First eigenvalue of the Laplace operator and mean curvature
The main theorem of this paper states a relation between the first nonzero eigenvalue of the Laplace operator and the squared norm of mean curvature in irreducible compact homogeneous manifolds under spatial conditions. This statement has some consequences presented in the remainder of paper.
Gespeichert in:
| Datum: | 2008 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3217 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509269254733824 |
|---|---|
| author | Etemad, Dehkordy A. Етемад, Дегкорді А. |
| author_facet | Etemad, Dehkordy A. Етемад, Дегкорді А. |
| author_sort | Etemad, Dehkordy A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:23Z |
| description | The main theorem of this paper states a relation between the first nonzero eigenvalue of the Laplace operator and the squared norm of mean curvature in irreducible compact homogeneous manifolds under spatial conditions. This statement has some consequences presented in the remainder of paper. |
| first_indexed | 2026-03-24T02:38:25Z |
| format | Article |
| fulltext |
UDC 517.5
A. Etemad (Isfahan Univ. Technology, Iran)
FIRST EIGENVALUE OF THE LAPLACE OPERATOR
AND MEAN CURVATURE*
ПЕРШЕ ВЛАСНЕ ЗНАЧЕННЯ ОПЕРАТОРА ЛАПЛАСА
ТА СЕРЕДНЯ КРИВИНА
The main theorem of this paper states a relation between the first nonzero eigenvalue of Laplace operator
and the squared norm of mean curvature in irreducible compact homogeneous manifolds under spatial
conditions. This statement has some results that states in the remainder of paper.
Основна теорема цiєї статтi встановлює зв’язок мiж першим ненульовим власним значенням опе-
ратора Лапласа та нормою середньої кривини у квадратi у незвiдних компактних однорiдних мно-
говидах пiд дiєю просторових умов. Одержано також деякi iншi результати.
1. Introduction. Let M be an n-dimensional Riemannian manifold and p ∈ M. For
an orthonormal basis {e1, e2, . . . , en} of the tangent space TpM, the scalar curvature S
of p is defined to be S(p) =
∑
i<j
K(ei, ej), where K(ei, ej) is sectional curvature of
M associated with tangent plane generated by ej and ej at P. Let S(L) be the scalar
curvature of L, where L is a subspace of TpM of dimension r < n. Thus, the scalar
curvature S(M) of M at p is nothing but the scalar curvature tangent space of M at p
and if L is a 2-plane section, S(L) is nothing but the sectional curvature of L.
For an integer k ≥ 0, denote by γ(n, k) the finite set consisting of k-tuples
(n1, n2, . . . , nk) of integers grater than 1 satisfying n1 ≥ 2 and n1 + . . . + nk ≤ n.
Denote by γ(n) the set of k-tuples with k ≥ 0 for fixed n.
The cardinal number #γ(n) of γ(n) increases quite rapidly with n. For each k-tuples
(n1, n2, . . . , nk), we define an invariant δ(n1, . . . , nk) by
δ(n1, . . . , nk)(p) = S(p)− inf(S(L1) + . . . + S(Lk)),
where L1, . . . , Lk run over all k mutually orthogonal subspaces of TpM such that
dim Li = ni, i = 1, . . . , k. In particular, we have δ(φ) = S, δ(2) = S − inf K. The
invariants δ(n1, . . . , nk) with k > 0 and the scalar curvature S are very different in
nature.
For each k-tuples (n1, n2, . . . , nk) with in γ(n), we define following two positive
numbers that are used in the following:
c(n1, n2, . . . , nk) =
n2(n + k − 1−
∑
j
nj)
2(n + k −
∑
j
nj)
,
b(n1, n2, . . . , nk) =
1
2
n(n− 1)−
k∑
j=1
nj(nj − 1)
.
* This work was partially supported by IUT (CEAMA).
c© A. ETEMAD, 2008
1000 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 7
FIRST EIGENVALUE OF THE LAPLACE OPERATOR AND MEAN CURVATURE 1001
Now an isometric immersion f : Mn → Rm(c) from a Riemannian n-manifold into a
Riemannian space form of constant curvature c is called an ideal immersion if
δ(n1, n2, . . . , nk) = c(n1, . . . , nk)|H|2 + b(n1, . . . , nk)c
for some k-tuples (n1, n2, . . . , nk) in γ(n).
