Harmonic maps and the topology of manifolds with positive spectrum and stable minimal hypersurfaces

Publ. Mat. 50 (2006), 301–313 HARMONIC MAPS AND THE TOPOLOGY OF MANIFOLDS WITH POSITIVE SPECTRUM AND STABLE MINIMAL HYPERSURFACES Qiaoling Wang∗ Abstract In this paper, we prove two Liouville theorems for harmonic maps and apply them to study the topology of manifolds with positive spectrum and stable minimal hypersurfaces in Riemannian manifolds with non-negative bi-Ricci curvature. 1. Introduction Harmonic maps are natural generalizations of harmonic functions and are critical points of the energy functional defined on the space of maps between two Riemannian manifolds. The Liouville type properties for harmonic maps have been studied extensively in the past years (Cf. [Ch], [C], [EL1], [EL2], [ES], [H], [HJW], [J], [SY], [S], [Y1], etc.). In 1975, Yau [Y1] proved that any harmonic function bounded from one side on a complete Riemannian manifold with non-negative Ricci curvature must be a constant. Schoen and Yau [SY] have shown that a harmonic map of finite energy from a complete Riemannian manifold with non-negative Ricci curvature to a complete manifold with non-positive sectional curvature is constant. This Liouville theorem of Schoen-Yau was used [SY] to show the important result which states that any smooth map of finite energy from a complete Riemannian manifold with non-negative Ricci curvature to a compact manifold with non-positive sectional curvature is homotopic to constant on each compact set. In this paper, we use the same idea of Schoen-Yau to study complete non-compact manifolds with Ricci curvature bounded from below and stable minimal hypersurfaces 2000 Mathematics Subject Classification. 53C20, 53C42. Key words. Liouville theorems, harmonic maps, topology, manifolds, spectrum, minimal hypersurfaces. ∗Partially supported by CNPq. 302 Q. Wang in manifolds with non-negative bi-Ricci curvature. Our first result can be stated as follows. Theorem 1.1. Let Mn(n≥2) and Ns(s≥1) be two complete Rimannian manifolds. Suppose that Mnis non-compact and that Nshas nonpositive sectional curvature. Assume that the first eigenvalue λ1(Mn) of Mnis positive and that the Ricci curvature of Mnsatisfies (1.1) RicMn≥ −1 + 1 2nsλ1(Mn) + δ for some δ > 0. Then any harmonic map from Mnto Nswith finite energy is a constant. It has been shown by Schoen-Yau (Cf. [SY]) that for any smooth map fof finite energy from a complete Riemannian manifold Mto a compact manifold Nwith non-positive sectional curvature, there is a harmonic map h:M→Nwith finite energy such that fis homotopic to hon each compact set of M. Thus Theorem 1.1 implies immediately the following Corollary 1.2. Let sbe a positive integer and let Mn(n≥2) be a complete non-compact Rimannian manifold with λ1(Mn)>0and RicMn≥ −1 + 1 2nsλ1(Mn) + δ for some δ > 0. Let Nsbe a compact manifold with non-positive sectional curvature. If f:Mn→Nsis a smooth map with finite energy, then fis homotopic to constant on each compact set. As an application of this corollary, one has the following result. Corollary 1.3. Let Mn(n≥2) be as in Theorem 1.1 and let Dbe a compact domain in Mnwith smooth simply connected boundary. Then there exists no non-trivial homomorphism from π1(D)into the fundamental group of a compact manifold Nwith non-positive sectional curvature. Proof: We use the arguments in [SY]. Let h:π1(D)→π1(N) be a homomorphism. Since Nis K(π, 1), there is a smooth map f:D→N such that f∗=h. Observe that fis a homotopic to a constant map on ∂D because ∂D is simply connected. Thus, fcan be extended to be a smooth map ˜ f:M→Nsuch that outside a compact set, ˜ fis constant. As ˜ fhas finite energy, we conclude from