x.
Finally suppose y`" y´" x. The line l x " tx`tvu is the limit of secant lines passing through yǹ and yń , therefore l x is tangent to M at x. But then v P S x Ă U . Contradiction.
We extend the argument of Lemma 6.5 in the following way. Proof. The proof is a modification of the proof of Lemma 6.5. We leave the details for the reader.
Resuming the proof of Lemma 6.2 define the sets B loc`a nd B loc´b y B loc˘" tx 1 P B loc : for every v 1 P V loc we have˘v 1 R Sec x 1 pM X Bpx, rqqu.
Then clearly B loc`Y B loc´" B loc . Proof. The lemma follows from the fact that M X Bpx, rq is closed and Sec x 1 is continuous with respect to x 1 as long as x 1 R M . The next step in the proof of the Separation Lemma 6.2 is the following. Proof. Let U and r ą 0 be as in the statement of Lemma 6.4. The set Sec x pM zBpx, rqq is the image of the compact manifold M zBpx, rq of dimension n under a smooth map, hence its interior is empty. Therefore there exists a point v P S n`1 such that neither v nor´v is in the image Sec x pM zBpx, rqq and also neither v nor´v is in U . By the continuity of Sec x , there exist a small ball B gl Ă Bpx, rq with center x and a small ball V gl Ă S n`1 containing v such that if v 1 P V gl and x 1 P B gl , then neither v 1 nor´v 1 belongs to Sec x 1 pM zBpx, rqq. Let V loc and B loc be from Lemma 6.6. Define V x " V loc XV gl and B x " B loc X B gl . Then for any We define Bx as intersections of B loc˘w ith B x , where B loc˘a re as in Lemma 6.7 above.
We resume the proof of the Separation Lemma 6.2. Define As V x is an open set containing v x , we have δ x ă 1. This means that if x 1 P B x and y P Sec x pM q, then either xy, v x y ď δ x or xy,´v x y ď δ x .
Cover now M by open sets B x for x P M . As M is compact, there exists a finite set x 1 , . . . , x n such that M Ă B x 1 Y¨¨¨Y B xn . The compactness of M implies also that there exists ε 0 ą 0 such that the set M`Bp0, ε 0 q, that is, the set of points at distance less than ε 0 from M , is contained in B x 1 Y¨¨¨Y B xn . Define D " maxpδ x 1 , . . . , δ xn q and let N 0 " pM`Bp0, ε 0 qqzM . For i " 1, . . . , n define v i " v x i and Bȋ " Bx i XN 0 . Then Bȋ cover N 0 and for any i, if x 1 P Bȋ and y P Sec x 1 pM q, then xy,˘v i y ď D as desired.
For points x that are at distance greater than ε 0 from M , the statement of the Separation Lemma 6.2 holds as well. Theorem 6.9 (Separation Theorem). There exists a constant D ă 1 such that for any x P R n`2 zM there exists an open neighborhood U x Ă R n`2 of x and a point z P S n`2 , such that for any x 1 P U x and y P Sec x 1 pM q we have xy, zy ă D.
Proof. Denote by D close the constant D from the Separation Lemma 6.2. The constant D close works for points at distance less than ε 0 from M .
We work with points far from M . Choose R ą 0 large enough so that M Ă Bp0, Rq. For any x R Bp0, Rq we can take the point x }x} for z and then if y P Sec x pM q, then xy, zy ď 0. By the continuity of x Þ Ñ Sec x we may choose a neighborhood W x of x such that if x 1 P W x and y n P Sec x 1 pM q, then xy, zy is bounded from above by a small positive number, say 1 10 . This takes care of the exterior of the ball Bp0, Rq. We define D f ar " 1 10 . The constant D f ar works for points outside of the ball Bp0, Rq.
Let P " tx P Bp0, Rq : distpx, M q ě ε 0 u. For any point x P P , Sec x pM q is the image of an n-dimensional compact manifold under a smooth map, so it is a boundary closed subset of S n`1 . Thus there exist a point z x P S n`1 and a neighborhood of U x of z x such that U x X Sec x pM q " H. Shrinking U x if necessary we may guarantee that there exists a neighborhood W x Ă R n`2 of x such that if x 1 P W x and y P Sec x 1 pM q, then y R U x . We define again δ x " sup yPS n`2 zVx xz x , yy ă 1.