2. Main theorem. In this section, the main theorem is stated. This theorem
demonstrates a relation between the first nonzero eigenvalue of Laplace operator and
squared norm of mean curvature in a manifold with certain properties.
We define an ideal submanifold as a submanifold whose inclusion map is an ideal
immersion.
In the following, suppose that the first nonzero eigenvalue of Laplacian operator of
a manifold is denoted by λ1 and the tensor of mean curvature is denoted by H.
Theorem 2.1. Let M be an n-dimensional irreducible compact homogeneous
Riemannian manifold that is also an ideal submanifold of a space form Rm(c) with
c ≥ 0. Then
λ1 ≥ n|H|2,
where |H|2 is the squared norm of the mean curvature of M.
Proof. Since M is an n-dimensional submanifold in a Riemannian space form
of constant curvature c, by [1], at every point p ∈ M and for each of k-tuples
(n1, n2, . . . , nk) in γ(n), we have
δ(n1, n2, . . . , nk) ≤ c(n1, n2, . . . , nk)|H|2 + b(n1, n2, . . . , nk)c.
Since M is an irreducible compact homogeneous Riemannian n-manifold, again by
[1], for any of k-tuples (n1, n2, . . . , nk) in γ(n), λ1 satisfies
λ1 ≥ n∆(n1, . . . , nk),
where
∆(n1, . . . , nk) =
δ(n1, . . . , nk)
c(n1, . . . , nk)
.
Therefore, for a Riemannian n-manifold with an ideal immersion into a space form
with nonnegative constant curvature, we have ∆(n1, . . . , nk) ≥ |H|2 for some k-tuples
(n1, n2, . . . , nk) in γ(n). This enables us to have result.
From now on, let us denote an n-dimensional Riemannian irreducible compact and
homogeneous manifold by nRich-manifold.
Other direct applications of this theorem can be found in the following corollaries:
Corollary 2.1. Let M be a nRich-manifold that is also an ideal submanifold of a
space form Rm(c) with c ≥ 0. Then∫
M
|H|2dV ≤ λ1V
n
, (2.1)
where V is the volume of M. If c = 0, we have an equality in (2.1).
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 7
1002 A. ETEMAD
Proof. The first part is clear by Theorem 2.1. For the second part, by the Reilly
formula (see [2]), λ1 has also the upper bound
n
V (M)
∫
M
|H|2dV, so the equality is
obtained.
The Remark 2.1 as follows is needed for the next corollary and other results in the
rest.
Remark 2.1 [1]. Every totally umbilical submanifold of a real space form (Eucli-
dean spaces, real hyperbolic spaces, spheres and real projective spaces) is an ideal
immersion.
We also denote a k-dimensional Euclidean space and a k-dimensional Euclidean
sphere by Rk and Sk, respectively.
Corollary 2.2. Let M be an irreducible compact homogeneous surface of unit
sphere S3 such that the length of its mean curvature is constant the H0. If M is a
topological sphere, then
λ1 ≥ nH2
0 .
Proof. By [3], with this conditions M is totally umbilic, so by Remark 2.1, M is
an ideal hypersurface of Sn+1. Thus, this result is a spacial case of Theorem 2.1.
Using Theorem 2.1, each of following results states a relation between the nonzero
eigenvalue of Laplace operator and circumradius of certain submanifolds of Rn+1 and
Sn+1.
First, we need the definition of a circumradius for a manifold. For a given immersion
x : Mn → Rn+p or Sn+p, where Mn is an n-dimensional manifold, the circumradius
of Mn denoted by r = r(M) is the radius of the smallest closed ball containing x(M).
Theorem 2.2. Let M be an nRich-manifold that is also an ideal hypersurface of
Rn+1, n ≥ 2. In this case,
λ1 ≥
n
r2
.
Proof. Let x : Mn → Rn+1 be an ideal (inclusion) immersion. Then by [4] for x,
we have ∫
M
|H|2dV ≥ V
r2
. (2.3)
So, the result may be obtained by Theorem 2.1.
Theorem 2.3. Suppose that M is an nRich hypersurface of the unit sphere Sn+1,
n ≥ 2, with following conditions:
i) M is of constant scalar curvature S ≥ 0 and, in the case S = 0, the sign of mean
curvature is unchanged,
ii) Gaussian image of M is in a closed hemisphere of Sn+1, whence λ1 ≥
n
r2
.
