Let M be an ndimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature. For 2 ≤ n ≤ 6, we prove that if M satisfies the δstability inequality (0 < δ ≤ 1), then there is no nontrivial L2β harmonic 1form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy the δstability inequality. Moreover, we prove a vanishing theorem for L2 harmonic 1forms on M when M is an ndimensional complete noncompact submanifold in a complete simplyconnected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k.
JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.1 (1-16) J. Math. Anal. Appl. ••• (••••) •••–••• Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold Nguyen Thac Dung a , Keomkyo Seo b,∗ a Department of Mathematics, Mechanics, and Informatics (MIM), Hanoi University of Sciences (HUS–VNU), Vietnam National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam b Department of Mathematics, Sookmyung Women’s University, Hyochangwongil 52, Yongsan-ku, Seoul, 140-742, Republic of Korea a r t i c l e i n f o a b s t r a c t Article history: Received 30 June 2014 Available online xxxx Submitted by H.R. Parks Let M be an n-dimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature. For 2 ≤ n ≤ 6, we prove that if M satisfies the δ-stability inequality (0 < δ ≤ 1), then there is no nontrivial L2β harmonic 1-form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy the δ-stability inequality. Moreover, we prove a vanishing theorem for L2 harmonic 1-forms on M when M is an n-dimensional complete noncompact submanifold in a complete simply-connected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k. © 2014 Elsevier Inc. All rights reserved. Keywords: δ-Stability inequality L2 harmonic 1-form Traceless second fundamental form First eigenvalue 1. Introduction Let M n be an n-dimensional orientable minimal hypersurface in a Riemannian manifold N of nonnegative sectional curvature. We recall that a minimal hypersurface in a Riemannian manifold is called stable provided the second variation of the volume is nonnegative for any normal variation on a compact subset. More precisely, a minimal hypersurface M in a Riemannian manifold N is said to be called stable if for any f ∈ C0∞ (M ) |∇f |2 − |A|2 + Ric(ν, ν) f 2 dv ≥ 0, (1.1) M where A is the second fundamental form, Ric is the Ricci curvature of N , ν is the unit normal vector of M , and dv is the volume form on M . * Corresponding author. E-mail addresses: dungmath@yahoo.co.uk (N.T. Dung), kseo@sookmyung.ac.kr (K. Seo). URL: http://sookmyung.ac.kr/~kseo (K. Seo). http://dx.doi.org/10.1016/j.jmaa.2014.10.076 0022-247X/© 2014 Elsevier Inc. All rights reserved. JID:YJMAA 2 AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.2 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• On the other hand, for a number 0 < δ ≤ 1, it is called δ-stable if any function f ∈ C0∞ (M ) satisfies |∇f |2 − δ |A|2 + Ric(ν, ν) f 2 dv ≥ 0. (1.2) M It is obvious that δ1 -stability implies δ2 -stability for 0 < δ2 < δ1 ≤ 1. In particular, if M is stable, then M is δ-stable for 0 < δ ≤ 1. There have been plenty of works on δ-stable complete minimal hypersurfaces in a Riemannian manifold. (See [3,8,10,19,27] and references therein for more details.) It is well-known that the only complete orientable stable minimal surface in R3 is a plane [4,7]. For 1/8 < δ, Kawai [15] showed that a