The sets W x cover P and we take a finite subcover W x 1 , . . . , W x M . We define D mid as the maximum of δ x 1 , . . . , δ x M . The constant D mid works for points at distance between ε 0 and inside of Bp0, Rq.
It is enough to take D " maxpD close , D mid , D f ar q.
From now on we assume that D ă 1 is fixed.
6.2. The Drilled Ball Lemma. We begin to bound the value of B Bx i Φpxq. To this end we will differentiate the coefficients of Secx η z . The point z will always be chosen in such a way that xSec x 1 pyq, zy ă D for all y P M and for all x 1 sufficiently close to x.
Then, for any i " 1, . . . , n`2ˇˇˇˇB where γ " nβ´pn`1qα and C drill " C M C D n,1 is independent of α, β and x. Proof. By Lemma 2.23 and the Separation Lemma 6.2, the derivative of the pull-back B Bx i Secx η is an n-form whose coefficients are bounded from above by by C D n,1 }y´x} n`1 . If y P M αβε , then }y´x} ě ε α . The form B Bx i Secx η is integrated over M αβε . We use Proposition 6.1 twice. First to conclude that the volume of M αβε is bounded from above by C M ε nβ , second to conclude that the integral is bounded by C D n,1 C M ε nβ´pn`1qα . The next result shows that if M is locally parametrized by some Ψ then if we take a first order approximation, the contribution to the derivative of Φpxq from the local piece does not change much. We need to set up some assumptions.
Choose ε ą 0 and α P p 1 2 , 1q. For a fixed point x at distance ε from M we set M αε " M X Bpx, ε α q. We assume that ε, α are such that M αε can parametrized by Ψ : B 1 Ñ M αε , where B 1 is some bounded open subset in R n and Ψp0q is the point on M αε that is nearest to x. We also assume that B 1 is a star-shaped, that is, if w P B 1 , then tw also in B 1 for t P r0, 1s. For simplicity of the formulae we may transform B 1 in such a way that Choose σ ą 0 in such a way that B 1 is a subset of an n-dimensional ball Bp0, σq and B 1 is not a subset of Bp0, σ{2q. Let Ψ 1 be the first order approximation of Ψ, that is Ψ 1 pwq " Ψp0q`DΨp0qw. Let M 1 be the image of B 1 under Ψ 1 .
V M x Ψ 1 pwq Ψpwq y 1 ε Figure 7. Notation of the proof of the Approximation Lemma 6.12.
Write C 1 and C 2 for the supremum of the first and second derivative of Ψ on B 1 . Lemma 6.12 (Approximation Lemma). Suppose There exists a constant C app depending on Ψ such thaťˇˇˇB where δ " αpn`3q´pn`2q.
Proof. As our first step we relate σ with ε and α.
This means that Ψpt 0 w }w} q cannot possibly belong to Bpx, ε α q, hence it is not in the image ΨpB 1 q " M X Bpx, ε α q. This shows that t 0 w }w} cannot belong to B 1 . As w was an arbitrary point in B 1 , this implies that no element in B 1 can have norm 2ε α . As B 1 is connected, this implies that B 1 must be contained in Bp0, 2ε α q. By the definition of σ we immediately recover that σ ď 4ε α .
We have }w} ă 4ε α and 2α ą 1. Using the assumption that ε ă p32C 2 q´1 {p2α´1q we infer that C 2 p4ε α q 2 ď 1 2 ε. so that }Ψ 1 pwq´Ψpwq} ď 1 2 ε. Therefore, as distpx, M αε q " ε, we infer that for each point y 1 in the interval connecting y 0 and y 1 we have }x´y 1 } ě 1 2 ε. Write now for i " 1, . . . , n`1: By the mean value theorem, for any i, j there exists a point y 1 in the interval connecting y 0 and y 1 such that where B v is the directional derivative in the direction of the vector y 0´y1 }y 0´y1 } . Now }x´y 1 } ě 1 2 ε, hence by Lemma 2.21: From (6.15) we deduce that for some constant C depending on C 2 . Set G ij pwq and H ij pwq to be defined by The values of G ij and H ij are bounded by a constant depending on C 1 . Moreover the expression DΨpwq´DΨ 1 pwq has all entries bounded from above by }w} times a constant, hence an exercise in linear algebra shows that (6.20) |G ij pwq´H ij pwq| ď C G }w} for some constant C G depending on C 1 . Now write Ψ˚F ij px, yqdy 1^. . . { dy i , dy j . . .^dy n`2´Ψ1 F ij px, yqdy 1^. . . { dy i , dy j . . .^dy n`2 " pF ij px, ΨpwqqG ij pwq´F ij px, Ψ 1 qpwqH ij pwqq dw 1^. . .^dw n .
