Abstract
We study a complete noncompact submanifold \(M^n\) in a sphere \(\mathbb {S}^{n+p}\). We prove that the dimension of the space of \(L^2\) harmonic \(1\)-forms on \(M\) is finite and there are finitely many non-parabolic ends on \(M\) if the total curvature of \(M\) is finite and \(n\ge 3\). This result is an improvement of Fu–Xu theorem on submanifolds in spheres and a generalized version of Cavalcante, Mirandola and Vitorio’s result on submanifolds in Hadamard manifolds.
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1 Introduction
Suppose that \(x: M^n\rightarrow N^{n+p}\) is an isometric immersion of an \(n\)-dimensional manifold \(M\) in an \((n+p)\)-dimensional Riemannian manifold \(N\). Let \(A\) denote the second fundamental form and \(H\) the mean curvature vector of the immersion \(x\). Let
for all vector fields \(X\) and \(Y\), where \(\langle ,\rangle \) is the induced metric of \(M\). We say the immersion \(x\) has finite total curvature if
If \(M^n\) \((n\ge 3)\) is a complete minimal hypersurface in \(\mathbb {R}^{n+1}\) with finite index, Li and Wang [9] proved that \(M\) has finitely many ends. More generally, Zhu [12] showed that: suppose that \(N^{n+1}\) \((n\ge 3)\) is a complete simply connected manifold with non-positive sectional curvature and \(M^n\) is a complete minimal hypersurface in \(N\) with finite index. If the bi-Ricci curvature satisfies
for all orthonormal tangent vectors \(X, Y\) in \(T_pN\) for \(p\in M\), then M must has finitely many ends. Cavalcante et al. [1] considered a complete noncompact submanifold \(M^n\) \((n\ge 3)\) isometric immersed in a Hadamard manifold \(N^{n+p}\) with sectional curvature satisfying \(-k^2\le K_N\le 0\) for some constant \(k\) and obtained that if the total curvature is finite and the first eigenvalue of the Laplacian operator of \(M\) is bounded from below by a suitable constant, then the dimension of the space of the \(L^2\) harmonic \(1\)-forms on \(M\) is finite and \(M\) has finitely many non-parabolic ends. Fu and Xu [3] considered a complete submanifold \(M^n\) in a sphere \(\mathbb {S}^{n+p}\) with finite total curvature and bounded mean curvature and showed that the dimension of \(H^1(L^2(M))\) is finite and there are finitely many non-parabolic ends on \(M\) .
In this paper, we discuss a complete noncompact submanifold \(M^n\) in a sphere \(\mathbb {S}^{n+p}\) with finite total curvature and no restriction of mean curvature. We recall some relevant definitions. The Hodge operator \(*:\wedge ^k(M)\rightarrow \wedge ^{n-k}(M)\) is defined by
where \(\sigma (i_1,i_2,\ldots ,i_n)\) denotes a permutation of the set \((i_1,i_2,\ldots ,i_n)\) and \(\mathrm{sgn}\sigma \) is the sign of \(\sigma \). The operator \(d^*:\wedge ^k(M)\rightarrow \wedge ^{k-1}(M)\) is given by
The Laplacian operator is defined by
A \(k\)-form \(\omega \) is called \(L^2\)-harmonic if \(\triangle \omega =0\) and
We denote \(H^1(L^2(M))\) by the space of all \(L^2\) harmonic \(1\)-forms on \(M\). We obtain finiteness of non-parabolic ends for the submanifold in a sphere with finite total curvature:
Theorem 1.1
Let \(M^n\) \((n\ge 3)\) be an \(n\)-dimensional complete noncompact oriented manifold isometrically immersed in an \((n+p)\)-dimensional sphere \(\mathbb {S}^{n+p}\). If the total curvature is finite, then the dimension of \(H^1(L^2(M))\) is finite and there are finitely many non-parabolic ends on \(M\).
Remark 1.2
Theorem 1.1 generalizes Theorem 1.4 in [3] without the restriction of the mean curvature vector and is also an extension of finiteness of non-parabolic ends on submanifolds in Hadamard manifolds in [1].
2 Proof of main results
We initially introduce several results which will be used to prove Theorem 1.1.
Proposition 2.1
[8, 9] If M is a complete Riemannian manifold, then the number of non-parabolic ends of M is bounded from above by \(\dim H^1(L^2(M))+1\).
Proposition 2.2
[4, 13] Let \(M^n\) be a complete noncompact oriented manifold isometrically immersed in a sphere \(\mathbb {S}^{n+p}\). Then
for each \(f\in C_0^1(M)\), where \(C_0\) depends only on \(n\) and \(H\) is the mean curvature vector of \(M\) in \(\mathbb {S}^{n+p}\).
