Hypersurfaces with Lr-Pointwise 1-Type Gauss Map
In this paper, we study hypersurfaces in Еⁿ⁺¹ whose Gauss map G satisfies the equation LrG = f(G + C) for a smooth function f and a constant vector C, where Lr is the linearized operator of the (r+1)-st mean curvature of the hypersurface, i.e., Lr(f) = Tr(Pr ○∇²f) for f ∊ C∞(M), where Pr is the r-th...
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Akram Mohammadpouri 2019-02-01T18:31:07Z 2019-02-01T18:31:07Z 2018 Hypersurfaces with Lr-Pointwise 1-Type Gauss Map / Akram Mohammadpouri // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 1. — С. 67-77. — Бібліогр.: 23 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.01.067 Mathematics Subject Classification 2010: 53D02, 53C40, 53C42 https://nasplib.isofts.kiev.ua/handle/123456789/145859 In this paper, we study hypersurfaces in Еⁿ⁺¹ whose Gauss map G satisfies the equation LrG = f(G + C) for a smooth function f and a constant vector C, where Lr is the linearized operator of the (r+1)-st mean curvature of the hypersurface, i.e., Lr(f) = Tr(Pr ○∇²f) for f ∊ C∞(M), where Pr is the r-th Newton transformation, ∇²f is the Hessian of f, LrG = (LrG₁, . . . ,LrGn₊₁) and G = (G₁, . . . ,Gn₊₁). We focus on hypersurfaces with constant (r + 1)-st mean curvature and constant mean curvature. We obtain some classification and characterization theorems for these classes of hypersurfaces. У статтi вивчаються гiперповерхнi в Еⁿ⁺¹ гауссове вiдображення G яких задовольняє рiвняння LrG = f(G + C) для гладкої функцiї f i постiйного вектора C, де Lr є лiнеаризованим оператором (r + 1)-ої середньої кривизни гiперповерхнi, тобто Lr(f) = Tr(Pr ○∇²f) для f ∊ C∞(M), а Pr є r-им перетворенням Ньютона, ∇²f є гессiаном f, LrG = (LrG₁, . . . ,LrGn₊₁) i G = (G₁, . . . ,Gn₊₁). Наша увага зосереджена на гiперповерхнях з постiйною (r+1)-ою середньою кривизною i постiйною середньою кривизною. Для цих класiв гiперповерхонь отримано теореми класифiкацiЁ i характеризацiї. The author would like to thank gratefully the anonymous referee for useful comments. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Hypersurfaces with Lr-Pointwise 1-Type Gauss Map Гiперповерхнi з Lr-точковим типу 1 гауссовим вiдображенням Article published earlier |
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Hypersurfaces with Lr-Pointwise 1-Type Gauss Map |
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Hypersurfaces with Lr-Pointwise 1-Type Gauss Map Akram Mohammadpouri |
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Hypersurfaces with Lr-Pointwise 1-Type Gauss Map |
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Hypersurfaces with Lr-Pointwise 1-Type Gauss Map |
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Hypersurfaces with Lr-Pointwise 1-Type Gauss Map |
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hypersurfaces with lr-pointwise 1-type gauss map |
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Akram Mohammadpouri |
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Akram Mohammadpouri |
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Гiперповерхнi з Lr-точковим типу 1 гауссовим вiдображенням |
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In this paper, we study hypersurfaces in Еⁿ⁺¹ whose Gauss map G satisfies the equation LrG = f(G + C) for a smooth function f and a constant vector C, where Lr is the linearized operator of the (r+1)-st mean curvature of the hypersurface, i.e., Lr(f) = Tr(Pr ○∇²f) for f ∊ C∞(M), where Pr is the r-th Newton transformation, ∇²f is the Hessian of f, LrG = (LrG₁, . . . ,LrGn₊₁) and G = (G₁, . . . ,Gn₊₁). We focus on hypersurfaces with constant (r + 1)-st mean curvature and constant mean curvature. We obtain some classification and characterization theorems for these classes of hypersurfaces.