δ-stable complete minimal surface in R3 should be a plane. Furthermore, do Carmo and Peng [5] proved that if a stable complete minimal hypersurface M in the Euclidean space with M |A|2 dv < ∞, then M is a hyperplane. Later, Shen and Zhu [26] proved that an n-dimensional stable complete minimal hypersurface M in the Euclidean space with M |A|n dv < ∞ is a hyperplane. Recently, Tam and Zhou proved that a complete n−2 n -stable minimal hypersurface whose second fundamental form satisfies some decay conditions in the Euclidean space is either a hyperplane or a catenoid. In case of complete orientable stable minimal hypersurfaces, several results on the nonexistence of L2 harmonic forms are well-known. Palmer [22] proved that if there exists a codimension one cycle on a complete minimal hypersurface M in the Euclidean space which does not separate M , then M is unstable by using the nonexistence of L2 harmonic 1-form. Thereafter, using Bochner’s vanishing technique, Miyaoka [20] showed that a complete orientable noncompact stable minimal hypersurface in a nonnegatively curved manifold has no nontrivial L2 harmonic 1-forms. In [32], Yun proved that if M ⊂ Rn+1 is a complete minimal hypersurface with sufficiently small total scalar curvature M |A|n , then there is no nontrivial L2 harmonic 1-form on M . Yun’s result has been generalized into various ambient spaces [2,6,23–25]. For an n-dimensional complete orientable noncompact (not necessarily minimal) hypersurface M in a complete manifold N of nonnegative sectional curvature with 2 ≤ n ≤ 4, Kim and Yun [16] recently proved that if M satisfies the stability inequality (1.1), then there is no nontrivial L2 harmonic 1-form on M , which is an extension of a well-known fact in the case when M is a complete stable minimal hypersurface in N . In Section 2, motivated by this, we prove that if M is an n-dimensional complete noncompact (not necessarily minimal) hypersurface in a complete manifold N of nonnegative sectional curvature and M satisfies the δ-stability inequality (1.2) for a number 0 < δ ≤ 1, then there is no nontrivial L2β harmonic 1-form on M for some constant β. (See Theorem 2.6 for more details.) As a consequence, we extend Kim and Yun’s result into the case when n = 5, 6. In Section 3, we also provide sufficient condition for complete hypersurfaces to satisfy the δ-stability inequality in a Riemannian manifold. In Section 4, we deal with complete noncompact submanifold cases. For an n-dimensional complete noncompact submanifold M in a complete simply-connected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k, it turns out that if the L2 norm φ n of the traceless second fundamental form φ is sufficiently small and the first eigenvalue λ1 (M ) of the Laplacian is bigger than some constant depending only on k, n, and φ n , then there is no nontrivial L2 harmonic 1-form on M . 