We estimate using (6.19): Combining this with (6.19) we infer that where the factor ε´n´1 comes from the estimate of F ij px, Ψpwqq and the constant C depends on previous constants, that is, C depends on C 1 , C 2 .
We use now (6.21) together with (6.16) and the definitions of G ij , H ij . After straightforward calculations we obtain for some constant C:ˇˇˇż The last expression is bounded by C app pε pn`3qα´pn`2q`εpn`2qα´pn`1q q, where C app is a new constant. As α ă 1 and ε ! 1, the term ε pn`3qα´pn`2q is dominating.
In the following result we show that the constants in the Approximation Lemma 6.12 can be made universal, that is, depending only on M and α and not on x and ε. Proposition 6.22. For any α P p 1 2 , 1q there exist constants C α and ε 1 ą 0 such that for any x R M such that distpx, M q ă ε 1 we havěˇˇˇB Here Φ V is a map Φ defined relatively to the plane V that is tangent to M at a point y such that distpx, M q " }x´y}.
Proof. Cover M by a finite number of subsets U i such that each of these subsets can be parametrized by a map Ψ i : By the compactness of M , there exists ε 1 ą 0 such that if U Ă M has diameter less than ε 1 , then U is contained in one of the U i . Shrinking ε 1 if necessary we may and will assume that if distpx, M q ă ε 1 , then there is a unique point y P M such that distpx, M q " }x´y}. Set C 1 and C 2 to be the upper bound on the first and the second derivatives of all of the Ψ i . The derivative DΨ i pwq is injective for all w P V i . We assume that C 0 ą 0 is such that }DΨ i pwqv} ě C 0 }v} for all v P R n , i " 1, . . . , n and w P V i .
Choose a point x at distance ε ą 0 to M such that 2ε α ă ε 1 and ε ă ε 1 . Let M αε " Bpx, ε α q X M . As this set has diameter less than ε 1 , we infer that M αε Ă U i for some i. Let B " Ψ´1 i pM αε q Ă V i . Let y P M be the unique point realizing distpx, M q " }x´y}. We translate the set B in such a way that Ψ i p0q " y. Next we rotate the coordinate system in R n`2 in such a way that the image of DΨp0q has a block structure A ' p 0 0 0 0 q for some invertible matrix A. We know that A´1 is a matrix with coefficients bounded by an universal constant depending on c 1 and C 1 . Define now B 0 " A´1pBq and Ψ x " Ψ i˝A . Then Ψ has first and second derivatives bounded by a constant depending on C 1 , C 2 and c 1 . Denote these constants by C 1 pxq, C 2 pxq. Let also be C 0 pxq ą 0 be such that if w, w 1 P B, then }Ψ x pwq´Ψ x pw 1 q} ě C 0 pxq}w´w 1 }. Such constant exists because DΨpwq is injective and we use the mean value theorem. Moreover C 0 pxq is bounded below by a constant depending on C 1 , C 2 and C 0 .
It remains to ensure that the following two conditions are satisfied. First, the set B " Ψ´1 x pM αε q has to be star-shaped, second the inequality (6.13) is satisfied. The second condition is obviously guaranteed by taking ε 1 sufficiently small. We claim that the first condition can also be guaranteed by taking ε 1 . To see this we first notice that if ε 1 is sufficiently small, then M αε is connected for all ε ă ε 1 . Next, we take a closer look at the definition of B Ă V i . Namely we can think of B as the set of points w P V i satisfying the inequality Rpwq ď ε α , where Rpwq " }Ψ x pwq´x} 2 " xΨ x pwq´x, Ψ x pwq´xy.