У статтi вивчаються гiперповерхнi в Еⁿ⁺¹ гауссове вiдображення G яких задовольняє рiвняння LrG = f(G + C) для гладкої функцiї f i постiйного вектора C, де Lr є лiнеаризованим оператором (r + 1)-ої середньої кривизни гiперповерхнi, тобто Lr(f) = Tr(Pr ○∇²f) для f ∊ C∞(M), а Pr є r-им перетворенням Ньютона, ∇²f є гессiаном f, LrG = (LrG₁, . . . ,LrGn₊₁) i G = (G₁, . . . ,Gn₊₁). Наша увага зосереджена на гiперповерхнях з постiйною (r+1)-ою середньою кривизною i постiйною середньою кривизною. Для цих класiв гiперповерхонь отримано теореми класифiкацiЁ i характеризацiї.
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1812-9471 |
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https://nasplib.isofts.kiev.ua/handle/123456789/145859 |
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Hypersurfaces with Lr-Pointwise 1-Type Gauss Map / Akram Mohammadpouri // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 1. — С. 67-77. — Бібліогр.: 23 назв. — англ. |
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AT akrammohammadpouri hypersurfaceswithlrpointwise1typegaussmap AT akrammohammadpouri giperpoverhnizlrtočkovimtipu1gaussovimvidobražennâm |
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2025-11-26T02:06:10Z |
| last_indexed |
2025-11-26T02:06:10Z |
| _version_ |
1850608060654944256 |
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Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 1, pp. 67–77
doi: https://doi.org/10.15407/mag14.01.067
Hypersurfaces with Lr-Pointwise 1-Type
Gauss Map
Akram Mohammadpouri
In this paper, we study hypersurfaces in En+1 whose Gauss map G sati-
sfies the equation LrG = f(G+ C) for a smooth function f and a constant
vector C, where Lr is the linearized operator of the (r+1)-st mean curvature
of the hypersurface, i.e., Lr(f) = Tr(Pr ◦ ∇2f) for f ∈ C∞(M), where
Pr is the r-th Newton transformation, ∇2f is the Hessian of f , LrG =
(LrG1, . . . , LrGn+1) and G = (G1, . . . , Gn+1). We focus on hypersurfaces
with constant (r + 1)-st mean curvature and constant mean curvature. We
obtain some classification and characterization theorems for these classes of
hypersurfaces.
Key words: linearized operators Lr, Lr-pointwise 1-type Gauss map, r-
minimal hypersurface.
Mathematical Subject Classification 2010: 53D02, 53C40, 53C42
1. Introduction
The study of submanifolds of finite type began in the late seventies with
B.Y. Chen’s attempts to find the best possible estimate of the total mean cur-
vature of compact submanifolds of a Euclidean space and to find a notion of
“degree” for submanifolds of a Euclidean space (see [8] for details). Since then
the subject has had a rapid development and many mathematicians contributed
to it (see the excellent survey of B.Y. Chen [6]). By definition, an isometrically
immersed submanifold x : Mn → En+k is said to be of finite type if x has a finite
decomposition as x− x0 =
∑p
i=1 xi, for some positive integer p, such that ∆xi =
λixi, λi ∈ R, 1 ≤ i ≤ p, x0 is constant, xi, 1 ≤ i ≤ p, are non-constant smooth
maps xi : Mn → En+k and ∆ is the Laplace operator of M . In [10], this definition
was similarly extended to differentiable maps, in particular, to the Gauss map of
hypersurfaces. The notion of finite type Gauss map is an especially useful tool in
the study of hypersurfaces (cf. [2–5,9, 12,16,19]). If an oriented hypersurface M
of a Euclidean space has a 1-type Gauss map G, then G satisfies ∆G = λ(G+C)
for a constant λ ∈ R and a constant vector C. In [10], Chen and Piccinni made
a general study on compact hypersurfaces of Euclidean spaces with finite type
Gauss map; they proved that a compact hypersurface M of En+1 has a 1-type
Gauss map G if and only if M is a hypersphere in En+1.