2. Harmonic 1-forms on complete hypersurfaces of lower dimensions A complete manifold M is called non-parabolic if it has a positive Green function. Otherwise, M is called parabolic. We note that M is non-parabolic provided it has a non-constant positive superharmonic function on M . The following sufficient condition for parabolicity is well-known. Theorem. (See [11,12,14,28].) Let M be a complete manifold. If, for any point p ∈ M and a geodesic ball Bp (r) ⊂ M , JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.3 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• ∞ 3 r dr = ∞, Vol(Bp (r)) 1 then M is parabolic. Using the above theorem, we see that if M is non-parabolic, then ∞ r dr < ∞, Vol(Bp (r)) (2.1) 1 and hence M has an infinite volume. Definition 2.1. Let M n be an n-dimensional orientable hypersurface in a Riemannian manifold N . We say the δ-stability inequality holds on M for 0 < δ ≤ 1 if any f ∈ C0∞ (M ) satisfies |∇f |2 − δ |A|2 + Ric(ν, ν) f 2 dv ≥ 0. M It turns out that a complete orientable noncompact hypersurface in a complete manifold with nonnegative sectional curvature has an infinite volume if the δ-stability inequality holds for 0 < δ ≤ 1. Lemma 2.2. Let M n be a complete orientable noncompact hypersurface in a complete manifold N with nonnegative sectional curvature. If the δ-stability inequality holds on M for 0 < δ ≤ 1, then the volume of M is infinite. Proof. If M is non-parabolic, then M has an infinite volume by (2.1). We now assume that M is parabolic. Given 0 < δ ≤ 1, since the δ-stability inequality holds on M , we have |∇ϕ|2 ≥ δ M Ric(ν, ν) + |A|2 ϕ2 M for any f ∈ C0∞ (M ). Let q := δ(|A|2 + Ric(ν, ν)) and let D ⊂ M be any bounded domain with smooth boundary. Denote by λq1 (D) the first eigenvalue of the Schrödinger operator Δ + q acting on functions vanishing on ∂D. The assumption that the δ-stability holds on M is equivalent to that λq1 (D) ≥ 0 for any bounded domain D ⊂ M . From the result in [7], it follows that there is a positive function u such that the equation Δu + qu = 0 on M . Since the sectional curvature of N is nonnegative, u is a positive superharmonic function on M . The parabolicity of M implies that u is constant. Hence |A| ≡ 0, which shows that M is totally geodesic in N . Thus M has nonnegative Ricci curvature, which gives the conclusion that M has an infinite volume [31]. ✷ Let M n be an n-dimensional orientable submanifold in an (n + p)-dimensional Riemannian manifold N . Fix a point x ∈ M and choose any local orthonormal frame {e1 , · · · , en+p } such that {e1 , · · · , en } is an orthonormal basis of the tangent space Tx M and {en+1 , · · · , en+p } is an orthonormal basis of the normal space Nx M . For each α ∈ {n + 1, · · · , n + p}, define a linear map Aα : Tx M → Tx M by n+p ¯ X Y, eα , Aα X, Y = ∇ JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.4 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• 4 ¯ denotes the Levi-Civita connection on N . Then the (unwhere X, Y are tangent vector fields and ∇ normalized) mean curvature vector H is defined by n+p H= (trace Aα )eα . α=n+1 Define a linear map φα : Tx M → Tx M by φα X, Y = Aα X, Y − X, Y H, eα and a traceless bilinear map φ : Tx M × Tx M → Nx M by n+p φ(X, Y ) = φα X, Y eα . α=n+1 This map φ is called the traceless second fundamental form of M . Denote by A the second fundamental form. Then |φ|2 = |A|2 − H2 . n Note that H2 ≤ n. |A|2 In particular, if p = 1, then φ=A− H g, n where g is the induced metric on M . Lemma 2.3. Let b := √ (n−2)2 n−1 √ . 