On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁
One primary objective in submanifold geometry is to discover fascinating and significant classical examples of ₁. In this paper, which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant -m...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2025 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212880 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁. Hung-Lin Chiu, Sin-Hua Lai and Hsiao-Fan Liu. SIGMA 21 (2025), 011, 25 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860297428175945728 |
|---|---|
| author | Chiu, Hung-Lin Lai, Sin-Hua Liu, Hsiao-Fan |
| author_facet | Chiu, Hung-Lin Lai, Sin-Hua Liu, Hsiao-Fan |
| citation_txt | On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁. Hung-Lin Chiu, Sin-Hua Lai and Hsiao-Fan Liu. SIGMA 21 (2025), 011, 25 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | One primary objective in submanifold geometry is to discover fascinating and significant classical examples of ₁. In this paper, which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant -mean curvature surfaces, we have identified intriguing examples of such surfaces. Notably, we present a complete description of rotationally invariant surfaces of constant p-mean curvature and shed light on the geometric interpretation of the energy with a lower bound.
|
| first_indexed | 2026-03-21T18:31:18Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 011, 25 pages
On Invariants of Constant p-Mean Curvature Surfaces
in the Heisenberg Group H1
Hung-Lin CHIU ab, Sin-Hua LAI c and Hsiao-Fan LIU de
a) Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan
E-mail: hlchiu@math.nthu.edu.tw
b) National Center for Theoretical Sciences, Taipei, Taiwan
c) Fundamental Education Center, National Chin-Yi University of Technology,
Taichung, Taiwan
E-mail: shlai@ncut.edu.tw
d) Department of Applied Mathematics and Data Science, Tamkang University,
New Taipei City, Taiwan
E-mail: hfliu@mail.tku.edu.tw
e) Department of Applied Mathematics, National Chung Hsing University, Taiwan
Received April 15, 2024, in final form February 04, 2025; Published online February 18, 2025
https://doi.org/10.3842/SIGMA.2025.011
Abstract. One primary objective in submanifold geometry is to discover fascinating and
significant classical examples ofH1. In this paper which relies on the theory we established in
[Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we
provided for constructing constant p-mean curvature surfaces, we have identified intriguing
examples of such surfaces. Notably, we present a complete description of rotationally invari-
ant surfaces of constant p-mean curvature and shed light on the geometric interpretation of
the energy E with a lower bound.
Key words: Heisenberg group; Pansu sphere; p-minimal surface; Codazzi-like equation; ro-
tationally invariant surface
2020 Mathematics Subject Classification: 53A10; 53C42; 53C22; 34A26
1 Introduction
This article is an extension of the previous paper [5], in which we studied the constant p-mean
curvature surfaces in the Heisenberg group H1. In [5], we focused on the foundation of the
theory and paid more attention to the investigation of p-minimal surfaces. However, in the
present article, instead of theory, we mainly focus on the examples, including an approach to
construct constant p-mean curvature surfaces.
Recall that the Heisenberg group H1 is the space R3 with the associated group multiplication
(x1, y1, z1) ◦ (x2, y2, z2) = (x1 + x2, y1 + y2, z1 + z2 + y1x2 − x1y2),
which is a 3-dimensional Lie group. The space of all left-invariant vector fields is spanned by
the following three vector fields:
e̊1 =
∂
∂x
+ y
∂
∂z
, e̊2 =
∂
∂y
− x
∂
∂z
and T =
∂
∂z
.
The Heisenberg dilation (scaling) by the factor δ > 0 is the map Dδ : H1 → H1 defined
by Dδ(x, y, z) =
(
δx, δy, δ2z
)
for any (x, y, z) ∈ H1 (see [6]).
mailto:hlchiu@math.nthu.edu.tw
mailto:shlai@ncut.edu.tw
mailto:hfliu@mail.tku.edu.tw
https://doi.org/10.3842/SIGMA.2025.011
2 H.-L. Chiu, S.-H. Lai and H.-F. Liu
The standard contact bundle on H1 is the subbundle ξ of the tangent bundle TH1 spanned
by e̊1 and e̊2. It is also defined to be the kernel of the contact form Θ = dz + xdy − ydx. The
CR structure on H1 is the endomorphism J : ξ → ξ defined by J (̊e1) = e̊2 and J (̊e2) = −e̊1.
One can view H1 as a pseudo-hermitian manifold with (J,Θ) as the standard pseudo-hermitian
structure. There is a naturally associated connection ∇ if we regard all these left-invariant
vector fields e̊1, e̊2, and T as parallel vector fields. A naturally associated metric on H1 is the
adapted metric gΘ, which is defined by gΘ = dΘ(·, J ·) + Θ2. It is equivalent to defining the
metric regarding e̊1, e̊2, and T as an orthonormal frame field. We sometimes use ⟨·, ·⟩ to denote
the adapted metric. In this paper, we use the adapted metric to measure the lengths, angles of
vectors, and so on.
Suppose Σ is a surface in the Heisenberg group H1. There is a one-form I on Σ induced from
the adapted metric gΘ. This induced metric is defined on the whole surface Σ and is called the
first fundamental form of Σ. The intersection TΣ∩ξ is integrated to be a singular foliation on Σ
called the characteristic foliation. Each leaf is called a characteristic curve. A point p ∈ Σ is
called a singular point if the tangent plane TpΣ coincides with the contact plane ξp; otherwise,
p is called a regular (or non-singular) point. Generically, a point p ∈ Σ is a regular point, and
the set of all regular points is called the regular part of Σ. In this paper, we always assume that
the surface Σ is of class C2, but of class C∞ on the regular part. On the regular part, we can
choose a unit vector field e1 such that e1 defines the characteristic foliation. The vector e1 is
determined up to a sign. Let e2 = Je1. Then {e1, e2} forms an orthonormal frame field of the
contact bundle ξ. We usually call the vector field e2 a horizontal normal vector field. Then the
p-mean curvature H of the surface Σ is defined by ∇e1e2 = −He1. The p-mean curvature H is
only defined on the regular part of Σ. If H = c, which is a constant on the whole regular part,
we call the surface a constant p-mean curvature surface. In particular, if c = 0, it is a p-minimal
surface. There also exists a function α defined on the regular part such that αe2+T is tangent to
the surface Σ. We call this function the α-function of Σ. It is uniquely determined up to a sign,
which depends on the choice of the characteristic direction e1. Define ê1 = e1 and ê2 =
αe2+T√
1+α2
,
then {ê1, ê2} forms an orthonormal frame field of the tangent bundle TΣ. Notice that ê2 is
uniquely determined and independent of the choice of the characteristic direction e1. In [3, 4],
it was shown that these three invariants, I, e1, and α, form a complete set of invariants for
constant p-mean curvature surfaces with H = c in H1. Namely, for any two surfaces with the
same constant p-mean curvature having the same I, α, e1, they are differed only by a Heisenberg
symmetry. In particular, if Σ ⊂ H1 is a constant p-mean curvature surface with H = c, then
in terms of a compatible coordinate system (U ;x, y), which means e1 = ∂
∂x , the integrability
condition (see [5]) is reduced to
−ax + a
bx
b
=
cα(
1 + α2
)1/2 , −bx
b
= 2α+
ααx
1 + α2
,
αxx + 6ααx + 4α3 + c2α = 0, (1.1)
where the two functions a and b are a representation of the first fundamental form I in the
following sense that they describe the vector field
ê2 = a(x, y)
∂
∂x
+ b(x, y)
∂
∂y
.
In other words, there exists the α satisfying the Codazzi-like equation
αxx + 6ααx + 4α3 + c2α = 0, (1.2)
which is a nonlinear ordinary differential equation. In [5], we normalized a and b such that
they can be uniquely determined by the function α, and hence we obtained the result that the
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 3
existence of a constant p-mean curvature surface (without singular points) is equivalent to the ex-
istence of a solution to a nonlinear second-order ODE (1.2), which is a kind of Liénard equations
(cf. [7]). They are one-to-one correspondences in some sense. For a detailed description, see [5,
Theorems 1.1, 1.3 and 6.3]. This result tells us that the investigation of the geometry of con-
stant p-mean curvature surfaces in H1 is equal to the study of the solution of the equation (1.2).
More specifically, we obtained a complete set of solutions (see [5, Theorems 1.2 and 1.4] or
Theorem 2.1) and used the types of the solutions to the equation to characterize the constant
p-mean curvature surfaces as several classes, which are vertical, special type I, special type II and
general type (see [5, Definitions 5.1 and 5.2] for p-minimal cases and see Definitions 2.2 and 2.3
for the cases with c ̸= 0 in the present article). After the process of normalization, we obtained
a complete set of invariants from the normal form of the α-function. It is worth of our mention
that these invariants in some sense measure how different a constant p-mean curvature surface is
from the model case, which is the horizontal plane in the p-minimal case, and the Pansu sphere
in the case c > 0.