c© Akram Mohammadpouri, 2018
https://doi.org/10.15407/mag14.01.067
68 Akram Mohammadpouri
As is well known, the Laplace operator of a hypersurface M immersed into
En+1 is an (intrinsic) second-order linear differential operator which arises natu-
rally as the linearized operator of the first variation of the mean curvature for nor-
mal variations of the hypersurface. From this point of view, the Laplace operator
∆ can be seen as the first one of a sequence of n operators L0 = ∆, L1, . . . , Ln−1,
where Lr stands for the linearized operator of the first variation of the (r + 1)-
st mean curvature arising from normal variations of the hypersurface (see [22]).
These operators are given by Lr(f) = Tr(Pr ◦ ∇2f) for any f ∈ C∞(M), where
Pr denotes the r-th Newton transformation associated to the second fundamental
form of the hypersurface, and ∇2f is the Hessian of f (see the next section for
details).
From this point of view, S.M.B. Kashani introduced the notion of Lr-finite
type hypersurface in the Euclidean space [15], as an extension of the finite
type theory. One can find our results in the last section of the last chapter
of B.Y. Chen’s book [8].
Notice that sometimes the symbol � is used to denote the operator L1 which is
the Cheng–Yau operator introduced in [11]. Later, in [17], the notion of pointwise
1-type Gauss map for the surfaces of the Euclidean 3-space E3 was extended in
a natural way in terms of the Chen–Yau operator � as follows:
Definition 1.1. A surface M of the Euclidean space E3 is said to have an
L1-pointwise 1-type Gauss map if its Gauss map satisfies
�G = f(G+ C) (1.1)
for a smooth function f ∈ C∞(M) and a constant vector C ∈ E3. More precisely,
an L1-pointwise 1-type Gauss map is said to be of the first kind if (1.1) is satisfied
for C = 0; otherwise, it is said to be of the second kind. Moreover, if (1.1) is
satisfied for a constant function f , then we say that M has an L1-(global) 1-type
Gauss map.
Rotational, helicoidal and canal surfaces in E3 with L1-pointwise 1-type Gauss
map were studied in [18,21]. Motivated by this study, we define the hypersurfaces
with Lr-pointwise 1-type Gauss map in this paper. In Section 2, we give the
definition of a hypersurface with Lr-pointwise 1-type Gauss map and the basic
definitions of the theory of hypersurfaces in En+1. In Section 3, we focus on
the hypersurfaces with constant (r + 1)-st mean curvature and constant mean
curvature. We obtain some classification and characterization theorems for the
hypersurfaces with Lr-pointwise 1-type Gauss map.
2. Preliminaries
In this section, we recall the basic concepts of the theory of hypersurfaces [1].
Let x : Mn → En+1 be an isometrically immersed hypersurface in the Euclidean
space with Gauss map G. We denote by ∇0 and ∇ the Levi-Civita connections
on En+1 and Mn, respectively. The Gauss and Weingarten formulae are given
by ∇0
XY = ∇XY + 〈SX, Y 〉G and SX = −∇0
XG for all tangent vector fields
Hypersurfaces with Lr-Pointwise 1-Type Gauss Map 69
X,Y ∈ X (Mn), where S : X (Mn)→ X (Mn) is the shape operator (Weingarten
endomorphism) of Mn with respect to the Gauss map G.