Proof of Theorem 1.1
Suppose that \(\omega \in H^1(L^2(M))\). Then we have
Note that the following Bochner’s formula holds [6]:
Equalities (2.1) and (2.2) imply that
There exists the Kato inequality [2, 11]:
Combining (2.3) and (2.4), we get that
Take \(h=|\omega |\). There is an estimate for the Ricci curvature of the submanifold \(M\) in [5, 10]:
By (2.5), we obtain that
Suppose that \(\eta \) is a compactly supported piecewise smooth function on \(M\). Then
Integrating by parts on \(M\), we obtain that
By (2.6), we get
That is,
Note that
for any positive real number \(a\). By (2.7) and (2.8), we obtain that
That is,
where
and
Now we estimate the term \(\int _M|\Phi |^2\eta ^2h^2\): take \(\phi (\eta )=\left( \int _{\mathrm{Supp} \eta }|\Phi |^{n}\right) ^{\frac{1}{n}}\). Then
for any positive real number \(b\), where the second inequality holds because of Proposition 2.2. Note that
for any positive real number \(c\). By (2.9)–(2.11), we have
That is,
where
and
Next, we prove there exists a positive constant \(\delta \) such that if \(\Vert \Phi \Vert _{L^n(M)}<\delta \) then \(C,D,E\) and \(F\) are positive. Obviously, \(\phi (\eta )\le \Vert \Phi \Vert _{L^n(M)}<\delta \). Choose \(d\in (0,\frac{1}{2})\) and let \(a=a(d)\), \(\delta =\delta (d)\) such that
Choosing \(0<c<d\) and \(0<b<d\), we obtain that
and
Since the total curvature \(\Vert \Phi \Vert _{L^n(M)}\) is finite, we can choose a fixed \(r_0\) such that
Set
and
Thus,
for any \(\eta \in C^{\infty }_0(M-B_{r_0})\), where \(\tilde{C}\), \(\tilde{D}\), \(\tilde{E}\) and \(\tilde{F}\) are positive. Proposition 2.2 implies that
for any \(\eta \in C^{\infty }_0(M-B_{r_0})\) and any positive real number \(s\). Inequality (2.12) implies that
Combining with (2.14), we get
for any \(\eta \in C^{\infty }_0(M-B_{r_0})\). Choose a sufficiently large \(s\) such that \(n^2-\frac{(1+s)\tilde{D}}{\tilde{C}}<0\) and \(n^2-\frac{(1+s)\tilde{E}}{\tilde{C}}<0\). Then (2.15) implies that
for any \(\eta \in C^{\infty }_0(M-B_{r_0})\), where \(\tilde{A}\) is a positive constant depending only on \(n\). From now on, the proof follows standard techniques [for instance, see [1] after inequality (33)] and uses a Moser iteration argument and lemma 11 in [7]. We only include a concise proof here for the sake of completeness. Choose \(r>r_0+1\) and \(\eta \in C_0^{\infty }(M-B_{r_0})\) such that
for some positive constant \(c_1\). Then (2.16) becomes that
Letting \(r\rightarrow \infty \) and noting that \(|\omega |\in H^1(L^2(M))\), we obtain that
Combining with Hölder inequality , we have that
Let
Fix \(x\in M\) and take \(\tau \in C_0^1(B_1(x))\). (2.6) implies that
Note that
Combining with (2.19), we obtain that
By Cauchy–Schwarz inequality and (2.20), we have
where \(\mathcal {A}=\frac{p(p+1)}{4(p-1)}\le p<2n^2p\) and \(\mathcal {B}=p+1+(n-1)\mathcal {A}\le 1+np<2n^2p\). Choosing \(f=\tau h^{\frac{p}{2}}\) in Proposition 2.2 and combining with (2.21), we have
where \(\mathcal {C}=|H|^2+1+\Psi \). Let \(p_k=\frac{2n^k}{(n-2)^k}\) and \(\rho _k=\frac{1}{2}+\frac{1}{2^{k+1}}\) for \(k=0,1,2,\ldots \). Take a function \(\tau _k\in C_0^\infty (B_{\rho _k(x)})\) satisfying:
Choosing \(p=p_k\) and \(\tau =\tau _k\) in (2.22), we have
where \(k_0\) is a positive integer such that \(2C_0n^2(4^3+\sup _{B_1(x)}\mathcal {C})\le 4^{k_0}\). By recurrence, we have
where \(\mathcal {D}\) is a positive constant depending only on \(n\) and \(\sup _{B_1(x)}{\Psi }\). Letting \(k\rightarrow \infty \), we get
Now, choose \(y\in \overline{B}_{r_0+1}\) so that \(\sup _{B_{r_0+1}}h^2={h(y)}^2\). Note that \(B_1(y)\subset B_{r_0+2}\). (2.25) implies that
By (2.18), we have
where \(\mathcal {F}\) depends only on \(n\), \(Vol{(B_{r_0+2})}\) and \(\sup _{B_{r_0+2}}{\Psi }\). In order to show the finiteness of the dimension of \(H^1(L^2(M))\), it suffices to prove that the dimension of any finite dimensional subspaces of \(H^1(L^2(M))\) is bounded above by a fixed constant. By (2.26) and Lemma 11 in [7], we get \(\dim H^1(L^2(M))<+\infty \). By Proposition 2.1, we obtain that the number of non-parabolic ends of \(M\) is finite.
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Acknowledgments
Both authors would like to thank professors Hongyu Wang and Detang Zhou for useful suggestions. P. Zhu was partially supported by NSFC Grants 11101352, 11371309, Fund of Jiangsu University of Technology Grants KYY13005, KYY 13031 and Qing Lan Project. S. Fang was partially supported by the University Science Research Project of Jiangsu Province 13KJB110029 and the Fund of Yangzhou University 2013CXJ001.
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Zhu, P., Fang, S. Finiteness of non-parabolic ends on submanifolds in spheres. Ann Glob Anal Geom 46, 187–196 (2014). https://doi.org/10.1007/s10455-014-9418-0
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DOI: https://doi.org/10.1007/s10455-014-9418-0