We first study rotationally invariant surfaces in H1 with constant p-mean curvature H = c
using the Codazzi-like equation (1.2). In [8], M. Ritoré and C. Rosales made an investigation
on such kinds of surfaces by a first-order ODE system. In the present paper, we shall study
them again from the point of view of our theory established in the previous paper [5] and the
present one. Let Σ(s, θ) be a rotationally invariant surface in H1 with H = c, generated by the
curve γ(s) = (x(s), 0, t(s)), x(s) ≥ 0, on the xt-plane, that is, Σ is parametrized by
Σ(s, θ) = (x(s) cos θ, x(s) sin θ, t(s)),
where x′2 + t′2 = 1. Here ′ means taking a derivative with respect to s. Recall the energy
E =
xt′√
x2x′2 + t′2
+ λx2, (1.3)
which was introduced in [8] and was shown to be a constant. Here 2λ = c and notice that our
p-mean curvature differs from the one defined in [8] by a sign. Hence, we have Theorems A
and B as follows.
Theorem A. A curve γ = (x, t) is the generating curve of a rotationally invariant surface Σ
in H1 with H = c ̸= 0 if and only if γ = (x, t) is defined by x2 = k
c2
+r cos (cs̃), t = − s̃
c−
r
2 sin (cs̃),
up to a constant, for some horizontal arc-length parameter s̃ and some k, r ∈ R such that
k ≥ 1 and r =
2
c2
√
k − 1.
In addition, we have k = 2cE + 2. If r = 0, then Σ is a cylinder. If r ̸= 0, then, in terms of
normal coordinates
(
s̄, θ̄
)
, the two invariants for Σ are
ζ1(θ̄) = −2Eθ̄
cr
, up to a constant, which is linear on θ̄,
ζ2(θ̄) = −2cE + 2
c2r
, which is a constant.
Theorem B. A curve γ = (x, t) is the generating curve of a rotationally invariant p-minimal
surface Σ in H1 if and only if either t is a constant, and hence Σ is a part of the horizontal
plane, or γ = (x, t) is defined by x2 = s̃2 + c2, t = ms̃, up to a constant, for some horizontal
arc-length parameter s̃ and some c2,m ∈ R, m ̸= 0. In addition, we have E = m. In terms of
normal coordinates (s̄, θ̄), the two invariants for Σ are
ζ1(θ̄) = Eθ̄, up to a constant, which is linear on θ̄,
ζ2(θ̄) = c2, which is a constant.
4 H.-L. Chiu, S.-H. Lai and H.-F. Liu
For more interesting examples, in Section 4, we provide an approach to construct a constant
p-mean curvature surface. This approach is an analog of the one we performed in the previ-
ous paper [5] for p-minimal surfaces. Actually, in [5], we deformed the horizontal plane along
a curve C(θ) = (x1(θ), x2(θ), x3(θ)) to obtain a p-minimal surface. More specifically, in [5, Sec-
tion 9], depending on a parametrized curve C(θ) = (x1(θ), x2(θ), x3(θ)) for θ ∈ R, we deformed
the graph u = 0 to obtain a p-minimal surface parametrized by
Y (r, θ) = (x1(θ) + r cos θ, x2(θ) + r sin θ, x3(θ) + rx2(θ) cos θ − rx1(θ) sin θ),
for r ∈ R. It is easy to check that Y is an immersion if and only if either Θ
(
C′(θ)
)
−(
x′2(θ) cos θ−x′1(θ) sin θ
)2 ̸= 0 or r +
(
x′2(θ) cos θ − x′1(θ) sin θ
)
̸= 0 for all θ. In particular, the
surface Y defines a p-minimal surface of special type I if the curve C satisfies
x′3(θ) + x1(θ)x
′
2(θ)− x2(θ)x
′
1(θ)−
(
x′2(θ) cos θ − x′1(θ) sin θ
)2
= 0,
for all θ. In addition, the corresponding ζ1-invariant [5, formula (9.9)] reads
ζ1(θ) = x′2(θ) cos θ − x′1(θ) sin θ −
∫ [
x′1(θ) cos θ + x′2(θ) sin θ
]
dθ. (1.4)
Similarly, the surface Y defines a p-minimal surface of general type if the curve C satisfies
x′3(θ) + x1(θ)x
′
2(θ)− x2(θ)x
′
1(θ)−
(
x′2(θ) cos θ − x′1(θ) sin θ
)2 ̸= 0,
for all θ. In addition, the corresponding ζ1- and ζ2-invariant read
ζ1(θ) = x′2(θ) cos θ − x′1(θ) sin θ −
∫ [
x′1(θ) cos θ + x′2(θ) sin θ
]
dθ,
ζ2(θ) = x′3(θ) + x1(θ)x
′
2(θ)− x2(θ)x
′
1(θ)−
(
x′2(θ) cos θ − x′1(θ) sin θ
)2
. (1.5)
In Section 4, we construct a constant p-mean curvature surface by perturbing the Pansu
sphere along a given curve C(θ). In Section 2.1, we see that the Pansu sphere (2.2) can be
parametrized by
X(s, θ) =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
x(s)
y(s)
t(s)
=
x(s) cos θ − y(s) sin θ
x(s) sin θ + y(s) cos θ
t(s)
,
where
x(s) =
1
2λ
sin(2λs), y(s) = − 1
2λ
cos(2λs) +
1
2λ
,
t(s) =
1
4λ2
sin(2λs)− 1
2λ
s+
π
4λ2
,
with X(0, θ) =
(
0, 0, π
4λ2
)
and X
(
π
λ , θ
)
=
(
0, 0,− π
4λ2
)
as the North pole and South pole, respec-
tively. We deform it along C(θ) = (x1(θ), x2(θ), x3(θ)) to obtain a constant p-mean curvature
surface Y (s, θ) as follows
Y (s, θ) = (x1(θ) + (x(s) cos θ − y(s) sin θ), x2(θ) + (x(s) sin θ + y(s) cos θ),
x3(θ) + t(s) + x2(θ)(x(s) cos θ − y(s) sin θ)− x1(θ)(x(s) sin θ + y(s) cos θ)).
We also give a condition for Y to be an immersion. The coordinate system (s, θ) for Y is
a compatible one. We have (see (4.4))
α = λ
A(θ) cos 2λs+
(
1
2λ −B(θ)
)
sin 2λs(
B(θ)− 1
2λ
)
cos 2λs+A(θ) sin 2λs+D(θ)
, (1.6)
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 5
where
A(θ) = x′2(θ) cos θ − x′1(θ) sin θ, B(θ) = x′2(θ) sin θ + x′1(θ) cos θ,
D(θ) = λΘ(C′(θ)) +
(
1
2λ
−B(θ)
)
, V (θ) =
(
A(θ),
1
2λ
−B(θ)
)
.
It is obvious that V (θ) = 0 implies α = 0, and hence Y is a cylinder. For nonzero V (θ), we then
define
∥V (θ)∥ =
√
[A(θ)]2 +
[
1
2λ
−B(θ)
]2
and write
G(θ) =
D(θ)
∥V (θ)∥
,
V (θ)
∥V (θ)∥
= (sin ζ(θ), cos ζ(θ)),
for some function ζ(θ). From (1.6), we thus have
α = λ
sin
(
2λs+ ζ(θ)
)
G(θ)− cos
(
2λs+ ζ(θ)
) .
Finally, we normalize the α-function to obtain the two invariants for Y , stated in Theorem 4.1.
We list it here as the third main result of the present paper.
Theorem C. If V = 0, then α = 0. If V ̸= 0, we consider the new coordinates s̃ = s + Γ(θ),
θ̃ = Ψ(θ), where
Γ(θ) =
θ
2λ
−
∫
D(θ)dθ, Ψ(θ) = 2λ
∫
∥V (θ)∥ dθ,
then the coordinate system
(
s̃, θ̃
)
is normal. In terms of the normal coordinates, the invariants
of Y are given by
ζ1
(
θ̃
)
= ζ(θ)− 2λΓ(θ), ζ2
(
θ̃
)
= G(θ), (1.7)
where θ = Ψ−1
(
θ̃
)
.
In Section 5, we use formula (1.4), (1.5) and (1.7) to construct various examples of surfaces
of constant p-mean curvature including degenerate p-minimal surfaces of special type I.
2 Solutions to the Codazzi-like equation
The Codazzi-like equation for a surface in H1 with constant p-mean curvature H = c > 0 is
αxx + 6ααx + 4α3 + c2α = 0. (2.1)
Theorem 2.1 ([5]). Besides the following three special solutions to (2.1),
α(x) = 0, − c
2
tan(cx+ cK1), α(x) = − c
2
tan
( c
2
x− c
2
K2
)
,
we have the general solution to (2.1) of the form
α(x) =
c
2
sin
(
cx+ c1
)
c2 − cos
(
cx+ c1
) ,
which depends on constants K1, K2, c1, and c2.
6 H.-L. Chiu, S.-H. Lai and H.-F. Liu
Note that all the solutions are a periodic function with α
(
x+ 2π
c
)
= α(x) for all x. We give
some remarks as follows.