As is well known, for every point p ∈ Mn, S defines a linear self-adjoint en-
domorphism on the tangent space TpM
n, and its eigenvalues λ1(p), λ2(p), . . .,
λn−1(p), λn(p) are the principal curvatures of the hypersurface. The characteris-
tic polynomial QS(t) of S is defined by
QS(t) = det(tI − S) = (t− λ1)(t− λ2) . . . (t− λn−1)(t− λn) =
n∑
k=0
(−1)kakt
n−k,
where ak is given by
ak =
∑
1≤i1<···<ik≤n
λi1 . . . λik , with a0 = 1.
The r-th mean curvature Hr of Mn in En+1 is defined by
(
n
r
)
Hr = ar, with
H0 = 1.
If Hr+1 = 0, then we say that Mn is an r-minimal hypersurface. The r-th
Newton transformation of Mn is the operator Pr : X (Mn)→ X (Mn) defined by
Pr =
r∑
j=0
(−1)j
(
n
r − j
)
Hr−jS
j =
r∑
j=0
(−1)jar−jS
j .
Equivalently,
P0 = I, Pr =
(
n
r
)
HrI − S ◦ Pr−1.
Along with each Newton transformation Pr, we consider the second-order lin-
ear differential operator Lr : C∞(Mn) → C∞(Mn) given by Lr(f) = Tr(Pr ◦
∇2f). Here, ∇2f : X (Mn) → X (Mn) denotes the self-adjoint linear operator
metrically equivalent to the Hessian of f and it is given by 〈∇2f(X), Y 〉 =
〈∇X(∇f), Y 〉, X,Y ∈ X (Mn).
Now we state the following lemma from [1], which we will need later.
Lemma 2.1. Let x : Mn → En+1 be a connected orientable hypersurface
immersed into the Euclidean space with Gauss map G. Then the Gauss map G
of M satisfies
LrG = −
(
n
r + 1
)
∇Hr+1 −
(
n
r + 1
)
(nH1Hr+1 − (n− r − 1)Hr+2)G. (2.1)
Next we will give the definition for a hypersurface with Lr-pointwise 1-type
Gauss map.
Definition 2.2. An oriented hypersurface M of a Euclidean space En+1 is
said to have an Lr-pointwise 1-type Gauss map if its Gauss map satisfies
LrG = f(G+ C) (2.2)
70 Akram Mohammadpouri
for a smooth function f ∈ C∞(M) and a constant vector C ∈ En+1. More
precisely, an Lr-pointwise 1-type Gauss map is said to be of the first kind if (2.2)
is satisfied for C = 0; otherwise, it is said to be of the second kind. Moreover, if
(2.2) is satisfied for a constant function f , then we say M has a (global) 1-type
Gauss map.
A function (or mapping) φ defined on M is said to be harmonic if its Laplacian
vanishes identically, i.e., if ∆φ = 0. After changing the Laplace operator ∆ by
the operator Lr, we give the following definition.
Definition 2.3. An oriented hypersurface M of a Euclidean space En+1 is
said to have an Lr-harmonic Gauss map if its Gauss map satisfies LrG = 0.
We also need the following remark, theorem and lemma for later use.
Remark 2.4 ([7]). A hypersurface of a Euclidean space En+1 is called isopara-
metric if its principal curvatures are constant counting multiplicities. An isopara-
metric hypersurface of En+1 has q distinct principal curvatures with q ≤ 2. If q =
2, one of principal curvatures must be 0. Isoparametric hypersurfaces of En+1
are locally hyperspheres, hyperplanes or a standard product embedding of Sk ×
En−k. This result was proved in [20] for n = 2, and in [23], for arbitrary n.
Theorem 2.5 ([14]). Let M3 be an oriented 3-dimensional complete Rie-
mannian manifold, and x : M3 → E4 be a minimal isometric immersion with
constant Gauss–Kronecker curvature. Then the Gauss–Kronecker curvature is
identically zero.
