aa r X i v : . [ m a t h . DG ] F e b GEHRING LINK PROBLEM, FOCAL RADIUS ANDOVER-TORICAL WIDTH
JIAN GE
Abstract.
In this note, we study the Gehring link problem in the roundsphere, which motives our study of the width of a band in positively curvedmanifolds. Using the same idea, we are able to get a sphere theorem forhypersurface in in the round S n in terms of its focal radius as well as therigidity of Clifford hypersurface in S n . The 3-dimension case of our theoremsconfirm two conjectures raised by Gromov in [Gro18]. Introduction
Let ( Y = T × [0 , , g ) be a Riemannian torical band, where T is the 2-dimensional torus. In [Gro18], Gromov studied the width of Y , which by definitionis the distance between two boundary components of Y , under a lower scalar cur-vature bound of g . He also conjectured the upper bound of width of such bandisometrically embedded in the round 3-sphere is π/
2. A related conjecture posedin [Gro18] is the focal radius of an embedded torus in the round n -sphere.Our starting point is to find the geometric intuition behind the conjectural upperbound π/ If two sets are linked in the sphere, they cannot be more than π/ -apart. ”The following example serves as our motivation. We take T Cl ⊂ S , where S := { ( z , z ) ∈ C || z | + | z | = 1 } is the unit 3-sphere and the 2-dimensionalClifford Torus is defined by T Cl := { ( z , z ) ∈ C || z | = | z | = 1 / √ } . Let B ( T Cl , r ) be the r -tubular neighborhood of T Cl inside S . It is clear that S \ B ( T Cl , r ) has two connected component if r < π/
4, each of which is a solidtorus. These tori form a Hopf link in S . Their distance is clearly ≤ π/
2. Thisreminds us the classical Gehring linking problem in R . Theorem 0.1 ([Ort75], [BS83]) . Let A and B are closed curves which are differ-entiably embedded in R such that they are linked and d( A, B ) ≥ , then the lengthof A and B must great or equal to π . We now state our first result.
Mathematics Subject Classification.
Primary: 53C23; Secondary: 51K10.
Key words and phrases.
Focal radius, over-toric band, sphere theorem, Gehring link problem,width.*Partially supported by NSFC.
Theorem 0.2 (Gehring Link Problem in 3-sphere) . Let
A, B be two disjoint linkedJordan curves in the unit -sphere S . Then d( A, B ) ≤ π/ , with equality holds if and only if A and B are the dual great circles in the Hopffibration. Our method of the proof can be easily generlized to higher dimension. Theclassical Gehring Link problem in R n is proved in [Gag80]. Theorem 0.3 (Spherical Gehring Link Problem) . If A k and B l are k and l sphereswhich are embedded in S n such that they are linked, where n = k + l + 1 . Then d( A, B ) ≤ π/ , with equality holds if and only if A and B are the dual great sub-spheres in S n , i.e. S n = A ∗ B , where ∗ denotes the spherical join.Remark . Our proof of Theorem 0.3 remains valid if we replace the n -sphereby a closed Riemannian manifold ( M n , g ) with sectional curvature ≥ A and B ). For simplicity wefocus only on the n -sphere.Theorem 0.2 explains heuristically why the width of a torical band cannot betoo large: The complement of the band in S are ‘linked’. However, this is onlypartially correct. In general the complement of a torical band is not a classicallink. For example when the torus is the boundary of a nontrivial knot in S , onlyone component of the complement is homotopic to a cicle. Nevertheless, we canstill make use the idea of ‘link’ as an obstruction to have large width, we call thisobstruction ‘boundary irreducible’ see Definition 2.1. In fact, using this idea weprove the following theorem, which confirms a conjecture of Gromov in [Gro18]. Theorem 0.5.