(1) In terms of the following identities
− tan
(
θ +
π
2
)
= cot θ =
sin 2θ
1− cos 2θ
, − tan 2θ = − sin 2θ
cos 2θ
=
sin 2θ
0− cos 2θ
,
we see that the two nontrivial special solutions in Theorem 2.1 correspond to the general
solution in Theorem 2.1 with c2 = 0 and c2 = 1, respectively.
(2) From the following identity
sin (θ + π)
c2 − cos (θ + π)
=
sin θ
−c2 − cos θ
,
we can assume without loss of generality that c2 ≥ 0 in the general solution.
Due to Theorem 2.1, we are able to use the types of the solutions to (2.1) to classify the
constant p-mean curvature surfaces into several classes, which are vertical, special type I, special
type II and general type. In terms of compatible coordinates (x, y), the function α(x, y) is
a solution to the Codazzi-like equation (2.1) for any given y. By Theorem 2.1, the function α(x, y)
hence has one of the following forms of special types
0,
c
2
sin(cx+ c1)
0− cos(cx+ c1)
,
c
2
sin (cx+ c1)
1− cos
(
cx+ c1)
,
and general types c
2
sin (cx+c1)
c2−cos (c(x+c1)
, where, instead of constants, both c1 and c2 are now functions
of y. Notice that it is convenient at some point to assume that c2(y) ≥ 0 for all y. We now use
the types of the function α(x, y) to define the types of constant p-mean curvature surfaces as
follows.
Definition 2.2. Locally, we say that a constant p-mean curvature surface is
(1) vertical if α vanishes (i.e., α(x, y) = 0 for all x, y);
(2) of special type I if α = c
2
sin (cx+c1(y))
1−cos (cx+c1(y))
;
(3) of special type II if α = c
2
sin(cx+c1(y))
0−cos(cx+c1(y))
;
(4) of general type if α = c
2
sin (cx+c1(y))
c2(y)−cos (cx+c1(y))
with c2(y) /∈ {0, 1} for all y.
We further divide constant p-mean curvature surfaces of general type into three classes as
follows.
Definition 2.3. A constant p-mean curvature surface of general type is
(1) of type I if c2(y) > 1 for all y;
(2) of type II if 0 < c2(y) < 1 for all y, and −c1+cos−1 c2
c < x < 2π−c1−cos−1 c2
c ;
(3) of type III if 0 < c2(y) < 1 for all y, and either −c1
c ≤ x < −c1+cos−1 c2
c or 2π−c1−cos−1 c2
c <
x ≤ −c1+2π
c ,
where cos−1 is the inverse of the function cos : [0, π] → [−1, 1].
We notice that the type is invariant under the action of a Heisenberg rigid motion and the
regular part of a constant p-mean curvature surface Σ ⊂ H1 is a union of these types of surfaces.
The corresponding paths of each type of α are shown on the phase plane (see Figure 1). We
express some basic facts as follows.
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 7
c2 > 1
c2 = 1
c2 = 0
3
α
α′
Direction Field of a first order system:α'=v,v'=-6αv-4α3-(1.5)2α
Figure 1. Direction field for c = 1.5.
� If α vanishes, then it is part of a vertical cylinder.
� The two concave downward parabolas in red represent
α =
c
2
sin (cx+ c1)
1− cos (cx+ c1)
,
c
2
sin (cx+ c1)
0− cos (cx+ c1)
,
respectively. The one for α = c
2
sin (cx+c1)
1−cos (cx+c1)
is above the one for α = c
2
sin (cx+c1)
0−cos (cx+c1)
. For
surfaces of special type I, we have that α
(
π−c1
c
)
= 0, α′ (π−c1
c
)
= − c2
4 and
α →
∞, if x → −c1
c
from the right,
−∞, if x → 2π − c1
c
from the left,
and, for surfaces of special type II in which α has period π, we have that
α
(
−c1
c
)
= α
(
π − c1
c
)
= 0, α′
(
−c1
c
)
= α′
(
π − c1
c
)
= −c2
2
and
α →
∞, if x → π − 2c1
2c
from the right,
−∞, if x → 3π − 2c1
2c
from the left.
� The closed curves in orange on the phase plane correspond to the family of solutions
α(x) =
c
2
sin (cx+ c1)
c2 − cos (cx+ c1)
,
8 H.-L. Chiu, S.-H. Lai and H.-F. Liu
where c1, c2 are constants and c2 > 1, which are of type I. There exist zeros for α-function
at x = −c1
c , π−c1
c , at which we have that
α′
(
−c1
c
)
=
1
2
c2
c2 − 1
> 0, α′
(
π − c1
c
)
= −1
2
c2
c2 + 1
< 0.
There are no singular points for surfaces of type I.
� The curves in between the two red concave downward parabolas are of type II. The α-
function of type II has a zero at x = π−c1
c , and
α′
(
π − c1
c
)
= −1
2
c2
c2 + 1
< 0.
For surfaces of type II, it can be checked that
α →
∞, if x → −c1 + cos−1 c2
c
from the right,
−∞, if x → −c1 + 2π − cos−1 c2
c
from the left.
� The curves beneath the lower concave downward parabola are of type III. There exists
a zero for α-function at x = −c1
c , and
α′
(
−c1
c
)
=
1
2
c2
c2 − 1
< 0.
For surfaces of type III, we have
α →
−∞, if x → −c1 + cos−1 c2
c
from the left,
∞, if x → −c1 + 2π − cos−1 c2
c
from the right.
2.1 The Pansu sphere
Lemma 2.4 (Pansu sphere). A Pansu sphere given in [1] by
f(z) =
1
2λ2
(
λ|z|
√
1− λ2|z|2 + cos−1(λ|z|)
)
, |z| ≤ 1
λ
, (2.2)
of constant p-mean curvature c = 2λ has its α-function of special type I. In fact, we have
α =
λ sin(2λs)
(1− cos(2λs))
, a =
−λ√
1 + α2
, b =
2λ2
√
1 + α2(1− cos 2λs)
.
Proof. We parametrize a Pansu sphere by
X(s, θ) =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
x(s)
y(s)
t(s)
=
x(s) cos θ − y(s) sin θ
x(s) sin θ + y(s) cos θ
t(s)
, (2.3)
where
x(s) =
1
2λ
sin(2λs), y(s) = − 1
2λ
cos(2λs) +
1
2λ
,
t(s) =
1
4λ2
sin(2λs)− 1
2λ
s+
π
4λ2
, (2.4)
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 9
with X(0, θ) =
(
0, 0, π
4λ2
)
and X
(
π
λ , θ
)
=
(
0, 0,− π
4λ2
)
as the North pole and South pole, re-
spectively. We then have e1 := Xs, which means that X(s, θ) defines a compatible coordinate
system. Moreover, e2 = Je1, where
e1 = Xs =
(
x′(s) cos θ − y′(s) sin θ, x′(s) sin θ + y′(s) cos θ, t′(s)
)
=
(
x′(s) cos θ − y′(s) sin θ
)̊
e1 +
(
x′(s) sin θ + y′(s) cos θ
)̊
e2,
e2 = −
(
x′(s) sin θ + y′(s) cos θ
)̊
e1 +
(
x′(s) cos θ − y′(s) sin θ
)̊
e2.
We note that α is a function satisfying
αe2 + T = AXs + BXθ, (2.5)
for some functions A and B. Direct calculation shows that
Xθ = (−x(s) sin θ − y(s) cos θ)̊e1 + (x(s) cos θ − y(s) sin θ)̊e2 + (x2(s) + y2(s))T.
Therefore, (2.5) implies
−α
(
x′(s) sin θ + y′(s) cos θ
)
= A
(
x′(s) cos θ − y′(s) sin θ
)
+ B(−x(s) sin θ − y(s) cos θ),
α
(
x′(s) cos θ − y′(s) sin θ
)
= A
(
x′(s) sin θ + y′(s) cos θ
)
+ B(x(s) cos θ − y(s) sin θ),
1 = B
(
x2(s) + y2(s)
)
. (2.6)
The last equation of (2.6) yields B = 1
x2(s)+y2(s)
, and hence
b =
B√
1 + α2
=
2λ2
√
1 + α2(1− cos 2λs)
.
The first two equations of (2.6) indicate −α
((
x′2
)
+
(
y′2
))
= B
(
−xx′ − yy′
)
, which implies
α =
xx′ + yy′
x2 + y2
. (2.7)
In what follows, we claim the above α is one of special solutions. Notice that (2.4) shows
x2 + y2 =
1
4λ2
(2− 2 cos(2λs)),
and hence (2.7) can be rewritten as
α =
1
2
(
lnx2 + y2
)′
=
λ sin(2λs)
(1− cos(2λs))
.
Substituting b and α into (2.6), we have a = A√
1+α2
= −λ√
1+α2
. ■
Given a α-function, we have shown [5] that the first fundamental form (a, b) is determined
up to two functions h(y) and k(y) as follows.