Lemma 2.6. Let M be an oriented hypersurface in En+1 with at most 2
distinct principal curvatures of multiplicities q and n − q (1 6 q 6 n). Suppose
that {e1, . . . , en} is an orthonormal frame corresponding to the principal directions
and the principal curvatures κ1, κ2 such that Sei = κ1ei, 1 6 i 6 q and Sej =
κ2ej, q + 1 6 j 6 n. If a vector field C ∈ C∞(M,En+1) is constant, then
ei(Cn+1) = −κ1Ci, 1 6 i 6 q, (2.3)
ei(Cn+1) = −κ2Ci, q + 1 6 i 6 n, (2.4)
where Ci = 〈C, ei〉 and Cn+1 = 〈C,G〉.
Proof. By the definition above, we have C =
∑n
i=1Ciei + Cn+1G. Suppose
that ωkij = 〈∇eiej , ek〉, 1 6 i, j, k 6 n; by a direct calculation, we have
∇0
eiC =
n∑
j=1
ei(Cj)ej + ei(Cn+1)G+
n∑
j=1
Cj∇0
eiej + Cn+1∇0
eiG
=
n∑
j=1
ei(Cj)ej + ei(Cn+1)G+
n∑
j,k=1
Cjω
k
ijek + CiκlG− Cn+1κlei,
where l = 1 if 1 6 i 6 q and l = 2 if q + 1 6 i 6 n. Since ∇0
eiC = 0, we get the
result.
Hypersurfaces with Lr-Pointwise 1-Type Gauss Map 71
3. Characterization theorems on hypersurfaces with Lr-point-
wise 1-type Gauss map
In this section, we will give some characterization theorems on the hypersur-
faces of En+1 in terms of their Gauss map. We focus on the hypersurfaces with
constant (r + 1)-st mean curvature and on hypersurfaces with constant mean
curvature.
3.1. Hypersurfaces with constant (r + 1)-st mean curvature.
Theorem 3.1. If an oriented hypersurface M of a Euclidean space En+1 has
Lr-harmonic Gauss map, then the (r + 1)-st mean curvature of M is constant,
in particular, if n = r + 1, then M is minimal or (n− 1)-minimal, i.e., Hn = 0.
Proof. By Lemma 2.1, M has the Lr-harmonic Gauss map if the (r + 1)-st
mean curvature of M is constant. If n = r + 1, then Lemma 2.1 implies that
HHn = 0, hence M is minimal or (n− 1)-minimal.
In particular, when n = 2, we deduce from Theorem 3.1 and Remark 2.4 the
following corollary proved by Kim and Turgay in [17].
Corollary 3.2. An oriented surface M in E3 has an L1-harmonic Gauss
map if and only if it is flat, i.e., its Gaussian curvature vanishes identically.
From Theorem 3.1 and Theorem 2.5 we can easily deduce the following corol-
lary.
Corollary 3.3. If an oriented complete hypersurface M in E4 has an L2-
harmonic Gauss map, then M is 2-minimal.
In [10], Chen and Piccinni proved that there is no compact hypersurface in
En+1 with harmonic Gauss map. To extend this result to the case of Lr-harmonic
Gauss map, we state and prove the following theorem.
Theorem 3.4. There is no compact hypersurface in En+1 with Lr-harmonic
Gauss map.
Proof. Let M be a compact hypersurface in En+1 with Lr-harmonic Gauss
map. By Lemma 2.1, the (r + 1)-st mean curvature of M is constant. It is well
known that every compact hypersurface in a Euclidean space has elliptic points,
that is, the points where all the principal curvatures are positive (or negative).
In particular, this implies that there exists no compact hypersurface in En+1 with
vanishing (r + 1)-st mean curvature for every r = 0, . . . , n − 1. Since Mn has
elliptic points, after an appropriate choice of the Gauss map G of Mn, if r is
odd, we can suppose that Hr+1 > 0. Also, if Hr+1 > 0, then Hj > 0 for all j =
1, . . . , r. Moreover,
Hi−1Hi+1 ≤ H2
i and H1 ≥ H1/2
2 ≥ H1/3
3 ≥ · · · ≥ H1/i
i , i = 1, . . . , r
72 Akram Mohammadpouri
(see page 52 of [13]). Thus, the above inequalities yield
H1Hr+1 ≥ Hr+2. (3.1)
On the other hand, by using formula (2.1), when r = n−1, we get H1 = 0, which
is impossible. When r < n − 1, we have H1Hr+1 < Hr+2, which contradicts
(3.1).