Generalizing this for w P B and v P R n we have D 2 Rpwqpv, vq " }DRpwqv} 2`2 xD 2 Ψ x pwqpv, vq, Ψ x pwq´xy. Now }D 2 Ψ x pwqpvq} ď C 2 pxq}v} 2 and by the mean value theorem also }DRpwqD Rp0q} ď C 2 pxq}w}. Suppose }x´Ψ x pwq} ď ε α . Then }Ψ x pwq´Ψ x p0q} ď 2ε α and so }w} ď 2C 0 pxqε α . Hence This shows that R is a convex function if ε is sufficiently small. Hence B is a convex subset, in particular, it is also star-shaped. Therefore all the assumptions of the Approximation Lemma 6.12 are satisfied, the statement follows.
6.3. Approximation Theorem. Combining the Drilled Ball Lemma 6.10 and Proposition 6.22, we obtain a result which is the main technical estimate. Theorem 6.23 (Approximation Theorem). For any θ P p n`2 n`4 , 1q there exists a constant C θ such that if x is at distance ε ą 0 to M and ε ă ε 1 , y 0 P M is the point realizing the minimum of distpx, M q and V is the tangent space to M passing through y 0 , then for any j " 1, . . . , n`2ˇˇˇB Here, Φ V is the map Φ defined relatively to the hyperplane V .
Remark 6.24. In Section 4, we have shown that Φ V is not well defined, but if we restrict to 'Seifert hypersurfaces' for V which are half-spaces (and that is what we in fact do), then Φ V is defined up to an overall constant. In particular, its derivatives do not depend on the choice of the half-space.
Recall that α k 0 " 0. Equation (6.26) does not cover the part of M outside of Bpx, 1q. However, on M zBpx, 1q, the form ξ is easily seen to have coefficients bounded above by a constant independent of ε and x, hence for some constant C ext depending on M but not on x and ε. It remains to show We cannot use the Drilled Ball Lemma 6.10 directly, because V is unbounded. However, we will use similar ideas as in the proof of the Drilled Ball Lemma 6.10. The form ξ is an n-form whose coefficients on V are bounded by C D 1 n,1 }y´x}´p n`1q , where D 1 is such that π´1 x pV q Ă tu n`2 ă D 1 u. Its restriction to V is equal to some function F x pyq times the volume form on V , where |F x pyq| ď C V C D 1 n,1 }yx }´p n`1q (it is easy to see that as V is a half-plane, C V exists). Therefore we need to bound The method is standard. Introduce radial coordinates on V centered at y 0 and notice that }y´x} ď 2}y 0´y } as long as y P V zBpx, ε α 0 q. Perform first the integral (6.29) over radial coordinates obtaining the integral over the radius only, that is ż where σ n´1 is the volume of a unit sphere of dimension n´1. This proves (6.28) with C f lat " 2 n´1 σ n´1 C V C D 1 n,1 . Combining (6.25), (6.26),(6.27) and (6.28) we obtain the desired statement.
6.4. The main estimate for the derivative. This section extends the intuitions given in Section 5.3.
It is a result of Erle [8] that the normal bundle of M Ă R n`2 is trivial, but there might be many different trivializations, one class for each element of rM, SOp2qs " rM, S 1 s " H 1 pM q. Choose a pair of two normal vectors v 1 , v 2 on M such that at each point y P M , v 1 pyq and v 2 pyq form an oriented orthonormal basis of the normal space N y M . Choose ε 0 ă ε 1 and let N be the tubular neighborhood of M of radius ε 0 . By taking ε 0 ą 0 sufficiently small we may and will assume that each y 1 P N can be uniquely written as y`t 1 v 1`t2 v 2 for y P M and t 1 , t 2 P R.
Let y P M . Choose a local coordinate system w 1 , . . . , w n in a neighborhood of y such thatˇˇB w j Bx iˇď C w for some constant C w . The local coordinate system w 1 , . . . , w n on M induces a local coordinate system on N given by w 1 , . . . , w n , t 1 , t 2 . Let r, φ be such that t 1 " r cosp2πφq, t 2 " r sinp2πφq.
where ǫ P t˘1u depending on the orientation of M .