Proposition 2.5. For any α(x, y) = c
2
sin (cx+c1(y))
c2(y)−cos (cx+c1(y))
, the explicit formula for the induced
metric on a constant p-mean curvature surface with c as its p-mean curvature and this α as its
α-function is given by
a =
(
− c
2
+
c
2c2
(c2 − cos (cx+ c1))
+
h(y)
|c2 − cos (cx+ c1)|
)
1(
1 + α2
)1/2 ,
and
b =
(
ek(y)
|c2 − cos (cx+ c1)|
)
1(
1 + α2
)1/2 ,
for some functions h(y) and k(y).
10 H.-L. Chiu, S.-H. Lai and H.-F. Liu
2.2 The normalization
As we normalize the induced metric a and b to be close as much as possible to the metric induced
on the horizontal p-minimal plane, we would like to normalize a and b so that they look like the
induced metric of the Pansu sphere. Indeed, from the transformation law [5, formula (2.20)], it
is easy to see that there exist another compatible coordinates (x̃, ỹ), called normal coordinates
such that
ã = −
c
2(
1 + α2
)1/2 , (2.8)
b̃ =
c2
2
|c2 − cos (cx+ c1)|
(
1 + α2
)1/2 , (2.9)
where c = 2λ. Such normal coordinates are uniquely determined up to a translation. We thus
have the following theorem.
Theorem 2.6. In normal coordinates (x, y), the functions c1(y) and c2(y) in the expression
α(x) = c
2
sin (cx+c1)
c2−cos (cx+c1)
are unique in the following sense: up to a translation on y, c2(y) is unique,
and c1(y) is unique up to a constant. We denote these two unique functions by ζ1(y) = c1(y),
ζ2(y) = c2(y). Therefore, the set {ζ1(y), ζ2(y)} constitutes a complete set of invariants for those
surfaces (α not vanishing).
It is worth our attention that, for the surfaces with c2 > 1, the denominator of the formula
for α is never zero. That means the surfaces won’t extend to a surface with singular points.
Moreover, if the surface is closed, it must be a closed constant p-mean curvature surface without
singular points, which means the surface is of type of torus. This indicates that it is possible to
find a Wente-type torus in this class of surfaces.
2.3 The structure of the singular sets
In this subsection, we study the structure of the singular set. For the general type, we choose
a normal coordinate system (s, θ) such that
α = λ
sin (2λs+ ζ1(θ))
ζ2(θ)− cos (2λs+ ζ1(θ))
,
and
a = − λ√
1 + α2
, b =
2λ2
|ζ2(θ)− cos (2λs+ ζ1(θ))|
√
1 + α2
.
Then the singular set is the graph of the function x(θ) = cos−1(ζ2(θ))−ζ1(θ)
2λ . The induced metric I
(or the first fundamental form) on the regular part reads
I = ds⊗ ds− a
b
ds⊗ dθ − a
b
dθ ⊗ ds+
(
1 + a2
)
b2
dθ ⊗ dθ.
Now we use the metric to compute the length of the singular set
{( cos−1(ζ2(θ))−ζ1(θ)
2λ , θ
)}
, where θ
belongs to some open interval.
Case ζ2(θ) ̸= 1. Let γ(θ) =
( cos−1(ζ2(θ))−ζ1(θ)
2λ , θ
)
, which is a parametrization of the singular
set. Then the square of the velocity at θ is
∣∣γ′(θ)∣∣2 = [q′(θ)]2 − 2aq′(θ)
b
+
a2 + 1
b2
=
[a− bq′(θ)]2 + 1
b2
> 0 for all θ, (2.10)
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 11
where
q′(θ) =
[
cos−1(ζ2(θ))− ζ1(θ)
2λ
]′
=
−ζ
′
2(θ)
2λ
√
1− ζ22 (θ)
− ζ
′
1(θ)
2λ
.
Formula (2.10) shows that the parametrized curve γ(θ) of the singular set has a positive length.
Case ζ2(θ) = 1. We parametrize the singular set by γ(θ) =
( cos−1(1−ϵ)−ζ1(θ)
2λ , θ
)
for ϵ > 0. It
is easy to see γ′(θ) =
(−ζ′1(θ)
2λ , 1
)
. When ε → 0, the metric
I = ds⊗ ds− a
b
ds⊗ dθ − a
b
dθ ⊗ ds+
(
1 + a2
)
b2
dθ ⊗ dθ
degenerates to Ĩ = ds⊗ ds.
Then the square of the velocity at θ is
∣∣γ′(θ)∣∣2 = [ζ′1(θ)]
2
4λ2 . Thus, if ζ1(θ) = c1 and ζ2(θ) = 1,
the length of the parametrized curve γ(θ) of the singular set is zero. This result coincides with
the singular set for the Pansu sphere being isolated. We conclude the above discussion with the
following theorem, an analog of [5, Theorem 1.7].
Theorem 2.7. The singular set of a constant p-mean surface with H = c ̸= 0 is either
(1) an isolated point; or
(2) a smooth curve.
In addition, an isolated singular point only happens on the surfaces of special type I with ζ1 =
const, namely, a part of the Pansu sphere containing one of the poles as the isolated singular
point.
Theorem 2.7 together with [5, Theorem 1.7] are just special cases of [2, Theorem 3.3]. How-
ever, we give a computable proof of this result for constant p-mean surfaces. We also have
the description of how a characteristic leaf goes through a singular curve, which is called a “go
through” theorem in [2]. Suppose p0 is a point in a singular curve. From the above basic facts,
we see that a characteristic curve γ always reaches the singular point p0 going a finite distance.
From the opposite direction, suppose γ̃ is another characteristic curve that reaches p0. Then
the union of γ, p0 and γ̃ forms a smooth curve (we also refer the reader to the proof of [5,
Theorem 1.8], they are similar). We thus have the following theorem.
Theorem 2.8. Let Σ ⊂ H1 be a constant p-mean surface with H = c ̸= 0. Then the character-
istic foliation is smooth around the singular curve in the following sense that each leaf can be
extended smoothly to a point on the singular curve.
Making use of Theorem 2.8, we have the following result.
Theorem 2.9. Let Σ be a constant p-mean surface of type II (III) with H = c ̸= 0. If it can
be smoothly extended through the singular curve, then the other side of the singular curve is of
type III (II).
Therefore, we see that a surface of general type II is always pasted together with a surface
of general type III at a singular curve and vice versa.
3 Rotationally invariant surfaces in H1
Let Σ(s, θ) be a rotationally invariant surface in H1 generated by a curve γ(s) = (x(s), 0, t(s))
on the xt-plane, that is, Σ is parametrized by Σ(s, θ) = (x(s) cos θ, x(s) sin θ, t(s)), where
x′2 + t′2 = 1. Here ′ means taking a derivative with respect to s.
12 H.-L. Chiu, S.-H. Lai and H.-F. Liu
3.1 The computation of H, α, a and b
Now we consider the horizontal (see [3, Definition 1.1]) generating curve
γ̃(s) = (x(s) cos θ(s), x(s) sin θ(s), t(s)).
Lemma 3.1. γ̃ is horizontal if and only if t′ + x2θ′ = 0.
Proof. Note that at the point γ̃(s),
e̊1 =
∂
∂x1
+ y1
∂
∂z
=
∂
∂x1
+ x(s) sin θ(s)
∂
∂z
,
e̊2 =
∂
∂y1
− x1
∂
∂z
=
∂
∂y1
− x(s) cos θ(s)
∂
∂z
,
and direct computations imply
γ̃′(s) =
(
x′ cos θ − xθ′ sin θ
)̊
e1 +
(
x′ sin θ + xθ′ cos θ
)̊
e2 +
(
t′ + x2θ′
)
T,
and hence γ̃′(s) ∈ ξ if and only if t′ + x2θ′ = 0. ■
Let s̃ be the horizontal arc-length of γ̃(s). We can thus re-parametrize the surface Σ(s, θ)
to be
Σ(s̃, θ̃) =
(
x(s) cos θ(s) cos θ̃ − x(s) sin θ(s) sin θ̃, x(s) cos θ(s) sin θ̃
+ x(s) sin θ(s) cos θ̃, t(s)
)
,
with a compatible coordinate system
e1 =
∂
∂s̃
= Σs
∂s
∂s̃
, where Σs =
(
x′ cosϕ− xθ′ sinϕ
)̊
e1 +
(
x′ sinϕ+ xθ′ cosϕ
)̊
e2.