Using Definition 2.2 and equation (2.1), we now state the following theorems
which characterize the hypersurfaces of Euclidean spaces with Lr-1-type Gauss
map of the first kind.
Theorem 3.5. An oriented hypersurface M in En+1 has an Lr-pointwise 1-
type Gauss map of the first kind if and only if it has a constant (r + 1)-st mean
curvature.
Theorem 3.6. An oriented hypersurface M in En+1 has an Lr-(global) 1-
type Gauss map of the first kind if and only if both Hr+1 and nH1Hr+1 − (n −
r − 1)Hr+2 are constant.
We can deduce the following corollary on hypersurfaces with Lr-1-type Gauss
map.
Corollary 3.7. All oriented isoparametric hypersurfaces of a Euclidean space
En+1 have an Lr-(global) 1-type Gauss map.
So, by Remark 2.4, hyperplanes, hyperspheres and the generalized cylinder
Sn−k × Ek of En+1 have the Lr-1-type Gauss map. We can also state some
characterization corollaries about hypersurfaces with at most 2 distinct principal
curvatures.
Corollary 3.8. An oriented hypersurface M in En+1 with at most 2 distinct
principal curvatures has an Lr-(global) 1-type Gauss map of the first kind, where
n 6= r + 1, if and only if it is an open domain of a hypersphere, a hyperplane or
a generalized cylinder.
Proof. By Theorem 3.6, we conclude that M has the Lr-(global) 1-type Gauss
map of the first kind if and only if it is isoparametric, and thus Remark 2.4 gives
the result.
Corollary 3.9. An oriented hypersurface M in En+1 with at most 2 distinct
principal curvatures has an Ln−1-(global) 1-type Gauss map of the first kind if
and only if it is either an (n− 1)-minimal hypersurface or an open domain of a
hypersphere, a hyperplane or a generalized cylinder.
Proof. By Theorem 3.6, M has the Ln−1-(global) 1-type Gauss map of the
first kind if and only if Hn and HHn are constant. If Hn 6= 0, then H is constant.
Therefore, M is isoparametric, so, Remark 2.4 gives the result.
Hypersurfaces with Lr-Pointwise 1-Type Gauss Map 73
In particular, when n = 2, we have the following result that was also proved
by Kim and Turgay in [17].
Corollary 3.10. An oriented surface M in E3 has an L1-(global) 1-type
Gauss map of the first kind if and only if it is either a flat surface or an open
domain of a sphere.
3.2. Hypersurfaces with constant mean curvature. Now, we study
the hypersurfaces in En+1 with at most 2 distinct principal curvatures and a
constant mean curvature which have an Lr-pointwise 1-type Gauss map. First,
we focus on minimal hypersurfaces.
Corollary 3.11. An oriented minimal hypersurface M in En+1 with at most
2 distinct principal curvatures has an Lr-pointwise 1-type Gauss map of the first
kind if and only if it is an open domain of a hyperplane.
Proof. The proof follows directly from Theorem 3.5 and Remark 2.4.
In particular, the following result follows directly from Corollary 3.11 that
was proved for minimal surfaces by Kim and Turgay [17].
Corollary 3.12. An oriented minimal surface M in E3 has an L1-pointwise
1-type Gauss map of the first kind if and only if it is an open domain of a plane.
Next, we prove the following proposition.
Proposition 3.13. Let M be a connected orientable hypersurface in En+1
with at most 2 distinct principal curvatures. Suppose that nH1Hr+1 = (n − r −
1)Hr+2. Then M has an Lr-pointwise 1-type Gauss map of the second kind if
and only if it is an open domain of a hyperplane.