Proof. Choose a point x " pw 1 , . . . , w n , t 1 , t 2 q. Let y 0 " pw 1 , . . . , w n , 0, 0q be the point minimizing the distance from x to M . We will use the Approximation Theorem 6.23. So let V be the n-dimensional plane tangent to M at y 0 . The map Φ V is the map Φ relative to V . By the explicit calculations in Section 4 we infer that BΦ V Bw j pxq " BΦ V Br pxq " 0 for j " 1, . . . , n and BΦ V Bφ " ǫ. Now BΦ Bw j differs from the derivatives of Φ V by at most C θ r´θ by the Approximation Theorem 6.23.
On the other hand, by chain rule BΦ Bφ "´r sin φ BΦ Bt 1`r cos φ BΦ Bt 2 . Applying Theorem 6.23 we infer thatˇˇB Φ Bφ pxq´B Φ V Bφ pxqˇˇď C θ r 1´θ . The same argument shows thatˇˇB Φ Br pxqˇˇď C θ r´θ. Notice that the derivatives with respect to r and θ do not depend on C w : this is so because the length of the framing vectors v 1 and v 2 is 1.
The composition Φ˝Π : X Ñ S 1 will still be denoted by Φ. We are going to show that this map is a locally trivial fibration whose fibers have bounded pn`1q-dimensional volume.
Proof. Choose r P p0, ε 0 s and x P M . Consider the map Φ r,x : S 1 Ñ S 1 given by Φ r,x " Φ| truˆS 1ˆt xu . The derivative of Φ r,x is equal to BΦ Bφ , by (7.2) it belongs either to the interval p´3 2 ,´1 2 q or to p 1 2 , 3 2 q depending on the ǫ. It follows that Φ r,x is a diffeomorphism. In particular, given r P p0, ε 0 s and x P M , for any t P S 1 , there exists a unique point Θ t pr, xq such that Φpr, Θ t pr, xq, xq " t. In this way we get a bijection Θ t pr, xq : p0, ε 0 sˆM Ñ Φ´1ptq.
Again by (7.2) | BΦ Bφ | ą 1 2 ą 0, so by the implicit function theorem we infer that Θ t is in fact a smooth map. Then Θ t is a smooth parametrization of the fiber of Φ. It remains to show that Φ is locally trivial.
To this end we choose a point t P S 1 and let U Ă S 1 be a neighborhood of t. Define the map r Θ : p0, ε 0 sˆUˆM Ñ Φ´1pU q by the formula r Θpr, t, xq " pr, Θ t pr, xq, xq.
Clearly r Θ is a bijection. As Θ t depends smoothly on the parameter t, we infer that r Θ is a smooth map and the map Φ´1pU q Ñ p0, ε 0 sˆUˆM given by pr, φ, xq Þ Ñ pr, Φpr, φ, xq, xq is its inverse. Therefore r Θ is a local trivialization.
The same argument as in the proof of Fibration Lemma 7.3 shows that Φ M is a locally trivial fibration with fiber p0, ε 0 s. For given pt, xq P S 1ˆM the map r Þ Ñ pr, Θ t pr, xq, xq parametrizes the fiber over pt, xq.
As a consequence of Fibration Lemma 7.3 we show that Φ : R n`2 zM Ñ S 1 does not have too many critical points. This is a consequence of Sard's theorem and the control of Φ near M provided by Lemma 7.3.
Remark 7.9. Bounded Volume Lemma 7.6 shows that the volume of the fibers Φ´1ptq is bounded near M by a constant that does not depend on t. This does not generalize to bounding a global volume of Φ´1ptq: one can show that the volume of Φ´1p0q is infinite using Corollary 3.2.
7.2. Extension to of Φ through r " 0. We pass to study the closure of the fibers of map Φ´1ptq X X. This is done by extending the map Φ. Set X " r0, ε 0 sˆS 1ˆM .
The manifold X can be regarded as an analytic blow-up of the neighborhood N zM . Proof. Let f r : S 1ˆM Ñ S 1 be given by f r pφ, xq " Φpr, φ, xq. We shall show that as r Ñ 0 the functions f r converge uniformly. The limit, f 0 , will be the desired extension.
As r Þ Ñ r 1´θ is a uniformly continuous function taking value 0 at 0, we obtain that f r uniformly converge to some limit, which we call f 0 . This amounts to saying that Φ extends to a continuous function on X. We can also calculate the function Φ for r " 0.