Moreover, we see |γ̃|′2 = x2x′2+t′2
x2 , so that we may choose s̃ such that
∣∣dγ̃(s̃)
ds̃
∣∣ = 1, that is,
ds̃
ds
= |γ̃′(s)| =
√
x2x′2 + t′2
x
. (3.1)
Manipulating Σ to be
Σ
(
s̃, θ̃
)
=
(
x(s) cos
(
θ(s) + θ̃
)
, x(s) sin
(
θ(s) + θ̃
)
, t(s)
)
,
(
denote ϕ = θ(s) + θ̃
)
and obtain
Σs̃ =
ds
ds̃
(
x′ cosϕ− xθ′ sinϕ
)̊
e1 +
ds
ds̃
(
x′ sinϕ+ xθ′ cosϕ
)̊
e2,
Σθ̃ = −x sinϕe̊1 + x cosϕe̊2 + x2
∂
∂z
.
Then
e1 = Σs̃ =
x√
x2x′2 + t′2
(
x′ cosϕ− xθ′ sinϕ
)̊
e1 +
ds
ds̃
(
x′ sinϕ+ xθ′ cosϕ
)̊
e2, (3.2)
e2 = Je1 =
x√
x2x′2 + t′2
(
x′ cosϕ− xθ′ sinϕ
)̊
e2 −
ds
ds̃
(
x′ sinϕ+ xθ′ cosϕ
)̊
e1. (3.3)
The fact that αe2+T√
1+α2
∈ TΣ implies αe2+T = a
√
1 + α2Σs̃+ b
√
1 + α2Σθ̃. Using (3.2), (3.3) and
comparing the coefficients of e̊1, e̊2, and T , respectively, one sees
a =
t′
x
√
1 + α2
√
x2x′2 + t′2
, b =
1
x2
√
1 + α2
, α =
x′√
x2x′2 + t′2
, (3.4)
and hence, from the first equation of the integrability conditions (1.1), we have
H = −
x3
(
x′t′′ − x′′t′
)
+ t′3
x
{
x2x′2 + t′2
}3/2 . (3.5)
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 13
3.2 Another understanding of energy E
In this subsection, we assume moreover that the rotationally invariant surface Σ is of constant
p-mean curvature. We consider the relation between the integrability condition and the energy
discussed in Ritoré and Rosales’ paper [8]. The integrability condition as̃ − a
b bs̃ +
cα√
1+α2
= 0
indicates that∫ (
as̃
b
− a
b2
bs̃ +
cα
b
√
1 + α2
)
ds̃ (3.6)
is a constant. Then we have (3.6) computed as
a
b
+ c
∫
x2x′√
x2x′2 + t′2
ds̃ =
a
b
+ c
∫
xx′ds =
a
b
+ λx2, up to a constant,
which clearly says that a
b +λx2 is constant. The constant a
b +λx2 interprets the energy E based
on Ritoré’s discussion. Indeed, we have
E =
a
b
+ λx2 =
xt′√
x2x′2 + t′2
+ λx2 = ts̃ + λx2, (3.7)
that is, ts̃ = E − λx2. One sees
t = Es̃− λ
∫
x2ds̃. (3.8)
3.3 The Coddazi-like equation
For later use, we calculate 1 + α2 = 1+x2x′2
x2x′2+t′2 and convert α to be of the general form
α =
x′√
x2x′2 + t′2
=
xs̃
x
. (3.9)
Note that α satisfies the Coddazi-like equation αs̃s̃ + 6ααs̃ + 4α3 + c2α = 0, where c = 2λ. Then
this ODE immediately shows
xs̃s̃s̃
x
+ 3
xs̃xs̃s̃
x2
+ c2
xs̃
x
= 0. (3.10)
The equation (3.10) is manipulated to be
(
xxs̃s̃ + (xs̃)
2 + c2
2 x
2
)
s̃
= 0, which gives(
x2
)
s̃s̃
+ c2x2 = k, for some constant k. (3.11)
Let u = x2, then (3.11) becomes a second-order inhomogeneous constant coefficient ODE
us̃s̃ + c2u = k. (3.12)
(I) Suppose c ̸= 0, the homogeneous ODE us̃s̃ + c2u = 0 has the general solution uh given by
uh = k1 sin(cs̃) + k2 cos(cs̃) = r cos(cs̃− c1), (3.13)
where r =
√
k21 + k22 and r sin c1 = k1. One also notes that up =
k
c2
is a particular solution
to (3.12), and hence
x2 = u =
k
c2
+ r cos(cs̃− c1). (3.14)
14 H.-L. Chiu, S.-H. Lai and H.-F. Liu
(II) When c = 0, it is clear that (3.11) becomes
(
x2
)
s̃s̃
= k, which implies
x2 = ks̃2 + 2k1s̃+ k2, (3.15)
for some constants k, k1 and k2.
Example 3.2. If k = 0, k1 = 0, then (3.15) yields x =
√
k2. On the other hand, (3.16)
suggests 0 = t′3√
k2t′3
= 1√
k2
> 0, which is a contradiction. We conclude that there are no such
kinds of p-minimal surfaces (k = 0, k1 = 0) which are rotationally symmetric. In this case,
α vanishes so that it corresponds a vertical cylinder surface which is absolutely not p-minimal.
3.4 The relation between k and E
Assume that c = 2λ ̸= 0. We write (3.5) as
−2λ =
x3
(
x′t′′ − x′′t′
)
+ t′3
x
{
x2x′2 + t′2
}3/2
=
x2
(
t′
x′
)′
x′2{
x2x′2 + t′2
}3/2 +
1
x4
(
xt′√
x2x′2 + t′2
)3
= I1 + I2, (3.16)
where
I1 =
x2
(
t′
x′
)′
x′2{
x2x′2 + t′2
}3/2 , I2 =
1
x4
(
xt′√
x2x′2 + t′2
)3
.
From (3.14), taking a derivative with respect to s, we have
x′ =
−cr sin (cs̃− c1)
√
x2x′2 + t′2
2x2
. (3.17)
On the other hand, (3.7) implies
t′ =
(
E − λx2
)√x2x′2 + t′2
x
. (3.18)
By means of (3.14), (3.17),(3.18) and (3.1), after direct computations, we have
I1 =
x2
(
t′
x′
)′
x′2{
x2x′2 + t′2
}3/2
=
(
2x
(
E − λx2
)
−cr sin (cs̃− c1)
)′(
c2r2 sin2 (cs̃− c1)
(
x2x′2 + t′2
)
4x4
)
=
1
4x2
(
−2x2x′
(
4λx2x′ − 2x′
(
E − λx2
))(x2 −
(
E − λx2
)2
x4x′2
)
+ 2
(
E − λx2
)(
c2x2 − k
))
,
and from (3.7), we have
I2 =
1
x4
(
xt′√
x2x′2 + t′2
)3
=
1
x4
(
E − λx2
)3
.
Therefore,
−2λ = I1 + I2 =
1
4x4
((
−4c− 2Ec2 − 2c+ ck
)
x4 +
(
8λE2 + 4E − 2Ek
)
x2
)
,
which implies k = 2cE + 2.
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 15
3.5 Horizontal generating curves for c ̸= 0
In this subsection, we will show that k, and hence the energy E, has a lower bound. A horizontal
generating curve of a rotationally invariant constant p-mean curvature surface is a geodesic curve,
which is parametrized by
γ̃(s̃) =
1
c
sin(cs̃) + x0
−1
c
cos(cs̃) + 1
c + y0(
1
c2
+
cy0
c2
)
sin(cs̃) +
x0
c
cos(cs̃)− s̃
c
+
π
c2
− x0
c
+ t0
(3.19)
for some (x0, y0, t0), where s̃ is a horizontal arc length parameter.
Suppose that γ(s) = (x(s), 0, t(s)) with x ≥ 0 is the corresponding generating curve, we have
x2 =
(
1
c
sin(cs̃) + x0
)2
+
(
−1
c
cos(cs̃) +
1
c
+ y0
)2
= 2
x0
c
sin(cs̃)− 2
(
1 + cy0
c2
)
cos(cs̃) + x20 +
(
1 + cy0
c
)2
+
1
c2
= r cos(cs̃− c1) +
k
c2
,
where
k = 1 + (cx0)
2 + (1 + cy0)
2 ≥ 1,
r =
√(
2
x0
c
)2
+
(
2
(
1 + cy0
c2
))2
=
2
c2
√
(cx0)2 + (1 + cy0)2 =
2
c2
√
k − 1,
and c1 is a real number such that sin c1 =
2
x0
c
r , cos c1 = −
2(
1+cy0
c2
)
r .
3.6 The invariants ζ1 and ζ2 for surfaces with c ̸= 0
If r = 0, then k = 1, x0 = 0, y0 = −1
c . Thus, (3.19) implies
γ̃(s̃) =
(
1
c
sin(cs̃),−1
c
cos(cs̃),− s̃
c
+
π
c2
+ t0
)
,
which generates a cylinder. We assume from now on that r ̸= 0. Taking the derivative with
respect to s̃ on both sides of (3.14) to have 2xxs̃ = −rc sin(cs̃ − c1). Together with (3.9), we
have α of the general form as follows:
α =
xxs̃
x2
=
−rλ sin(cs̃− c1)
k
c2
+ r cos(cs̃− c1)
=
λ sin(cs̃− c1)
c2 − cos(cs̃− c1)
,
where c2 = − k
rc2
.