The over-torical width of S denoted by width ˆ T ( S ) , is π/ . Another estimate can be drawn from the proof of Theorem 0.3 is the estimateof focal radius of torus in S . Let Σ ⊂ S be a smoothly embedded closed surface.Recall that the focal radius of Σ ⊂ S is the largest number r > { ( p, v ) ∈ ν Σ | p ∈ Σ , | v | < r } → S is a diffeomorphism onto its image, where ν Σ is the normal bundle of Σ. The focalradius of Σ ⊂ S will be denoted by rad J (Σ) = rad J ( M ⊂ S ). By the standardcomparison argument, it is well known that any hypersurface Σ in S has focalradius ≤ π/
2. In fact if a closed surface Σ in S has focal radius equal to π/ has to be the equatorial 2-sphere, cf. [GW18]. The following question is askedin [Gro18]: Let Σ n be a smoothly embedded n -torus in S n − , what is the largestpossible focal radius? Gromov conjectured that the Clifford Torus T nCl is the onlytorus realizes the conjectured upper bound arcsin(1 / √ n ). The upper bound for n = 2 actually follows from Corollary C in [GW20]. We give a different proof ofthis result as well as the rigidity. Theorem 0.6.
Let Σ be a smoothly embedded -torus in S . Then rad J (Σ) ≤ π , with equality holds if and only if Σ is the Clifford Torus T Cl . EHRING LINK PROBLEM, FOCAL RADIUS AND OVER-TORICAL WIDTH 3
In fact, we prove the following stronger result
Theorem 0.7.
Let Σ n − be an orientable hypersurface embedded in S n . Suppose Σ is not homeomorphic to S n − , then rad J (Σ) ≤ π/ , with equality holds if and onlyif Σ is isometric to the Clifford hypersurface S k (1 / √ × S l (1 / √ for some k, l ∈ N such that n = k + l + 1 . Next theorem shows that spheres are the only hypersurfaces with large focalradius in positively cruved manifold.
Theorem 0.8 (A Topological Sphere Theorem) . Let ( M n , g ) be a simply connectedclosed Riemannian manifold with sec ≥ and Σ n − be a smoothly embedded ori-entable hypersurface in M . Suppose rad J (Σ) > π . Then M is homeomorphic to S n and Σ is homeomorphic to S n − .Remark . By h -cobordism theorem, ( n − R n for n = 5. Therefore Σ is diffeomorphic to S n − for n = 5 inTheorem 0.8.In Section 1 we prove the Gehring Link problem in sphere. The ideas of the proofare developed further in Section 2, where we provide the proofs of Theorem 0.8 andTheorem 0.5. Another short geometric proof of Theorem 0.6 is provided in Section3. Acknowledgement : It is my great pleasure to thank Professor Misha Gromov forhis comments and interests in our work and Professor Luis Guijarro for commentsafter reading the first draft of this paper.I would like to thank Professor Yuguang Shi for bringing J. Zhu’s paper [Zhu20] tomy attention where the author estimated the 3-dimension width using a completelydifferent method.1.
Proof of the Spherical Gehring Link Problem
Let A k , B l be two submanifold of S n , we call A and B are unlinked if there existsan embedded ( n − S n − in S n such that A and B lie in complementaryhemispheres; otherwise A and B are called linked . Note that we do not require k + l + 1 = n in the definition of ‘linked’, it is only required if we want to definethe ‘Linking number’.The key idea goes back to Grove-Shiohama’s proof of the Diameter Sphere The-orem, cf [GS77]. Namely we have the following Proposition 1.1.
Let ( M, g ) be a closed Riemannian manifold with sectional cur-vature sec( g ) ≥ . Suppose diam( M, g ) ≥ π/ . Then for any point x ∈ M and r > π/ , the set N := M \ B ( x, r ) is either empty or a totally geodesic submanifoldwith strictly convex (possible empty) boundary. For any closed set A ⊂ M , it is clear that M \ B ( A, r ) = ∩ x ∈ A { M \ B ( x, r ) } . It follows that M \ B ( A, r ) is a totally geodesic submanifold with strictly convex(possible empty) boundary. Moreover if ∂N is nonempty, N is homeomorphic tothe standard n -disk. J. GE
Proof of Theorem 0.3 .