Theorem 7.14. For any t P S 1 , the maps Θ t : p0, ε 0 sˆM Ñ Φ´1ptq extend to a continuous map Θ t : r0, ε 0 sˆM Ñ Φ´1ptq Ă X. The map Θ t is injective. Figure 8. The picture indicates the necessity of proving the surjectivity of Θ t , the map Θ t is not onto. In Theorem 7.15 we show that the situation as on the picture cannot happen.
Proof. By (7.8) BΘt Br is bounded by 2C θ r´θ, so we the same argument as in the proof of Lemma 7.10 shows that Θ t pr, xq converges as r Ñ 0 uniformly with respect to x. Therefore Θ t is well defined.
We next prove the surjectivity of Θ t . Before we state the proof, we indicate a possible problem in Figure 8. Proof. By the Fibration Lemma 7.3 the map Θ t is onto Φ´1ptq Ă X. Hence it is enough to show that Θ t | t0uˆM is onto Φ´1ptq X pXzXq.
Observe that by Proposition 7.12 the intersection Φ´1ptq X t0uˆS 1ˆt xu consist of one point for any x P M and t.
On the other hand, since Φ´1ptq X pt0uˆS 1ˆt xuq is a single point, this point has to be equal to Θ t p0, xq. Therefore, Θ t p0, xq is onto Φ´1ptq X tr " 0u so Θ t pr, xq is onto Φ´1ptq.
As a corollary we will show the following result. Proof. We show that for any closed interval I Ă S 1 , the preimage X I :" Φ´1pIq is homeomorphic to the product Y I : " Iˆr0, ε 0 sˆM by a homeomorphism that preserves the fibers. Consider the map Θ I : Y I Ñ X I given by Θ I pt, xq " Θ t pxq for x P r0, ε 0 sˆM . By Theorems 7.14 and 7.15, this map is a bijection. Moreover, its inverse is Φ, which is continuous by the Continuous Extension Lemma 7.10. A continuous bijection between compact sets is a homeomorphism. It is clear that Θ I preserves the fibers. 8. Constructing Seifert hypersurfaces based on Φ Theorem 8.1. Let t P S 1 be a non-critical value of the map Φ and t ‰ 0. Then the closure of Φ´1ptq is a Seifert hypersurface for M which is smooth up to boundary. Moreover, the pn`1q-dimensional volume of Φ´1ptq is finite.
Proof. Let Σ " Φ´1ptq. By the implicit function theorem Σ is a smooth open submanifold of R n`2 zM . By Theorem 3.1 we infer that Σ is contained in some ball Bp0, Rq for large R. This implies that ΣzN is compact.
The main problem is to show that boundary of the closure of Σ is M . To this end we study the intersection Σ 0 :" Σ X pN zM q. Notice that we have a diffeomorphism Σ 0 -Φ´1ptq X X via the map X » Ñ pN zM q. Now Σ 0 is a smooth surface diffeomorphic to p0, ε 0 sˆM . By Theorem 7.14 the closure Σ 0 of Σ 0 in X is homeomorphic to the product r0, ε 0 sˆM . Under the map X Ñ N the closure Σ 0 is mapped to the closure of Σ in N . It follows that the boundary of the closure of Σ X N is M itself.
To show the finiteness of the volume of Σ, notice that the area of ΣzN is finite, because ΣzN is smooth and compact. The finiteness of the volume of Σ X N follows from the Bounded Volume Lemma 7.6.
In numerical applications calculating the map Φ in N can be challenging due to the lack of a good bound for derivatives of Φ in N . Therefore the following corollary should be useful.
Proposition 8.2. Choose t P S 1 , t ‰ 0 to be a non-critical value of Φ. Define M 1 " Φ´1ptq X BN . Let Σ 1 " Φ´1ptqzN . Then M 1 is diffeomorphic to M , isotopic to M as knots in S n`2 and Σ 1 is a smooth surface for M 1 .
Proof. The fact that M 1 is diffeomorphic to M follows from the Fibration Lemma 7.3. The isotopy is given by M r " π˝Φ t ptruˆM q, where Φ is as in Theorem 7.14 and π : X Ñ N is the projection. By definition BΣ 1 " M 1 and as Σ 1 is closed and bounded it is also compact.