In this subsection, we want to normalize a and b such that they have the forms looking
as (2.8) and (2.9), respectively. Together with (3.4), (3.14) and (3.7), we have
a =
t′
x
√
1 + α2
√
x2x′2 + t′2
=
(
E
x2
− λ
)
1√
1 + α2
,
b =
1
x2
√
1 + α2
=
1
k
c2
+ r cos(cs̃− c1)
1√
1 + α2
.
16 H.-L. Chiu, S.-H. Lai and H.-F. Liu
Thus we choose the normal coordinates
{
s̄, θ̄
}
with s̄ = s̃ + Γ
(
θ̃
)
, θ̄ = Ψ
(
θ̃
)
, such that
Γ′(θ̃) = −E, Ψ′(θ̃) = −2λ2r. Then we have
ā =
−λ√
1 + ᾱ2
, b̄ =
2λ2(
− k
rc2
− cos(cs̃− c1)
) 1√
1 + ᾱ2
,
with
ᾱ =
λ sin(cs̃− c1)
c2 − cos(cs̃− c1)
=
λ sin
(
cs̄− c1 − Eθ̄
rλ
)
c2 − cos
(
cs̄− c1 − Eθ̄
rλ
) ,
that is,
ζ1
(
θ̄
)
= −c1 −
2Eθ̄
cr
, ζ2
(
θ̄
)
= c2 = − k
rc2
= −2cE + 2
c2r
. (3.20)
If E = 0, then k = 2, thus the surface has the generating curve defined by
x2 =
2
c2
+ r cos (cs̃− c1) = r
(
2
c2r
+ cos (cs̃− c1)
)
,
t = −λ
(
2
c2
s̃+
r
c
sin (cs̃− c1)
)
,
with ζ1
(
θ̄
)
= −c1, ζ2
(
θ̄
)
= − 2
c2r
< 0. Therefore, we see that x2 ≥ 0 ⇔ cos (cs̃− c1) ≥ ζ2
(
θ̄
)
,
which means that the generating curve (x, t) is defined on the whole R if and only if ζ2
(
θ̄
)
≤ −1.
In particular, if ζ2
(
θ̄
)
= −1, it is the Pansu sphere.
If E ̸= 0, then k = 2cE + 2 and (3.20) implies that
ζ ′1
(
θ̄
)
= −2E
cr
> ζ2
(
θ̄
)
. (3.21)
For any constants η1 and η2 with η1 > η2, we obtain the unique solution to the equation system
−2E
cr
= η1, −2cE + 2
c2r
= η2.
3.7 The allowed values of k and E with c = 0
In this subsection, we shall show what possible values can k and E attain. Assume that c =
2λ = 0. We write (3.5) as
0 =
x3
(
x′t′′ − x′′t′
)
+ t′3
x
{
x2x′2 + t′2
}3/2 =
x2
(
t′
x′
)′
x′2{
x2x′2 + t′2
}3/2 +
1
x4
(
xt′√
x2x′2 + t′2
)3
= I1 + I2.
Taking a derivative of (3.15) with respect to s, we get
x′ =
(ks̃+ k1)
√
x2x′2 + t′2
x2
. (3.22)
On the other hand, from (3.7), we have
t′ =
E
√
x2x′2 + t′2
x
. (3.23)
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 17
By means of (3.15), (3.22),(3.23) and (3.1), a direct computation gives
I1 =
x2
(
t′
x′
)′
x′2{
x2x′2 + t′2
}3/2 =
1
x2
(
E
(
x2 − E2
)
x2
− kE
)
,
and (3.7) implies
I2 =
1
x4
(
xt′√
x2x′2 + t′2
)3
=
1
x4
E3.
Therefore, 0 = I1 + I2 =
E(1−k)
x2 , which says that
E = 0 or k = 1. (3.24)
The equation (3.7) says that t and Es̃ are differed only by a constant. If E = 0, then t is
constant, which gives us that Σ is a plane that is perpendicular to t-axis.
3.8 The invariants ζ1 and ζ2 for surfaces with c = 0 and E ̸= 0
If E = 0, the surface is a perpendicular plane to the t-axis. Therefore, in this subsection, we
assume that E ̸= 0, and thus, from (3.24), we have k = 1. Then one rewrites α to be
α =
xxs̃
x2
=
s̃+ k1
(s̃+ k1)2 +
(
k2 − k21
) .
From (3.15), we have 1
x2 = α
s̃+k1
> 0. We want to normalize a and b such that they have the
form specified in [5, Theorem 1.3]. Together with (3.4) and (3.7), we have
a =
t′
x
√
1 + α2
√
x2x′2 + t′2
=
(
E
x2
)
1√
1 + α2
= Eb,
b =
1
x2
√
1 + α2
=
α
(s̃+ k1)
√
1 + α2
=
|α|
(|s̃+ k1|)
√
1 + α2
.
Thus we choose the normal coordinates
{
s̄, θ̄
}
with s̄ = s̃+Γ
(
θ̃
)
, θ̄ = Ψ
(
θ̃
)
such that Γ′(θ̃) = −E,
Ψ′(θ̃) = 1. Then we have
ā = 0, b̄ =
|ᾱ|
|s̃+ k1|
√
1 + ᾱ2
with
ᾱ =
s̃+ k1
(s̃+ k1)2 +
(
k2 − k21
) =
s̄− Γ
(
θ̃
)
+ k1
(s̄− Γ
(
θ̃
)
+ k1)2 +
(
k2 − k21
) ,
that is,
ζ1
(
θ̄
)
= k1 + Eθ̄, which is linear in θ̄,
ζ2
(
θ̄
)
= k2 − k21, which is a constant, denoted as ζ2.
From (3.15) and (3.7), we conclude that the generating curve is defined by
x2 = (s̃+ k1)
2 + ζ2, t = Es̃, up to a constant. (3.25)
18 H.-L. Chiu, S.-H. Lai and H.-F. Liu
Remark 3.3. We remark that for λ = 0, two kinds of p-minimal surfaces are presented depend-
ing on the energy E. When E = 0, t in (3.16) is constant and then one obtains a plane that is
perpendicular to the t-axis. On the other hand, if E ̸= 0, we have p-minimal surfaces generated
by curves defined by (3.25). For λ ̸= 0, substituting (3.14) in (3.8), we see
t = Es̃− λ
∫ (
k
c2
+
√
k21 + k22 cos(cs̃− c1)
)
ds̃
=
(
E − k
4λ
)
s̃−
√
k21 + k22
2
sin(2λs̃− c1) + const.
In the case λ ̸= 0, we give the following two examples.
Example 3.4. We choose k1, k2 in (3.13) so that
√
k21 + k22 = − k
c2
, and then (3.14) implies
x = ±
√
2k
2λ
sin
(
λs̃− c1
2
)
.
Moreover, if E = 0, then t = − k
4λ s̃+
k
8λ2 sin(2λs̃− c1), which is a scaling sphere.
The other two integrability conditions (see [5, equation (2.13)]) are
−bs̃
b
= 2α+
ααs̃
1 + α2
, aHs̃ + bHθ̃ =
αs̃s̃ + 6ααs̃ + 4α3 + αH2
√
1 + α2
. (3.26)
We rewrite the first equation in (3.26) as
2α+
ααs̃
1 + α2
+
bs̃
b
= 0.
Integrating on both sides to see that∫ (
2α+
ααs̃
1 + α2
+
bs̃
b
)
ds̃
is a constant. More precisely, in terms of x, x′, t, t′, we write∫
2
(
α+
ααs̃
1 + α2
+
bs̃
b
)
ds̃ =
∫ (
2
xs̃
x
+
ααs̃
1 + α2
+
bs̃
b
)
ds̃ = ln
(
bx2
√
1 + α2
)
+ const.
The conclusion is that ln
(
bx2
√
1 + α2
)
is a constant, which also follows from (3.4).
Suppose H is constant. The second equation of (3.26) is exactly αs̃s̃+6ααs̃+4α3+αH2 = 0.
Using (3.9), this ODE becomes (3.10), which has been discussed previously.