Suppose d(
A, B ) > π/
2, then B ⊂ S n \ B ( A, π/ A isnot a great sub-sphere S k ⊂ S n , then the set N := S \ B ( A, π/
2) is a connectedconvex set with nonempty boundary. It follows that N is homeomorphic to a disk.By smoothing the distance function d A ( · ), we conclude ∂N is diffeomorphic to thesphere S n − . But B ⊂ S n \ B ( A, π/ A and B are linked in S . It follows that d( A, B ) ≤ π/ A, B ) = π/
2, then A anti := S n \ B ( A, π/
2) must has empty boundary.Since A anti is convex and in particular totally geodesic, B is isometric to S l for some l ∈ { , · · · , ( n − } . Apply the same discussion to B , we know A is isometric to S k for some k ∈ { , · · · , ( n − } . Since A and B are linked, n ≤ k + l + 1. (cid:3) The proof can be carried verbatim to the case when the sphere S n is replacedby a closed Riemannian manifold ( M, g ) with sec ≥ The complement of a non-sphercial band are linked
One crucial step in the proof of Theorem 0.3 in the previous section is to pro-duce an embedded sphere S n − in S n which separates A and B , provide they aremore than π/ π n − ( S n \ { A, B } ). This put a restriction on the topology of thecomplement. In this section, we focus on the complement instead of the set A and B . Let’s recall several definitions in [Gro18]. A proper band is a connected manifoldwith two distinguished disjoint non-empty subset in the boundary ∂Y − , ∂Y + suchthat ∂Y = ∂Y − ∪ ∂Y + . A proper band Y is called over-torical if there exists a map f : Y → Y := T n − × [0 , , with nonzero degree and respect the boundaries: ∂Y ± → ∂Y ± . We introduce thefollowing definition: Definition 2.1.
Let Y be a n -dimensional proper band. We call Y is boundaryreducible if there exists an embedded ( n − -sphere S n − ⊂ Y , such that S separates ∂Y − from ∂Y + . Otherwise Y is called boundary irreducible .Remark . Note that if Y is boundary irreducible, it is still possible to find anembedded ( n − Y that does not bound a n -ball. Namely it is possible Y is reducible in the classical sense. For example, take connected sum of a toricalband with any 3-manifold.For a closed orientable manifold Σ n − , let’s consider the band Y := Σ n − × [0 , Y is isometrically embedded in ( M n , g ), a closed Riemannian manifoldwith sec( M, g ) ≥
1. We will identify Y with its image in M . Let Y − and Y + be theboundaries Y × { } and Y × { } of Y . Suppose R := d( Y − , Y + ) > π/
2. Let N := M n \ B ( Y + , R − ε ) , ε < ( R − π/ / . By Proposition 1.1, it is clear that Y − ⊂ N and N is homeomorphic to a disk,whence ∂N is homeomorphic to S n − . In particular, we find an embedded ( n − Y which separates Y − from Y + . Namely Y can be written as a connectedsum of two manifolds, this is only possible if Σ n − itself is homeomorphic to S n − .Therefore, we just proved EHRING LINK PROBLEM, FOCAL RADIUS AND OVER-TORICAL WIDTH 5
Proposition 2.3.
Let ( M n , g ) be a closed orientable manifold with sec ≥ . Let Y = Σ n − × [0 , be a band isometrically embedded in M n . Suppose d( Y − , Y + ) >π/ , then Σ is homeomorphic to S n − . Clearly Proposition 2.3 and the Diameter Sphere Theorem (cf. [GS77]) implyTheorem 0.8 immediately. Now we move to the study of the non-spherical band.
Proof of Theorem 0.7.