Proof. Let M be a connected orientable hypersurface in En+1 with at most
2 distinct principal curvatures of multiplicities q and n− q, 1 6 q 6 n. If M has
an Lr-pointwise 1-type Gauss map of the second kind, then (2.2) is satisfied for
a constant vector C and a smooth function f . Let O = {p ∈M | f(p) 6= 0}. We
now suppose O 6= ∅. Since nH1Hr+1 = (n − r − 1)Hr+2, (2.1) and (2.2) imply
f(G+C) = −
(
n
r+1
)
∇Hr+1. Therefore, we have Cn+1 = 〈C,G〉 = −1 on O. Thus,
from (2.3) and (2.4), we obtain
κ1C1 = · · · = κ1Cq = κ2Cq+1 = · · · = κ2Cn = 0 on O.
Let O1 = {p ∈ O | κ1κ2(p) 6= 0}. Then, C1 = · · · = Cn = 0 on O1. Thus, the
constant vector C = −G on O1 and thus O1 is a part of a hyperplane, which
is a contradiction. Therefore, we have O1 = ∅, which implies κ1κ2 = 0. Since
nH1Hr+1 = (n − r − 1)Hr+2, O is an open domain of a hyperplane. Moreover,
by the continuity, we have M = O.
Conversely, suppose M is an open domain of a hyperplane. Then its Gauss
map G is a non-zero constant vector, which implies LrG = 0. Therefore, (2.2) is
satisfied for C = −G 6= 0 and an arbitrary smooth function f . Hence, M has the
Lr-pointwise 1-type Gauss map of the second kind.
74 Akram Mohammadpouri
By combining Corollary 3.11 and Proposition 3.13, we obtain.
Corollary 3.14. An oriented connected minimal hypersurface M in En+1
with at most 2 distinct principal curvatures has an Ln−1-pointwise 1-type Gauss
map if and only if it is an open domain of a hyperplane.
In particular, when n = 2, by combining Corollary 3.12 and Proposition 3.13,
we obtain the following corollary proved by Kim and Turgay in [17].
Corollary 3.15. An oriented connected minimal surface M in E3 has an
L1-pointwise 1-type Gauss map if and only if it is an open domain of a plane.
Next, we give a complete classification of hypersurfaces with constant mean
curvature and at most 2 distinct principal curvatures whose Gauss map satisfies
LrG = λ(G+ C) for a constant λ and a constant vector C.
Theorem 3.16. Let M be a hypersurface with constant mean curvature and
at most 2 distinct principal curvatures in En+1. Then M has an Lr-(global) 1-
type Gauss map if and only if it is an open domain of a hypersphere, a hyperplane
or a generalized cylinder.
Proof. Let M be an oriented hypersurface in En+1 with at most 2 distinct
principal curvatures of multiplicities q and n − q, 1 6 q 6 n. Suppose that
{e1, . . . , en} is an orthonormal frame of its principal directions to the principal
curvatures κ1, κ2 such that Sei = κ1ei, 1 6 i 6 q, Sej = κ2ej , q + 1 6 j 6 n.
Let us consider an open set U = {p ∈ M : ∇Hr+1(p) 6= 0}. Our objective is to
show that U is empty. Since M has a constant mean curvature, we have qκ1 +
(n− q)κ2 = h0 for a constant h0, which implies
ei(κ1) =
q − n
q
ei(κ2), i = 1, . . . , n. (3.2)
Now we suppose that M has an Lr-(global) 1-type Gauss map. Therefore, from
(2.1) and (2.2), we obtain
−∇Hr+1 − (h0Hr+1 − (n− r − 1)Hr+2)G = λ(G+ C). (3.3)
From (3.2), we conclude that there exist polynomials f and g with constant
coefficients such that
ei(Hr+1) = f(κ1)ei(κ1), ei(Hr+2) = g(κ1)ei(κ1), i = 1, . . . , n. (3.4)
From (3.2)–(3.4), we get
λCi = −ei(Hr+1) = ei(κ1)f(κ1), i = 1, . . . , n, (3.5)
λ(Cn+1 + 1) = −h0Hr+1 + (n− r − 1)Hr+2. (3.6)
By using (3.4) and (3.6), we obtain
λei(Cn+1 + 1) = (−h0f(κ1) + (n− r − 1)g(κ1))ei(κ1).