4 The construction of constant p-mean curvature surfaces
In this section, we construct constant p-mean curvature surfaces by perturbing the Pansu sphere
in some way. Recall the parametrization of the Pansu sphere (2.3). For each fixed angle θ, the
curve lθ defined by lθ(s) = (x(s) cos θ − y(s) sin θ, x(s) sin θ + y(s) cos θ, t(s)) is a geodesic with
curvature 2λ. Let C be an arbitrary curve C : R → H1 given by C(θ) = (x1(θ), x2(θ), x3(θ)). For
each fixed θ, we translate lθ by C(θ), so that the curve LC(θ)(lθ) is also a geodesic curve with
curvature 2λ. Then the union of all these curves ΣC = ∪θLC(θ)(lθ) constitutes a constant p-mean
curvature surface with a parametrization
Y (s, θ) = (x1(θ) + (x(s) cos θ − y(s) sin θ
)
, x2(θ) + (x(s) sin θ + y(s) cos θ),
x3(θ) + t(s) + x2(θ)(x(s) cos θ − y(s) sin θ)
− x1(θ)(x(s) sin θ + y(s) cos θ)). (4.1)
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 19
By a straightforward computation, and notice that
x′(s) cos θ − y′(s) sin θ = cos (2λs+ θ), x′(s) sin θ + y′(s) cos θ = sin (2λs+ θ),
x(s) cos θ − y(s) sin θ = − 1
2λ
(sin θ − sin (2λs+ θ)),
x(s) sin θ + y(s) cos θ =
1
2λ
(cos θ − cos (2λs+ θ)), x2(s) + y2(s) =
1
2λ2
(1− cos 2λs),
we have
Ys =
(
x′(s) cos θ − y′(s) sin θ
)̊
e1 +
(
x′(s) sin θ + y′(s) cos θ
)̊
e2|Y (s,θ)
= cos (2λs+ θ)̊e1 + sin (2λs+ θ)̊e2,
Yθ =
(
x′1(θ)− x(s) sin θ − y(s) cos θ
)̊
e1 +
(
x′2(θ) + x(s) cos θ − y(s) sin θ
)̊
e2 +
(
Θ(C′(θ))
+ 2x′2(θ)
(
x(s) cos θ − y(s) sin θ
)
− 2x′1(θ)
(
x(s) sin θ + y(s) cos θ
)
+ x2(s) + y2(s)
)
T
=
(
x′1(θ)−
1
2λ
(cos θ − cos (2λs+ θ))
)
e̊1 +
(
x′2(θ)−
1
2λ
(sin θ − sin (2λs+ θ))
)
e̊2
+
(
Θ(C′(θ))− x′2(θ)
1
λ
(sin θ − sin (2λs+ θ))− x′1(θ)
1
λ
(cos θ − cos (2λs+ θ))
+
1
2λ2
(1− cos 2λs)
)
T.
Therefore,
Ys ∧ Yθ =
[
x′2(θ) cos (2λs+ θ)− x′1(θ) sin (2λs+ θ) +
sin 2λs
2λ
]
e̊1 ∧ e̊2
+ [cos (2λs+ θ)
〈
Yθ, T
〉
]̊e1 ∧ T + [sin (2λs+ θ)
〈
Yθ, T
〉
]̊e2 ∧ T
=
[
A(θ) cos 2λs+
(
1
2λ
−B(θ)
)
sin 2λs
]
e̊1 ∧ e̊2
+ [cos (2λs+ θ)
〈
Yθ, T
〉
]̊e1 ∧ T + [sin (2λs+ θ)
〈
Yθ, T
〉
]̊e2 ∧ T, (4.2)
where
A(θ) = x′2(θ) cos θ − x′1(θ) sin θ, B(θ) = x′2(θ) sin θ + x′1(θ) cos θ,〈
Yθ, T
〉
=
1
λ
[(
B(θ)− 1
2λ
)
cos 2λs+A(θ) sin 2λs+D(θ)
]
,
D(θ) = λΘ(C′(θ)) +
(
1
2λ
−B(θ)
)
. (4.3)
From (4.2), we conclude that Y is an immersion if and only if either[
A(θ) cos 2λs+
(
1
2λ
−B(θ)
)
sin 2λs
]
̸= 0 or
〈
Yθ, T
〉
̸= 0.
For the constructed surface Y in (4.1), we always assume it is defined on a region such that Y is
an immersion and ΣC is the constant p-mean curvature surface defined by such an immersion Y .
A point p ∈ ΣC is a singular point if and only if
〈
Yθ, T
〉
= 0. Thus at a singular point, we must
have [
A(θ) cos 2λs+
(
1
2λ
−B(θ)
)
sin 2λs
]
̸= 0.
20 H.-L. Chiu, S.-H. Lai and H.-F. Liu
Now, we proceed to compute the invariants for Y . From the construction of Y , we see
that (s, θ) is a compatible coordinate system and we are able to choose the characteristic direc-
tion e1 = Ys, and hence
e2 = Je1 = − sin (2λs+ θ)̊e1 + cos (2λs+ θ)̊e2.
The α-function is a function defined on the regular part that satisfies
αe2 + T = a
√
1 + α2Ys + b
√
1 + α2Yθ = a
√
1 + α2e1 + b
√
1 + α2Yθ
for some functions a and b. This is equivalent to, comparing the alike terms,
−α sin (2λs+ θ) = a
√
1 + α2 cos (2λs+ θ)
+ b
√
1 + α2
(
x′1(θ)−
1
2λ
(cos θ − cos (2λs+ θ))
)
,
α cos (2λs+ θ) = a
√
1 + α2 sin (2λs+ θ)
+ b
√
1 + α2
(
x′2(θ)−
1
2λ
(sin θ − sin (2λs+ θ))
)
,
1 = b
√
1 + α2
〈
Yθ, T
〉
.
We thus have
a =
−2λ
(
x′1(θ) cos(2λs+ θ) + x′2(θ) sin(2λs+ θ)
)
− (1− cos 2λs)
2λ
√
1 + α2⟨Yθ, T ⟩
,
b =
1√
1 + α2
〈
Yθ, T
〉 ,
α =
x′2(θ) cos (2λs+ θ)− x′1(θ) sin (2λs+ θ) + sin 2λs
2λ〈
Yθ, T
〉
= λ
A(θ) cos 2λs+
(
1
2λ −B(θ)
)
sin 2λs(
B(θ)− 1
2λ
)
cos 2λs+A(θ) sin 2λs+D(θ)
. (4.4)
Let V =
(
A(θ), 1
2λ−B(θ)
)
and ∥V ∥ =
√
[A(θ)]2 +
[
1
2λ −B(θ)
]2
. If V = 0, then α = 0. If V ̸= 0,
then we can write V
∥V ∥ = (sin ζ(θ), cos ζ(θ)), for some function ζ(θ). The functions α, a, and b
can be further written as
α = λ
sin ζ(θ) cos 2λs+ cos ζ(θ) sin 2λs
− cos ζ(θ) cos 2λs+ sin ζ(θ) sin 2λs+ D(θ)
∥V ∥
= λ
sin
(
2λs+ ζ(θ)
)
G(θ)− cos
(
2λs+ ζ(θ)
) ,
a =
−2λ[sin ζ(θ) sin(2λs)− cos ζ(θ) cos 2λs]− 1
∥V ∥
2
√
1 + α2
[D(θ)
∥V ∥ − cos(2λs+ ζ(θ))
]
=
2λ∥V ∥ cos(2λs+ ζ(θ))− 1
2∥V ∥
√
1 + α2[G(θ)− cos(2λs+ ζ(θ))]
,
b =
λ
∥V ∥√
1 + α2
[D(θ)
∥V ∥ − cos(2λs+ ζ(θ))
] = λ
∥V ∥√
1 + α2[G(θ)− cos(2λs+ ζ(θ))]
,
where
G(θ) =
D(θ)
∥V ∥
=
D(θ)√
(A(θ))2 +
(
1
2λ −B(θ)
)2 . (4.5)
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 21
Next, we normalize the three invariants α, a, and b. Firstly, we choose another compatible
coordinates (s̃ = s+ Γ(θ), θ̃ = Ψ(θ)), for some Γ(θ) and Ψ(θ). From the transformation law of
the induced metric ã = a+ bΓ′(θ), b̃ = bΨ′(θ), this can be chosen so that
Γ′(θ) =
−2λ∥V (θ)∥G(θ) + 1
2λ
=
1
2λ
−D(θ),
or equivalently,
Γ(θ) =
θ
2λ
−
∫
D(θ)dθ.
If we further choose Ψ such that θ̃ = Ψ(θ) = 2λ
∫
∥V (θ)∥dθ, then in terms of the compatible
coordinates
(
s̃, θ̃
)
, the three invariants read
ã =
−λ√
1 + α̃2
, b̃ =
2λ2
√
1 + α̃2
{
G
(
Ψ−1
(
θ̃
))
− cos
[
2λs̃− 2λΓ
(
Ψ−1
(
θ̃
))
+ ζ
(
Ψ−1
(
θ̃
))]} ,
α̃ = λ
sin
(
2λs̃− 2λΓ(θ) + ζ
(
Ψ−1
(
θ̃
)))
G
(
Ψ−1
(
θ̃
))
− cos
[
2λs̃− 2λΓ
(
Ψ−1
(
θ̃
))
+ ζ
(
Ψ−1
(
θ̃
))] ,
where D and G are defined in (4.3) and (4.5), respectively.
We summarize the above discussion as a theorem in the following.
Theorem 4.1. The coordinate system (s, θ) for Y in (4.1) is compatible. If V = 0, then α = 0.