We consider the set N := S n \ B (Σ , π/ . Since Σ is orientable, the set N has two connected components, denote them by { A, B } . Since d( A, B ) ≥ d( A, Σ) + d( B, Σ) = π/ . Then there are two possibilities. One is diam(
M, g ) = π/
2. It follows from thediameter rigidity theorem of Gromoll-Grove that (
M, g ) is isometric to a CROSSor homeomorphic to S n . If ( M, g ) is not heomorphic to a sphere, then A and B arethe dual pairs ofBy our assumption that Σ is not homeomorphic to S n − , it follows that A and B are both totally geodesic submanifolds in S n with empty boundary. Therefore A and B are great sub-spheres in S n , namely, A = S k and B = S l for some k, l ∈ { , · · · , n − } . it follows that k + l + 1 ≤ n . On the other hand A = S n \ ( B ( B, π/ , and vice versa, then k + l + 1 = n . i.e. A and B are the dual great sub-spheres in S n . It follows that Σ is the Clifford hypersurface S k (1 / √ × S l (1 / √ (cid:3) Now we turn to the estimate of the over-toric width. The proof is similar asabove. Since π ( Y ) = 0, we have the following Lemma 2.4.
Let Y be a -dimensional proper over-torical band, then Y is boundaryirreducible. Recall that the over-torical width of S , denoted by width ˆ T ( S ) is defined asthe supremum of numbers d such that there exists a proper over-torical band Y ofwidth d and an isometric immersion φ : Y → S . Since the r -neighborhood of Clifford Torus in S provides a torus band of widtharbitrarily close to π/
2, width ˆ T ( S ) ≥ π/
2. To prove Theorem 0.5, it suffices toshow width ˆ T ( S ) ≤ π/
2. Let Y be as above, we have Proposition 2.5.
The width of Y is less than or equal to π/ .Proof. Suppose the width d := d( Y − , Y + ) > π/
2, where we equipped Y with the pullback of the round metric on the sphere. Consider the distance function ρ : Y → R ,defined by ρ ( x ) = d( x, Y − ) , where the distance d is defined by the shortest curve connecting x to Y − . Sincewe make no assumption on the convexity of Y ± , such a curve might intersect theboundary Y + . However, if the width d = d( Y − , Y + ) > π/
2, we know for any point x ∈ Σ := ρ − (cid:18) π/ d (cid:19) , J. GE any geodesic connecting x to Y − that realizes ρ ( x ) does not intersect Y + . Since Y has constant curvature 1, the standard comparison argument shows that Σ islocally strictly convex in Y . Therefore under the isometric immersion φ : Y → S ,its image Σ := φ (Σ) ⊂ S is also locally strictly convex in S . By the classicaltheorem of Hadamard (cf. [Had97]), all such surfaces must be embedded, henceΣ is an embedded S . Since φ is an immersion, it follows that φ is a coveringmap. Therefore Σ is a disjoint union of embedded separating 2-spheres in Y (Itmight be disjoint only if Y − or Y + has more than one connected components). Byconnecting these spheres via cylinders we get an embedded separating 2-sphere in Y . This contradict to Lemma 2.4. Therefore d ≤ π/ (cid:3) Yet another proof of the Theorem 0.6
In this section, we give another proof of Theorem 0.6 based on Weyl’s Tubeformula and the solution of Willmore Conjecture. Let Σ ⊂ S be an embedded2-torus with focal radius r ∈ [ π/ , π/ B (Σ , r )) ≤ vol( S ) = 2 π . (3.1)By the volume estimate Lemma 4.1, which will be proved in next section, and thefact that sin(2 r ) ≥ cot( r ) if r ∈ [ π/ , π/ B (Σ , r )) = sin(2 r ) area(Σ) ≥ cot( r ) area(Σ) . (3.2)Combining (3.1) and (3.2), we havecot( r ) · area(Σ) ≤ π . (3.3)On the other hand by the solution of Willmore Conjecture we have Z Σ (cid:18) κ + κ (cid:19) dµ ≥ π , (3.4)where κ and κ are the principal curvatures of M with respect to the standardmetric on S . It follows by the Gauss-Bonnet formula that2 π ≤ Z Σ (cid:18) κ + κ (cid:19) dµ ≤ Z Σ κ + κ ! dµ. (3.5)Since the focal radius of M equals to r , it follows that | κ i | ≤ cot( r )for i = 1 ,
2. In fact this can be seen by touching each point p ∈ Σ by a geodesicsphere with radius r in S from both sides of Σ, or we can use the theorem of[GW20]. Plugging them back to (3.5) yields2 π ≤ cot r · area( M ) . (3.6)Using (3.3), we have 2 π ≤ cot r · area( M ) ≤ cot r · π , i.e. cot( r ) ≥