Hypersurfaces with Lr-Pointwise 1-Type Gauss Map 75
Therefore, from (2.3) and (3.5), we get
λCi
[
−κ1 + h0 − (n− r − 1)
g(κ1)
f(κ1)
]
= 0, i = 1, . . . , q, on U, (3.7)
λCi
[
−κ2 + h0 − (n− r − 1)
g(κ1)
f(κ1)
]
= 0, i = q + 1, . . . , n, on U. (3.8)
Note that if λ = 0, then we have LrG = 0, and it implies that the (r+1)-st mean
curvature is constant on U , which is a contradiction. If κ1 = h0− (n− r−1) g(κ1)f(κ1)
or κ2 = h0 − (n− r − 1) g(κ1)f(κ1)
, we conclude that Hr+1 is constant on U , which is
a contradiction. Therefore, C1 = · · · = Cn = 0 on U . Thus, (3.5) implies that
Hr+1 is constant on U , which is a contradiction. Hence, U is empty and Hr+1
is constant on M . Since M has a constant mean curvature, we get that M is
isoparametric, and therefore Remark 2.4 gives the result.
In particular, when n = 2, we obtain the following corollary that was proved
by Kim and Turgay in [17].
Corollary 3.17. Let M be a surface with constant mean curvature in E3.
Then M has an L1-(global) 1-type Gauss map if and only if it is an open domain
of a sphere, a plane or a cylinder.
Acknowledgment. The author would like to thank gratefully the anony-
mous referee for useful comments.
Supports. This research was partially supported by the University of
Tabriz, Iran, under Grant No. 5221.
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Received March 9, 2016, revised December 15, 2016.
Akram Mohammadpouri,
University of Tabriz, Department of Pure Mathematics, Faculty of Mathematical Sci-
ences, Tabriz, Iran,
E-mail: pouri@tabrizu.ac.ir
mailto:pouri@tabrizu.ac.ir
Hypersurfaces with Lr-Pointwise 1-Type Gauss Map 77
Гiперповерхнi з Lr-точковим типу 1 гауссовим
вiдображенням
Akram Mohammadpouri
У статтi вивчаються гiперповерхнi в En+1, гауссове вiдображення
G яких задовольняє рiвняння LrG = f(G + C) для гладкої функцiї f
i постiйного вектора C, де Lr є лiнеаризованим оператором (r + 1)-ої
середньої кривизни гiперповерхнi, тобто Lr(f) = Tr(Pr ◦ ∇2f) для f ∈
C∞(M), а Pr є r-им перетворенням Ньютона, ∇2f є гессiаном f , LrG =
(LrG1, . . . , LrGn+1) i G = (G1, . . . , Gn+1). Наша увага зосереджена на
гiперповерхнях з постiйною (r+1)-ою середньою кривизною i постiйною
середньою кривизною. Для цих класiв гiперповерхонь отримано теореми
класифiкацiї i характеризацiї.
Ключовi слова: лiнеаризованi оператори Lr, Lr-точкове типу 1 гаус-
сове вiдображення, r-мiнiмальна гiперповерхня.
Introduction
Preliminaries
Characterization theorems on hypersurfaces with Lr-pointwise 1-type Gauss map
Hypersurfaces with constant (r+1)-st mean curvature.
Hypersurfaces with constant mean curvature.
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