Corollary 1.2 that fis homotopic to a contant and that his trivial. Harmonic Maps and the Topology of Manifolds 303 Before stating our next result, we fix some notation. Let Mbe a complete oriented minimal hypersurface immersed in an oriented Riemannian manifold M. Let ∇be the gradient operator of Mand denote by |A|2the squared norm of the second fundamental form Aof Min M. Mis said to be stable if (1.2) ZM{|∇f|2−(|A|2+ Ric(µ, µ))f2} ≥ 0 for all f:M→Rwith compact support, where Ric(µ, µ) is the Ricci curvature of Min the unit normal direction µto M. Definition 1.4 ([ShY], [T]).Let Mbe an m-dimensional complete Riemannian manifold, and u,vbe orthonormal tangent vectors. We set b-Ric(u, v) = Ric(u, u) + Ric(v, v)−K(u, v), and call it the bi-Ricci curvature in the directions u,v. Here Ric and Kdenote the Ricci curvature and sectional curvature of M, respectively. If m= 3, then b-Ric = s/2, where sdenotes the scalar curvature of M. It is clear from the definition that the non-negativity of the sectional curvature implies the non-negativity of the bi-Ricci curvature of M. We then prove the following theorem which generalizes a main result in [SY]. Theorem 1.5. Let Mnbe an n-dimensional complete oriented noncompact stable minimal hypersurface in a complete (n+ 1)-dimensional Riemannian manifold Mn+1 with non-negative bi-Ricci curvature. Assume that Nis a complete Riemannian manifold with non-positive sectional curvature. If f:Mn→Nis a harmonic map with finite energy, then f is constant. As in Corollaries 1.2 and 1.3, we have Corollary 1.6. Let Mbe an n-dimensional complete oriented noncompact stable minimal hypersurface in a complete (n+ 1)-dimensional Riemannian manifold Mwith non-negative bi-Ricci curvature. Assume that Nis a compact Riemannian manifold with non-positive sectional curvature. If f:M→Nis a smooth map with finite energy, then fis homotopic to constant on each compact set. Corollary 1.7. Let Mbe as in Corollary 1.6 and let Dbe a compact domain in Mwith smooth simply connected boundary. Then there exists no non-trivial homomorphism from π1(D)into the fundamental group of a compact manifold with non-positive sectional curvature. 304 Q. Wang 2. Preliminaries Let Mnand Nsbe complete Riemannian manifolds. Let f:Mn→Ns be a harmonic map. Let {ei}n i=1 and {eα}s α=1 be local orthonormal frames of Mnand Ns, respectively. Suppose {ωi}n i=1 and {θα}s α=1 are the dual coframes of {ei}n i=1 and {eα}s α=1, respectively, and {ωij }n i,j=1 and {θαβ}s α,β=1 are the corresponding connection forms. Denote by Rijkl and Kαβγδ the curvature tensors of Mnand Ns, respectively. Then we have the structure equations:      dωi=Pjωij ∧ωj ωij +ωji = 0 dωij =Pkωik ∧ωkj −1 2Pk,l Rijklωk∧ωl.      dθα=Pβθαβ ∧θβ θαβ +θβα = 0 dθαβ =Pγθαγ ∧θγβ −1 2Pγ,δ Kαβγδθγθδ. Define fαi, 1 ≤α≤s, 1 ≤i≤nby (2.1) f∗(θα) = X i fαiωi. Then the energy density e(f) is given by e(f) = X α,i f2 αi. Taking the exterior differentiation of (2.1), we get f∗(dθα) = X i (dfαi ∧ωi+fαi dωi), which gives (2.2) X i dfαi −X j fαjωij −f∗(θαβ)fβi ∧ωi= 0. Define fαij by (2.3) dfαi +X β fβif∗(θβα) + X j fαjωji =X j fαijωj. Then (2.2) and (2.3) imply that fαij =fαji and fis harmonic means X i fαii = 0,∀α= 1, . . . , s. Harmonic Maps and the Topology of Manifolds 305 Exterior differentiating (2.3), we have (2.4) X l dfαil +X j (fαijωjl +fαjlωji) + X β fβilf∗(θβα) ∧ωl =1 2X j,k,l Rijklfαjωk∧ωl+1 2X β,δ,γ,k,l Kαβγδfβifγkfδlωk∧ωl. Define X k fαijkωk=dfαij +X k (fαikωkj +fαkj ωki) + X β fβij f∗(θαβ); then (2.4) implies that fαikl −fαilk =X j Rijlkfαj +X β,γ,δ Kαβγδfβifγlfδk. Set e=e(f) and let ∆ be the Laplacian operator acting on functions on Mn. From the above formula, one can easily get the following Bochner type formula for harmonic maps which was first derived by Eells-Sampson [ES]. (2.5) 1 2∆e=X α,i,j f2 αij +X α,i,j Rijfαifαj −X α,β,γ,δ,i,j Kαβγδfαifβjfγifδj, where Rij is the Ricci tensor of Mn. Since |∇√e|2=1 eX j X i,α fαifαij  2 , we have X α,i,j f2 αij −|∇√e|2=1 2eX i,j,k,α,β (fαifβkj −fβkfαij)2 ≥1 2eX i,j,α (fαifαjj −fαjfαij )2. 