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Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic 3-Manifolds

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Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic 3-Manifolds Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic 3-Manifolds Xiaolong Hans Han Abstract We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hy- perbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock and Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic 3-manifolds, and formulate several questions and conjectures. Contents 1 Introduction2 1.1 Motivation and previous results . .2 1.2 A glimpse at the non-compact case . .3 1.3 Main theorem . .4 1.4 Topology and the Thurston norm of harmonic forms . .5 1.5 Minimal surfaces and the least area norm . .5 1.6 Sharpness and the interaction between harmonic forms and minimal surfaces6 1.7 Acknowledgments . .6 2 L2 Harmonic Forms and Compactly Supported Cohomology7 2.1 Basic definitions of L2 harmonic forms . .7 2.2 Hodge theory on cusped hyperbolic 3-manifolds . .9 2.3 Topology of L2-harmonic 1-forms and the Thurston Norm . 11 2.4 How much does the hyperbolic metric come into play? . 14 arXiv:2011.14457v2 [math.GT] 23 Jun 2021 3 Minimal Surface and Least Area Norm 14 3.1 Truncation of M and the definition of Mτ ................. 16 3.2 The least area norm and the L1-norm . 17 4 L1-norm, Main theorem and The Proof 18 4.1 Bounding L1-norm by L2-norm . 18 4.2 Main theorem and the proofs . 21 1 5 Sharpness of the Inequalities and a Functional Point of View 22 5.1 Three conditions for the sharpness and their interactions . 23 5.2 Proof of non-sharpness using harmonic 1-forms and minimal foliations . 24 5.3 A functional point of view . 26 References 27 1 Introduction 1.1 Motivation and previous results A cusped hyperbolic manifold is a complete non-compact hyperbolic manifold with finite volume. All manifolds are assumed oriented unless stated otherwise. By Mostow rigid- ity, the topology of a closed or cusped hyperbolic manifold M of dimension at least 3 determines its geometry. Effective geometrization attempts to seek qualitative and quan- titative connections between the topological invariants and geometric invariants of the manifold. For example, given the fundamental group, what do we know about the injec- tivity radius, diameter, or 2-systole of the manifold (see [Whi02, BS11, BCW04, Bel13])? By a conjecture by Bergeron and Venkatesh, and similar conjectures independently pro- posed by L¨uck and L^ein [L^e18,L¨uc16], one can extract the volume of a closed (arithmetic) hyperbolic 3-manifold, by the following: Conjecture 1.1. ([BV13]) Let Mn be a sequence of congruence covers of a fixed arith- metic hyperbolic 3-manifold M. Denote the degree of the cover by [π1M : π1Mn]. Then we have log jH (M ) j vol(M) = 6π lim 1 n tor (1.1) n!1 [π1M : π1Mn] where H1(Mn) is the abelianization of π1(Mn) and (·)tor is the torsion part of an abelian group. For recent progress and other related works on this conjecture, see [Le14, L¨uc13, MM13, BD15, ABB+17]. In [BSV16], Bergeron, Seng¨unand Venkatesh have another striking conjecture that H2 of the Mn above can be generated by integral cycles in H2(Mn; R) with low topological complexity, as measured by the Thurston norm k · kT h (see Section 2 for detailed definitions). On the other hand, in a general closed Riemannian manifold 1 ∼ M, one can measure the complexity of a cycle in H (M; R) = H2(M; R) by the more geometric L2-norm: Z 1 kφkL2 = α ^ ∗α, for φ 2 H (M), M where α is the harmonic representative of φ from the classical Hodge theory. Bergeron, Seng¨unand Venkatesh proved a beautiful theorem regarding the two norms: Theorem 1.2. (BSV16, Theorem 4.1) If M varies through a sequence of finite coverings of a fixed closed orientable hyperbolic 3-manifold M0, then we have C1 1 k · k ≤ k · k 2 ≤ C k · k on H (M; ) (1.2) vol(M) T h L 2 T h R 2 where C1 and C2 depend only on M0. The above theorem is another example of effective geometrization, where one bounds the geometric complexity of φ 2 H1 by its topological complexity. In [BD17], Brock and Dunfield removed the assumption of Mn being covers of a fixed closed manifold M and obtained the following inequalities: Theorem 1.3. [BD17, Theorem 1.2] For all closed orientable hyperbolic 3-manifolds M one has π 10π 1 k · kT h ≤ k · kL2 ≤ k · kT h on H (M; R): (1.3) pvol(M) pinj(M) Compared to [BSV16], Brock and Dunfield introduced the tools of minimal surfaces and harmonic expansions of harmonic functions in H3 in the proof. In particular, the proof of the left-hand