1, which implies r = π/ r ≥ π/
4. The rigidity partof the theorem follows from the rigidity part of the Willmore Conjecture.
EHRING LINK PROBLEM, FOCAL RADIUS AND OVER-TORICAL WIDTH 7 Volume of the tube
In this section, we calculate the volume of the r -neighborhood of Σ ⊂ S . It isa special case of Weyl’s Tube Formula. In fact it implies the focal radius ≤ π/ r ). Lemma 4.1 (Tube Formula) . Let Σ be an embedded -torus in S . Then for any r ≤ rad J (Σ) , we have vol( B (Σ , r )) = sin(2 r ) area(Σ) . Proof.
Let Σ( t ) := { x ∈ S | x = exp p ( tv ) , p ∈ M, v ∈ ν p M } be the parallel hypersurface with signed distance t ∈ [ − r, r ] to M . For any p ∈ Σ let κ ( p ) , κ ( p ) be the two principal curvature of Σ at p . Using Fermi coordinateand co-area formula we havevol( B (Σ , r )) = Z r − r area(Σ( t )) dt = Z r − r cos t Z Σ (1 − κ tan t )(1 − κ tan t ) dµdt = Z Σ Z r − r (cos t − ( κ + κ ) cos t · tan t + κ κ sin t ) dtdµ = Z Σ Z r − r (cos t + κ κ sin t ) dtdµ = Z Σ Z r − r (cos t − sin t ) dt + Z r − r ((1 + κ κ ) sin t ) dt ! dµ = Z Σ Z r − r (cos t − sin t ) dtdµ = sin(2 r ) area(Σ) . (cid:3) References [BS83] Enrico Bombieri and Leon Simon. On the Gehring link problem. In
Seminar on minimalsubmanifolds , volume 103 of
Ann. of Math. Stud. , pages 271–274. Princeton Univ. Press,Princeton, NJ, 1983.[Gag80] Michael E. Gage. A proof of Gehring’s linked spheres conjecture.
Duke Math. J. ,47(3):615–620, 1980.[Gro18] Misha Gromov. Metric inequalities with scalar curvature.
Geom. Funct. Anal. , 28(3):645–726, 2018.[GS77] Karsten Grove and Katsuhiro Shiohama. A generalized sphere theorem.
Ann. of Math.(2) , 106(2):201–211, 1977.[GW18] Luis Guijarro and Frederick Wilhelm. Focal radius, rigidity, and lower curvature bounds.
Proc. Lond. Math. Soc. (3) , 116(6):1519–1552, 2018.[GW20] Luis Guijarro and Frederick Wilhelm. Restrictions on submanifolds via focal radiusbounds.
Math. Res. Lett. , 27(1):115–139, 2020.[Had97] J Hadmard. Sur certaines propietes des trajectoires en dynamique.
J. Math. Pures Appl. ,3:331–387, 1897.[Ort75] M Ortel. unpublished. 1975.[Zhu20] Jintian Zhu. Width estimate and doubly warped product, 2020.
J. GE
Jian GeBeijing International Center for Mathematical Research,Peking University, Beijing 100871, ChinaE-mail address: [email protected] (Ge)
Beijing International Center for Mathematical Research, Peking University,Beijing 100871, P. R. China.
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