306 Q. Wang By Schwartz inequality, X i,j,α (fαifαjj −fαjfαij )2≥1 ns X i X j,α (fαifαjj −fαjfαij )  2 =1 ns X i X αj fαjfαji  2 =1 ns|∇√e|2. Therefore, it holds [SY] (2.6) X α,i,j f2 αij ≥1 + 1 2ns|∇√e|2. 3. Proofs of the Theorems Proof of Theorem 1.1: Let f:Mn→Nsbe a harmonic map with finite energy and set e=e(f). Let λ1=λ1(Mn); then by definition, λ1ZMn ψ2≤ZMn|∇ψ|2 for any compactly supported function ψ∈H1,2(Mn). Replacing ψby φ√ewith φ∈C∞ 0(Mn), we get (3.1) λ1ZMn eφ2≤ZMn e|∇φ|2+ZMn φ2|∇√e|2+ 2 ZMn √eφ∇√e∇φ. Since Nshas non-positive sectional curvature, we conclude from (1.1), (2.5) and (2.6) that 1 2∆e≥1 + 1 2ns|∇√e|2+−1 + 1 2nsλ1+δe. Harmonic Maps and the Topology of Manifolds 307 Thus one gets from the divergence theorem that 2ZMn √eφ∇√e∇φ=1 2ZMn∇e∇φ2 =−1 2ZMn φ2∆e ≤ −1 + 1 2nsZMn φ2|∇√e|2 +1 + 1 2nsλ1−δZMn eφ2. (3.2) On the other hand, for any l > 0, we also have (3.3) 2 ZMn √eφ∇√e∇φ≤1 lZMn|∇√e|2φ2+lZMn e|∇φ|2. Take a sufficiently large lso that 2ns(1 + l)δ > λ1. We get from (3.2) and (3.3) that 2ZMn √eφ∇√e∇φ =4ns(1 + l) 2ns(1 + l) + l+2l 2ns(1 + l) + lZMn √eφ∇√e∇φ ≤2ns(1 + l) 2ns(1 + l) + l−1 + 1 2nsZMn φ2|∇√e|2 +1 + 1 2nsλ1−δZMn eφ2 +l 2ns(1 + l) + l1 lZMn|∇√e|2φ2+lZMn e|∇φ|2 =2ns(1 + l) 2ns(1 + l) + l1 + 1 2nsλ1−δZMn eφ2−ZMn φ2|∇√e|2 +l2 2ns(1 + l) + lZMn e|∇φ|2, 308 Q. Wang which, combining with (3.1), gives (3.4) λ1ZMn eφ2≤2ns(1 + l) 2ns(1 + l) + l(1 + 1 2ns)λ1−δZMn eφ2 +1 + l2 2ns(1 + l) + lZMn e|∇φ|2. Fix a point p∈M. For r > 0, we choose φto satisfy the properties that φ=(1 on B(p, r) 0 on Mn\B(p, 2r) and |∇φ| ≤ Cr−1 for some constant C > 0, where B(p, r) is the geodesic ball of radius r and center p. Thus, (3.4) becomes (2ns(1 + l)δ−λ1)ZB(p,r) e≤C2r−2((1 + l)(2ns +l)) ZB(p,2r)\B(p,r) e. Letting r→ ∞, the right hand side tends to 0 since fhas finite energy. Since 2ns(1 + l)δ > λ1, we conclude that e≡0 and consequently fis constant. This completes the proof of Theorem 1.1. Proof of Theorem 1.5: Let e1, . . . , en, µ be a local orthonormal frame on Msuch that e1, . . . , enwhen restricted to Mform a local orthonormal frame in a neighborhood of a point x0∈M. Let ω1, . . . , ωn+1 be the dual coframe. Let ωij be the connection 1-forms of Mand ωn+1i=Pn j=1 hijωjwhen restricted to Mdefines the second fundamental form of M. The squared norm of the second fundamental form of Mis then given by |A|2=X i,j h2 ij. The condition that Mis stable is characterized by the following inequality (3.5) ZMn X i,j h2 ij ψ2+ZMn ψ2Ric(µ, µ)≤ZMn|∇ψ|2 for any compactly supported function ψ∈H1,2(Mn). Harmonic Maps and the Topology of Manifolds 309 Set e=e(f).Replacing ψin (3.5) by √eφ with φ∈C∞ 0(Mn), we get ZMn eφ2 X i,j h2 ij + Ric(µ, µ)  ≤ZMn e|∇φ|2+ 2 ZMn √eφ∇√e∇φ+ZMn φ2|∇√e|2 =ZMn e|∇φ|2−1 2ZMn φ2∆e+ZMn φ2|∇√e|2. (3.6) Since Mis minimal, we conclude from the Gauss equation that the Ricci curvature tensor (Rij) of Mis given by (3.7) Rij = n X k=1 R(ei, ek, ej, ek)− n X k=1 hikhkj, where Ris the curvature tensor of M. Set uα=Pn k=1 fαkek; then we have from (2.5), (2.6), (3.7) and the non-positivity of the sectional curvature of Nthat 1 2∆e≥X α,i,j f2 αij +X α,i,j Rijfαifαj =X α,i,j f2 αij +X αRic(uα, uα)−|uα|2K(uα, µ)−X α,i  X j hijfαj  2 ≥X α,i,j f2 αij +X αRic(uα, uα)−|uα|2K(uα, µ)− X i,j h2 ij e. (3.8) Substituting the above inequality into (3.6), we obtain (3.9) ZM e|∇φ|2≥ZM φ2 eRic(µ, µ)+X αRic(uα, uα)−|uα|2K(uα, µ)! +ZM φ2 X α,i,j f2 αij −|∇√e|2 .