inequality uses the uniform bounds on the principal curvatures for a stable minimal surface in a closed hyperbolic 3-manifold and the Gauss-Bonnet theorem. It is interesting to note that the left-hand side has been approached by very different techniques. Kronheimer and Mrowka and Lin use estimates for the solutions of the Seiberg-Witten equations in [KM97, Lin20]. Bray and Stern in [BS19] use the harmonic harmonic level set techniques, Bochner technique and the traced Gauss equations. The proof of the right-hand inequality combines several ideas. It uses explicit computations for the series expansions of harmonic functions and an inequality bounding L1-norm of a 1-form in a ball by its L2-norm in the same ball, reminiscent of the mean value property of harmonic functions. Similar L1 estimates were done by Cerbo and Stern in [CS19a, CS19b, CS20] for real and complex cusped manifolds with dimension at least 3 with implicit constants. 1.2 A glimpse at the non-compact case It is natural to wonder whether a version of Inequalities (1.3) still holds when M is cusped or infinite-volume. However, the L2-norm can depend on the geometry near infinity in a subtle way, probably not captured by the topology. The geometry of the cusped case is well-understood and much simpler than that of the infinite-volume case. Even if we only consider the cusped case, this question turns out to be nontrivial and interesting. Problems one naturally encounters include: 1. One immediate challenge is that the injectivity radius of M is zero and the right- hand side of Inequalities (1.3) is trivially true. We have to find the right geometric invariants in place of the injectivity radius. This issue will be resolved in a nat- ural way after we have a deeper understanding of Hodge theory on non-compact hyperbolic 3-manifolds and minimal surface theory in this context. 1 ∼ 2. Originally, Inequalities (1.3) are for H (M) = H2(M) where M is closed. Brock and Dunfield use the stability of least area incompressible surfaces in closed hyperbolic 3-manifolds heavily, where one has uniform control on the principal curvatures on such surfaces and the area is bounded from above and below by some constant multiples of the Euler characteristic. In the non-compact case for H2(M), one can have closed surfaces homotopic to ones arbitrarily deep inside the cusps, and the 3 uniform control on curvature is no longer as clear. In addition, the Thurston norm k · kT h is no longer a genuine norm on H2(M) by the following example. Let T be a boundary-parallel torus. If M has more than one cusp, then [T ] 6= 0 in H2(M) but kT kT h = 0. In this case one can question how useful such an inequality is. Even if we restrict the consideration to closed surfaces with genus at least 2 in the thick part, it might have regions that can enter the cusp arbitrarily deep, resulting in unbounded area. 1.3 Main theorem It is perhaps surprising that the solution to all the issues mentioned above for a cusped M can be resolved by one simple condition: 2 H1(M): can be represented by an L2 harmonic 1-form α. Denote by H1 the space of L2 harmonic 1-forms on M. In [Zuc83] (see also equation (1.4) and comments before that in [MP90]), Zucker showed that H1 is isomorphic to the 1 image of the natural inclusion of the compactly supported first cohomology Im(H0 (M) ! H1(M)). We first provide a self-contained introduction to Hodge theory for non-compact hyperbolic manifolds in Section 2.2, based on [MP90], and then analyze the topology of the dual surfaces of the 1-forms in this space in Section 2.3, and show that the Thurston norm restricted to this space is an actual norm in Lemma 2.10. We also point out the beautiful coincidence that the dual surfaces are exactly those described recently in [HW17, Corollary 1.2]: Corollary 1.4. Let S be a closed orientable embedded surface in a cusped hyperbolic 3- manifold M 3 which is not a 2-sphere or a torus. If S is incompressible and non-separating, then S is isotopic to an embedded least area minimal surface. This corollary will provide us with enough regularity so that we obtain estimates similar to the ones for stable incompressible minimal surfaces in the closed hyperbolic case. In [HW17, Theorem 1.1], Huang and Wang show that for each cusped M, there is a canonical height τ to truncate cusps off M relative to the maximal volume cusp boundary to obtain a compact manifold Mτ with boundary so that any least area closed incompressible non- separating minimal surface stays in Mτ [HW17, Theorem 5.9, Corollary 5.7] (τ = τ3).
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