2n( n−1+1)2 Then we have √ 2(n − 1)|H|2 − (n − 2) n − 1|H| n|A|2 − |H|2 ≥ −bn2 |A|2 . (2.2) Proof. If |A| = 0, then H = 0. Thus the inequality (2.2) is trivial. Now we assume that |A| > 0. The inequality (2.2) is equivalent to √ (n − 2) n − 1 |H| n2 |A| We define fn (t) on [0, √ n− H2 2(n − 1) H 2 − ≤ b. |A|2 n2 |A|2 n ] by √ (n − 2) n − 1 fn (t) = t n2 n − t2 − 2(n − 1) 2 t . n2 Suppose that there is a positive constant B such that B ≥ max[0,√n ] fn (t). Then √ (n − 2) n − 1t n − t2 ≤ 2(n − 1)t2 + Bn2 , ∀t ∈ [0, √ n] JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.5 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• 5 or equivalently, (n − 2)2 (n − 1)x(n − x) ≤ 4(n − 1)2 x2 + 4B(n − 1)n2 x + B 2 n4 , (2.3) where x := t2 for all x ∈ [0, n]. A simple computation shows that the inequality (2.3) holds true if √ (n − 2)2 n − 1 √ B≥ = b, 2n( n − 1 + 1)2 which gives the conclusion. ✷ In the following, we need the Ricci curvature estimate for submanifolds in a Riemannian manifold which was done by Leung [17]. Lemma 2.4. (See [17].) Let M be an n-dimensional submanifold in a Riemannian manifold N with sectional curvature KN satisfying that K ≤ KN where K is a constant. Then the Ricci curvature RicM of M satisfies RicM ≥ (n − 1)K + √ 1 2(n − 1)|H|2 − (n − 2) n − 1|H| n2 n|A|2 − |H|2 − n−1 2 |A| . n Using Lemma 2.3 and the above Ricci curvature estimate, one can obtain the following. Lemma 2.5. Let M n be a complete orientable noncompact hypersurface in N of nonnegative sectional curvature. Then √ RicM ≥ − n−1 2 |A| . 2 (2.4) Proof. By Lemma 2.3 and Lemma 2.4, we see RicM √ (n − 2)2 n − 1 n−1 √ |A|2 + ≥− n 2n( n − 1 + 1)2 √ n−1 2 |A| , =− 2 which completes the proof. ✷ Theorem 2.6. Let M n (2 ≤ n ≤ 6) be a complete orientable noncompact hypersurface in a complete manifold n−2 N with nonnegative sectional curvature. If the δ-stability inequality holds on M for some 2 √ ≤ δ ≤ 1, n−1 2β then there is no nontrivial L harmonic 1-form on M for any constant β satisfying √ 2δ 1− n−1 1− n−2 √ 2δ n − 1 0, we have |ω|α Δ|ω|α = |ω|α α(α − 1)|ω|α−2 ∇|ω| α−1 ∇|ω|α α α−1 ≥ ∇|ω|α α = 2 2 + α|ω|α−1 Δ|ω| + α|ω|2α−2 ω|Δ|ω √ 1 n−1 2 2 2 ∇|ω| − |A| |ω| + α|ω| n−1 2 √ n−2 n − 1 2 2α α 2 ∇|ω| |A| |ω| . −α 1− (n − 1)α 2 ≥ 2 2α−2 (2.6) Choose any nonnegative number q and a smooth function φ with compactly support in M . Multiplying both sides of the inequality (2.6) by |ω|2qα φ2 and integrating over M , we obtain 1− n−2 (n − 1)α ≤ |ω| |ω|2qα φ2 ∇|ω|α M (2q+1)α 2 M =α √ √ α φ Δ|ω| + α 2 n−1 2 |A|2 φ2 |ω|2(q+1)α M n−1 2 M −2 φ|ω|(2q+1)α ∇φ, ∇|ω|α . M 2 |ω|2qα ∇|ω|α φ2 |A|2 φ2 |ω|2(q+1)α − (2q + 1) M JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.7 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• 7 Hence 2(q + 1) − √ ≤α n−2 (n − 1)α n−1 2 2 |ω|2qα ∇|ω|α φ2 M |ω|2(q+1)α |A|2 φ2 − 2 M φ|ω|(2q+1)α ∇φ, ∇|ω|α . (2.7) M On the other hand, since M satisfies the δ-stability inequality and N has nonnegative sectional curvature, we have |∇φ|2 ≥ δ M Ric(ν, ν) + |A|2 φ2 ≥ δ M |A|2 φ2 . M Replacing φ by |ω|(1+q)α φ in the above inequality gives 2 |ω|2qα ∇|ω|α φ2 + |ω|2(q+1)α |A|2 φ2 ≤ (q + 1)2 δ M M |ω|2(q+1)α |∇φ|2 M |ω|(2q+1)α φ ∇φ, ∇|ω|α . + 2(1 + q) (2.8) M Combining (2.7) and (2.8), we obtain 2(q + 1) − ≤ α δ + √ n−2 (n − 1)α 2 |ω|2qα ∇|ω|α φ2 M n−1 (q + 1)2 2 2α δ √ 2 |ω|2qα ∇|ω|α φ2 + M |ω|2(q+1)α |∇φ|2 M n−1 (q + 1) − 2 2 |ω|(2q+1)α φ ∇φ, ∇|ω|α . (2.9) M Given ε > 0, the Schwarz inequality implies 2α δ √ n−1 (q + 1) − 2 2 ≤ 1− α δ √ |ω|(2q+1)α φ ∇φ, ∇|ω|α M n−1 (q + 1) 2 2|ω|(2q+1)α φ|∇φ| ∇|ω|α M 2 |ω|2qα ∇|ω|α φ2 + ≤ |D| ε M 1 ε |ω|2(q+1)α |∇φ|2 , M where α D := 1 − δ √ n−1 (q + 1). 