Proof. By [5], conditions i) and ii) deduce that M is totally umbilic. So, by
Remark 2.1, M is an ideal hypersurface of Sn+1 and, therefore, the result follows from
Theorem 2.1.
Corollary 2.3. Let M be an nRich hypersurface of Rn+1 such that, for a positive
scalar a, the scalar curvature of M satisfies in S = a|H|. In this case,
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 7
FIRST EIGENVALUE OF THE LAPLACE OPERATOR AND MEAN CURVATURE 1003
λ1 ≥
n
r2
.
Proof. By Corollary 2.5 [6, p. 13], M is totally umbilic and so, by Remark 1.3, M
is an ideal hypersurface of Rn+1. Therefore, the result follows from Theorem 2.1.
1. Chen B. Y. Some new obstructions to minimal and Lagrangian isometric immersions // Jap. J. Math.
– 2000. – 26. – P. 105 – 127.
2. Colbois B., Grosjean J. F. A pinching theorem for the first eigenvalue of the Laplacian on
hypersurfaces of the Euclidean space. – Prepublication, 2003.
3. Hsu Y. J., Wang T. H. A global pinching theorem for surfaces with constant mean curvature in S3
// Proc. Amer. Math. Soc. – 2002. – 130. – P. 157 – 161.
4. Rotondaro G. On total curvature of immersions and minimal submanifolds of spheres // Comment.
math. Univ. carol. – 1993. – 3, № 34. – P. 459 – 463.
5. Alencar H., Rosenberg H., Santos W. On the Gauss map of hypersurfaces with constant scalar
curvature in spheres // Proc. Amer. Math. Soc. – 2004. – 12. – P. 3731 – 3739.
6. Li H. Global rigidity theorems of hypersurfaces // Ark. Mat. – 1997. – 35. – P. 327 – 351.
Received 22.02.07,
after revision — 12.02.08
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 7
|
| id | umjimathkievua-article-3217 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:38:25Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/22/b76316ad9b5d357e256df54368acef22.pdf |
| spelling | umjimathkievua-article-32172020-03-18T19:48:23Z First eigenvalue of the Laplace operator and mean curvature Перше власне значення оператора Лапласа та середня кривина Etemad, Dehkordy A. Етемад, Дегкорді А. The main theorem of this paper states a relation between the first nonzero eigenvalue of the Laplace operator and the squared norm of mean curvature in irreducible compact homogeneous manifolds under spatial conditions. This statement has some consequences presented in the remainder of paper. Основна теорема цієї статті встановлює зв'язок між першим ненульовим власним значенням оператора Лапласа та нормою середньої кривини у квадраті у незвідних компактних однорідних мно-говидах під дією просторових умов. Одержано також деякі інші результати. Institute of Mathematics, NAS of Ukraine 2008-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3217 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 7 (2008); 1000–1003 Український математичний журнал; Том 60 № 7 (2008); 1000–1003 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3217/3180 https://umj.imath.kiev.ua/index.php/umj/article/view/3217/3181 Copyright (c) 2008 Etemad Dehkordy A. |
| spellingShingle | Etemad, Dehkordy A. Етемад, Дегкорді А. First eigenvalue of the Laplace operator and mean curvature |
| title | First eigenvalue of the Laplace operator and mean curvature |
| title_alt | Перше власне значення оператора Лапласа та середня кривина |
| title_full | First eigenvalue of the Laplace operator and mean curvature |
| title_fullStr | First eigenvalue of the Laplace operator and mean curvature |
| title_full_unstemmed | First eigenvalue of the Laplace operator and mean curvature |
| title_short | First eigenvalue of the Laplace operator and mean curvature |
| title_sort | first eigenvalue of the laplace operator and mean curvature |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3217 |
| work_keys_str_mv | AT etemaddehkordya firsteigenvalueofthelaplaceoperatorandmeancurvature AT etemaddegkordía firsteigenvalueofthelaplaceoperatorandmeancurvature AT etemaddehkordya perševlasneznačennâoperatoralaplasataserednâkrivina AT etemaddegkordía perševlasneznačennâoperatoralaplasataserednâkrivina |