2 (2.10) JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.8 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• 8 From the inequalities (2.9) and (2.10), it follows that 2(q + 1) − √ ≤ n−2 − (n − 1)α n−1α 2 δ √ n − 1 α(q + 1)2 2 δ 2 |ω|2qα ∇|ω|α φ2 M |ω|2(q+1)α |∇φ|2 M 2 |ω|2qα ∇|ω|α φ2 + + |D|ε |D| ε M |ω|2(q+1)α |∇φ|2 , M or equivalently, 2(q + 1) − √ ≤ n−2 − (n − 1)α √ n − 1 α(q + 1)2 − |D|ε 2 δ 2 |ω|2qα ∇|ω|α φ2 M n − 1 α |D| + 2 δ ε |ω|2(q+1)α |∇φ|2 . (2.11) M Now let β := (1 + q)α and choose the numbers α and q such that n−2 − 2(q + 1) − (n − 1)α √ n − 1 α(q + 1)2 > 0. 2 δ Therefore, for a sufficiently small ε > 0, the inequality (2.11) implies that there is a constant C > 0 which depends on ε, δ, q, α such that 2 |ω|2qα ∇|ω|α φ2 ≤ C M |ω|2β |∇φ|2 , (2.12) M provided that n−2 − 2(q + 1) − (n − 1)α √ n − 1 α(q + 1)2 > 0. 2 δ (2.13) Note that the inequality (2.13) is equivalent to n−2 − 2β − n−1 √ n − 1 β2 > 0, 2 δ which is satisfied by the assumption √ 2δ 1− n−1 1− 2δ n−2 √ n−1 0 and a fixed point p ∈ M , we take a test function φ(r) defined on [0, ∞) such that φ ≥ 0, φ = 1 on [0, R] and φ = 0 in [2R, ∞) with |φ | ≤ R2 , where r(x) denotes the distance from p to x on M . Then the inequality (2.12) becomes |ω|2qα ∇|ω|α M 2 ≤ 4C R2 |ω|2β . M JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.9 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• 9 Letting R → ∞, we conclude that |ω| is constant since ω is an L2β harmonic 1-form. However, since the volume of M is infinite by Lemma 2.2, we obtain that ω ≡ 0, which completes the proof. ✷ As a direct consequence of Theorem 2.6, if δ = 1, that is, M satisfies the stability inequality (1.1), we have the following. Corollary 2.7. For 2 ≤ n ≤ 6, let M n be a complete orientable noncompact hypersurface in a complete manifold N with nonnegative sectional curvature. Let α1 and b be the same constants as in Theorem 2.6. If the stability inequality (1.1) holds on M , then there is no nontrivial L2β harmonic 1-form on M for any constant β satisfying √ 2 1− n−1 1− n−2 √ 2 n−1 then there is no nontrivial L2 harmonic 1-form on M . Note that the lower bound of λ1 (M ) depends on inf |H| in their result. In this section, we prove a similar vanishing theorem for L2 harmonic 1-forms on complete noncompact submanifolds under the same assumptions as in [2] except that the lower bound of λ1 (M ) depends on φ n . More precisely, we prove Theorem 4.1. Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in a complete simplyconnected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 where k is a constant. Assume that the traceless second fundamental form φ satisfies φ n < 1 . n(n − 1)CS In the case k = 0, assume further that the first eigenvalue λ1 (M ) of M satisfies λ1 (M ) > 2n2 (n − 1)2 k2 n3 − (n − 2)(n − 1) n(n − 1)CS φ n − 2n(n − 1)CS φ 2 n , where CS is a Sobolev constant in (3.1). Then there is no nontrivial L2 harmonic 1-form on M . Proof. As in the proof of Theorem 2.6, it follows from the inequality (2.5) that for any harmonic 1-form ω |ω|Δ|ω| ≥ 1 ∇|ω| n−1 2 + RicM (ω, ω). Moreover, using Lemma 2.4 and the fact that |φ|2 = |A|2 − Ric ≥ (n − 1) |H|2 − k2 n2 − n−2 n2 |H|2 n , we see n(n − 1)|φ||H| − n−1 2 |φ| . n JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.13 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• 13 Combining these two inequalities, we have 1 ∇|ω| n−1 |ω|Δ|ω| ≥ − n−2 n2 2 + (n − 1) |H|2 − k2 |ω|2 n2 n−1 2 2 |φ| |ω| . n n(n − 1)|φ||H||ω|2 − (4.1) Consider a geodesic ball B(R) of radius R centered at x ∈ M . Choose a test function f satisfying 0 ≤ f ≤ 1, f ≡ 1 on B(R), f ≡ 0 on M \ B(2R), and |∇f | ≤ R1 . Multiplying both sides by a compactly supported function f 2 in BR ⊂ M and integrating over BR , we have 1 2 ∇|ω| f 2 + n−1 f 2 |ω|Δ|ω| ≥ BR BR |H|2 − k2 |ω|2 f 2 n2 (n − 1) BR n−2 n2 − n−1 2 2 2 |φ| |ω| f . n n(n − 1)|φ||H||ω|2 f 2 − BR BR Applying the divergence theorem, we have 0 ≤ −2 f |ω| ∇f, ∇|ω| − BR + n−2 n2 n(n − 1) n n−1 2 ∇|ω| f 2 + n−1 n BR |φ|2 |ω|2 f 2 BR |φ||H||ω|2 f 2 + (n − 1) BR k2 − |H|2 |ω|2 f 2 . n2 BR For any positive numbers α, β > 0, it follows from the Schwarz inequality ≤α f |ω| ∇f, ∇|ω| 2 BR f 2 ∇|ω| 2 + BR 1 α |∇f |2 |ω|2 (4.2) BR and |φ||H||ω|2 f 2 ≤ β 2 BR |H|2 |ω|2 f 2 + 1 β BR |φ|2 |ω|2 f 2 . BR Combining these inequalities, we obtain 0≤ α− + n n−1 f 2 ∇|ω| 2 + BR β(n − 2) 2n2 1 α |∇f |2 |ω|2 BR n(n − 1) − n−1 n2 |H|2 |ω|2 f 2 BR + n−2 2βn2 n(n − 1) + n−1 n |φ|2 |ω|2 f 2 BR + k2 (n − 1) BR |ω|2 f 2 . (4.3) JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.14 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• 14 Similarly, we have the following estimate: |φ| |ω| f ≤ 2 2 n |φ| 2 2 n BR |ω||f | BR n−2 n 2n n−2 BR ≤ CS 2 n |φ| n 2 ∇ |ω|f BR BR BR 2 n |φ|n ≤ CS |H|2 |ω|2 f 2 + f 2 ∇|ω| (1 + α) BR 2 + 1+ 1 α BR 2 n |φ|n + CS BR |∇f |2 |ω|2 BR |H|2 |ω|2 f 2 . (4.4) BR In the case k = 0, we need an additional estimate. Using the monotonicity of the first eigenvalue λ1 (BR ) of a ball BR , we observe that BR λ1 (M ) ≤ λ1 (BR ) ≤ |∇f |2 BR (4.5) f2 for any f ∈ C0∞ (M ). Putting |ω|f for f in the inequality (4.5) and using the Schwarz inequality (4.2) gives f 2 ∇|ω| |ω|2 f 2 ≤ (1 + α) λ1 (M ) BR 2 1 α + 1+ |∇f |2 |ω|2 . BR (4.6) BR Combining the inequalities (4.3), (4.4) and (4.6), we have f 2 ∇|ω| A 2 |H|2 |ω|2 f 2 ≤ C +B BR BR |∇f |2 |ω|2 , (4.7) BR where the constants A, B, C are defined by k2 (n − 1) n −α− + CS A= n−1 λ1 (M ) B= n−1 n−2 − n2 2n2 |φ| n 2 n n−2 2βn2 n(n − 1) + n−1 n (1 + α) BR n(n − 1)β − CS |φ|n 2 n n−2 2βn2 n(n − 1) + n−1 n BR C= 1 k2 (n − 1) + + CS α λ1 (M ) |φ|n 2 n n−2 2βn2 n(n − 1) + n−1 n 1+ 1 . α BR Using our assumption on φ n and the following arithmetic–geometric mean inequality β + CS φ 2 n 1 ≥ 2 CS φ β n, (4.8) √ we see that B > 0 for any β > 0. Take β = CS φ n which makes equality in the inequality (4.8). By the assumptions on λ1 (M ) and φ n , we can choose the number α > 0 small enough such that A > 0. Furthermore, it automatically follows that C > 0. JID:YJMAA AID:18972 /FLA Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.15 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• 15 Now letting R → ∞ in the inequality (4.7), we obtain |∇|ω|| ≡ 0 and |H||ω| ≡ 0. Since |∇|ω|| ≡ 0, we get |ω| ≡ constant. Suppose that |ω| is a nonzero constant. From the fact that |H||ω| ≡ 0, it follows that M is a minimal submanifold. However, since the volume of a complete minimal submanifold in a Riemannian manifold of nonpositive sectional curvature is infinite, we have M |ω|2 = ∞. This is a contradiction to the assumption that ω is an L2 harmonic 1-form. Therefore ω ≡ 0, which completes the proof. ✷ As a consequence, if the ambient space N is the Euclidean space, we obtain the following result. Corollary 4.2. Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in the Euclidean space RN . If the traceless second fundamental form φ satisfies φ n 1 , n(n − 1)CS < then there is no nontrivial L2 harmonic 1-form on M . It immediately follows from the above result that such M must have only one end. We remark that the upper bound of φ n is less than the upper bound in Corollary 1.1 of [2], which is nonetheless a generalization of [9] and [18]. Moreover, when the ambient space N has a pinched nonpositive sectional curvature, we immediately obtain the following. Corollary 4.3. Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in a complete simplyconnected Riemannian manifold N with sectional curvature KN satisfying that −k2 ≤ KN ≤ 0 for some constant k = 0. If φ n < 1 n(n − 1)CS and λ1 (M ) > 2n2 (n − 1)2 k2 , n3 − n2 + 3n − 4 then there is no nontrivial L2 harmonic 1-form on M . We remark that the upper bound of φ n and the lower bound of λ1 (M ) depend only on the dimension of M and the curvature of the ambient space, which is different from [2]. Acknowledgments The authors would like to thank the referee for the helpful comments and suggestions. The first author was supported in part by NAFOSTED under grant number 101.02-2014.49. A part of this paper was written during a stay of the first author at Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to express his sincerely thanks to staffs there for excellent working conditions and financial support. The second author was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2013R1A1A1A05006277). JID:YJMAA AID:18972 /FLA 16 Doctopic: Miscellaneous [m3L; v1.143-dev; Prn:5/11/2014; 17:24] P.16 (1-16) N.T. Dung, K. Seo / J. Math. Anal. Appl. ••• (••••) •••–••• References [1] G. Carron, L2 -cohomologie et inegalites de Sobolev, Math. Ann. 314 (4) (1999) 613–639. [2] M.P. Cavalcante, H. Mirandola, F. Vitório, L2 harmonic 1-forms on submanifolds with finite total curvature, J. Geom. Anal. 24 (1) (2014) 205–222. [3] T.H. Colding, W.P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. II. 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The analogue of this result is also true for a complete minimal hypersurface in hyperbolic space [24] Later, it turned out that these gap theorems hold for more general submanifolds Given an n-dimensional complete noncompact submanifold in Euclidean space, Carron [1] proved that there exists a constant c(n) such that... 3451–3461 [28] N.T Varopoulos, Potential theory and diffusion of Riemannian manifolds, in: Conference on Harmonic Analysis in Honor of Antoni Zygmund, vols I, II, in: Wadsworth Math Ser., Wadsworth, Belmont, CA, 1983, pp 821–837 [29] X Wang, On conformally compact Einstein manifolds, Math Res Lett 8 (5–6) (2001) 671–688 [30] Q Wang, On minimal submanifolds in an Euclidean space, Math Nachr 261/262 (2003)... that if |A| ≤ c(n), then all spaces of L2 harmonic forms are trivial Fu and Li [9] showed that for a complete noncompact submanifold M n ⊂ RN there also exists a constant d(n) such that if the Ln norm of the traceless second fundamental form φ is less than d(n) then there is no nontrivial L2 harmonic 1-form on M More generally, let M be an n-dimensional complete noncompact submanifold in a complete. .. |A| 2 f 2 M −δ A |f | 2 n 2n n−2 n−2 n M ≥ 1 − nCS A CS 2 n −δ A 2n |f | n−2 2 n n−2 n M ≥ 1 − (n + δ) A CS 2n |f | n−2 2 n n−2 n M ≥ 0, which gives the conclusion ✷ 4 Vanishing theorem for L2 harmonic 1-forms on complete noncompact submanifolds As mentioned in the introduction, there are several vanishing theorems for L2 harmonic forms on complete noncompact stable minimal hypersurfaces Recall that... 703–723 [20] R Miyaoka, L2 harmonic 1-forms on a complete stable minimal hypersurface, Geom Global Anal (1993) 289–293 [21] L Ni, Gap theorems for minimal submanifolds in Rn+1 , Comm Anal Geom 9 (3) (2001) 641–656 [22] B Palmer, Stability of minimal hypersurfaces, Comment Math Helv 66 (1991) 185–188 [23] K Seo, Minimal submanifolds with small total scalar curvature in Euclidean space, Kodai Math J 31 (1)... ∞ in the inequality (4.7), we obtain |∇|ω|| ≡ 0 and |H||ω| ≡ 0 Since |∇|ω|| ≡ 0, we get |ω| ≡ constant Suppose that |ω| is a nonzero constant From the fact that |H||ω| ≡ 0, it follows that M is a minimal submanifold However, since the volume of a complete minimal submanifold in a Riemannian manifold of nonpositive sectional curvature is in nite, we have M |ω|2 = ∞ This is a contradiction to the assumption... 190–191 [12] A Grigor’yan, On the existence of positive fundamental solution of the Laplace equation on Riemannian manifolds, Mat Sb 128 (3) (1985) 354–363 (in Russian); Engl transl in Math USSR Sb 56 (1987) 349–358 [13] D Hoffman, J Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm Pure Appl Math 27 (1974) 715–727 [14] L Karp, Subharmonic functions, harmonic mappings and isometric... 1-forms on complete submanifolds in Euclidean space, Kodai Math J 32 (3) (2009) 432–441 [10] H.P Fu, D.Y Yang, Vanishing theorems on complete manifolds with weighted Poincaré inequality and applications, Nagoya Math J 206 (2012) 25–37 [11] A Grigor’yan, On the existence of a Green function on a manifold, Uspekhi Mat Nauk 38 (1) (1983) 161–162 (in Russian); Engl transl in Russian Math Surveys 38 (1)... minimal submanifolds in hyperbolic space, Arch Math (Basel) 94 (2) (2010) 173–181 [25] K Seo, L2 harmonic 1-forms on minimal submanifolds in hyperbolic space, J Math Anal Appl 371 (2) (2010) 546–551 [26] Y Shen, X Zhu, On stable complete minimal hypersurfaces in Rn+1 , Amer J Math 120 (1998) 103–116 [27] L.F Tam, D Zhou, Stability properties for the higher dimensional catenoid in Rn+1 , Proc Amer Math... assumption that ω is an L2 harmonic 1-form Therefore ω ≡ 0, which completes the proof ✷ As a consequence, if the ambient space N is the Euclidean space, we obtain the following result Corollary 4.2 Let M n (n ≥ 3) be an n-dimensional complete noncompact submanifold in the Euclidean space RN If the traceless second fundamental form φ satisfies φ n 1 , n(n − 1)CS < then there is no nontrivial L2 harmonic 1-form ... mentioned in the introduction, there are several vanishing theorems for L2 harmonic forms on complete noncompact stable minimal hypersurfaces Recall that ω is an L2 harmonic 1-form on M if it satisfies... curvature estimate for submanifolds in a Riemannian manifold which was done by Leung [17] Lemma 2.4 (See [17].) Let M be an n-dimensional submanifold in a Riemannian manifold N with sectional... sufficient condition for complete hypersurfaces to satisfy the δ-stability inequality in a Riemannian manifold In Section 4, we deal with complete noncompact submanifold cases For an n-dimensional complete