If V ̸= 0, then the new coordinate system
(
s̃, θ̃
)
, where s̃ = s+ Γ(θ), θ̃ = Ψ(θ), with
Γ(θ) =
θ
2λ
−
∫
D(θ)dθ, Ψ(θ) = 2λ
∫
∥V (θ)∥dθ,
is normal. In terms of the normal coordinates, the invariants of Y are given by
ζ1
(
θ̃
)
= ζ
(
Ψ−1
(
θ̃
))
− 2λΓ
(
Ψ−1
(
θ̃
))
, ζ2
(
θ̃
)
= G
(
Ψ−1
(
θ̃
))
. (4.6)
Particularly, in order to have constant ζ1
(
θ̃
)
and nonzero constant ζ2
(
θ̃
)
, Theorem 4.1 sug-
gests the constant p-mean curvature surfaces deformed by curves
C(θ) = (x1(θ), x2(θ), x3(θ)) =
(
r
λ
sin θ,− r
λ
cos θ,
r(1− r)
λ2
θ
)
, (4.7)
where r ̸= 1
2 . More precisely, we have the following proposition.
Proposition 4.2. For any curve C(θ) defined as (4.7), the deformed surface Y (s, θ) has both
constant invariants ζ1
(
θ̃
)
and ζ2
(
θ̃
)
̸= 0.
Proof. We argue by assuming ζ2
(
θ̃
)
= ζ2 is a constant, x1(θ) =
r
λ sin θ , and x2(θ) = − r
λ cos θ
for any r ̸= 1
2 . Then (4.3) implies A(θ) = 0, B(θ) = r
λ , which leads to ∥V ∥ = |1−2r|
2λ . The second
equation of (4.6) shows that D(θ) = ζ2∥V ∥, and hence
ζ1
(
θ̃
)
= sin−1
(
A(θ)
∥V ∥
)
− 2λ
(
θ
2λ
−
∫
D(θ)dθ
)
= −θ +
∫
ζ2|1− 2r|dθ
= (ζ2|1− 2r| − 1)θ + const.
In order to have ζ1
(
θ̃
)
being constant, we must have ζ2|1− 2r| = 1. It is clear to see that ζ2 ̸= 0
and r ̸= 0. The system (4.6) immediately shows D(θ) = 1
2λ , which gives x′3(θ) =
r(1−r)
λ2 , by (4.3).
Namely, x3(θ) =
r(1−r)
λ2 θ + const.
Moreover, the new coordinates can be obtained by
θ̃ = Ψ(θ) = |1− 2r|θ + const and Γ(θ),
up to a constant. ■
22 H.-L. Chiu, S.-H. Lai and H.-F. Liu
5 Examples
It is easy to see that the Pansu sphere can be obtained by deforming the following curves
C1(θ) = (0, 0, const) or C2(θ) =
(
1
λ
sin θ,− 1
λ
cos θ, const
)
.
Using a similar idea as Theorem 4.1 and Proposition 4.2, we obtain curves C(θ) that result
in constant p-mean curvature surfaces with constant ζ2 and linear ζ1
(
θ̃
)
in Sections 5.1 and 5.3.
We collect C(θ) in Tables 1 and 2 as follows.
Table 1. Examples of C(θ) for constant p-mean curvature surfaces.
C(θ) constant ζ1 linear ζ1
ζ2 > 1
(
r
λ sin θ,− r
λ cos θ, r(1−r)
λ2 θ
)
0 < r < 1, r ̸= 1
2
ζ2 =
1
|1−2r|
m ̸= 1: (x1(θ), x2(θ), x3(θ))
x1(θ) =
sin θ
2λ − sin((m−1)θ)
2λk(m−1)
x2(θ) = − cos θ
2λ − cos((m−1)θ)
2λk(m−1)
x3(θ) =
1+k2(m−1)
4λ2k2(m−1)
θ − sin(mθ)
4λ2k(m−1)
m = 1:
(
sin θ
2λ − θ
2λk ,−
cos θ
2λ , kθ−θ cos θ
4λ2k
)
ζ1 = mθ + const and ζ2 = k > 0
ζ2 = 1 Pansu sphere
0 < ζ2 < 1
(
r
λ sin θ,− r
λ cos θ, r(1−r)
λ2 θ
)
r < 0 or r > 1
ζ2 =
1
|1−2r|
ζ2 = 0
( β
4λ , 0,
β
4λ − θ
4λ2
)
β = ln | sec θ + tan θ|
ζ1 = 0
(
cos θ, sin θ,−
(
θ + θ
2λ2
))
ζ1 = −θ + const
Table 2. Examples of C(θ) for p-minimal surfaces.
C(θ) constant ζ1 linear ζ1
ζ2 > 0 type I
(r sin θ,−r cos θ, z(θ))
z′(θ) + r2 > 0
ζ1 = −rθ
ζ2 < 0 type II, III
(r sin θ,−r cos θ, z(θ))
z′(θ) + r2 < 0
ζ1 = −rθ
special type I
degenerate case:
(
−θ, 0, sin(2θ)−2θ
4
)
or
entire graph: u = 0
special type II u = xy + g(y)
5.1 Examples of constant p-mean curvature surfaces
Proposition 5.1. Given any curve
C(θ) =
(
1
λ
sin θ,− 1
λ
cos θ,
k − 1
2λ2
θ + const
)
,
the deformed surface Y (s, θ) has the invariants ζ1
(
θ̃
)
= (k − 1)θ̃ + const and ζ2
(
θ̃
)
= k,
where k ∈ R.
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 23
Remark 5.2. It is easy to see that the surfaces obtained by curves given in Proposition 5.1 are
not rotationally symmetric since ζ ′1 = k − 1 < ζ2 by (3.21).
Proposition 5.3. For any constant k > 0 and m, there exist constant p-mean curvature sur-
faces Y (s, θ) defined as (4.1) with invariants ζ1(θ) = mθ + const and ζ2 = k.
Proof. It suffices to solve the system (4.6). In order to obtain a surface with linear ζ1(θ) = mθ
for any given nonzero constant ζ2 = k, we assume
A(θ) =
1
2λk
sin(mθ) and
1
2λ
−B(θ) =
1
2λk
cos(mθ). (5.1)
It results in ∥V ∥ = 1
2λk , and
ζ1(θ) = sin−1
(
A(θ)
∥V ∥
)
− 2λ
(
θ
2λ
−
∫
D(θ) dθ
)
= mθ − θ + 2λ
∫
ζ2(θ)∥V ∥ dθ
= mθ − θ + 2λk
∫
1
2λk
dθ = mθ + const.
Next we solve for x′1(θ) and x′2(θ) from (4.3), that is,
1
2λk
sin(mθ) = x′2(θ) cos θ − x′1(θ) sin θ,
1
2λ
− 1
2λk
cos(mθ) = x′2(θ) sin θ + x′1(θ) cos θ.
It is easy to see that
x′1(θ) =
1
2λ
cos(θ)− 1
2λk
cos((m− 1)θ), x′2(θ) =
1
2λ
sin(θ) +
1
2λk
sin((m− 1)θ),
and hence for m ̸= 1,
x1(θ) =
1
2λ
sin(θ)− 1
2λk(m− 1)
sin((m− 1)θ) + const,
x2(θ) = − 1
2λ
cos(θ)− 1
2λk(m− 1)
cos((m− 1)θ) + const. (5.2)
The equation (4.3) also suggests
x′3(θ) =
1 + k2(m− 1)
4λ2k2(m− 1)
− m cos(mθ)
4λ2k(m− 1)
,
and then we have
x3(θ) =
1 + k2(m− 1)
4λ2k2(m− 1)
θ − sin(mθ)
4λ2k(m− 1)
+ const. (5.3)
Therefore, deforming such curves C(θ) = (x1(θ), x2(θ), x3(θ)) defined by (5.2) and (5.3) gives
surfaces with nonzero ζ2 = k and linear ζ1(θ) = mθ + const for all m ̸= 1.
When m = 1, direct computations from (5.1) imply
x1(θ) =
1
2λ
sin(θ)− θ
2λk
+ const, x2(θ) = − 1
2λ
cos(θ) + const,
x3(θ) =
kθ − θ cos θ
4λ2k
. ■
Example 5.4. If
C(θ) = (x1(θ), x2(θ), x3(θ))
=
(
1
4λ
ln | sec θ + tan θ|+ c3, c4,
c5
4λ
ln | sec θ + tan θ| − 1
4λ2
θ + c6
)
,
then ζ1
(
θ̃
)
= 0, ζ2
(
θ̃
)
= 0.
24 H.-L. Chiu, S.-H. Lai and H.-F. Liu
5.2 Basic properties of surfaces of special type I
For p-minimal surfaces of special type I, we have the first fundamental form, in terms of normal
coordinates (x, y),
I = dx⊗ dx+
(
1 + α2
α4
)
dy ⊗ dy,
so that I degenerates along the curve where α blows up. Recall that the parametrization of the
surface Y is
Y (r, θ) = (x(θ) + r cos θ, y(θ) + r sin θ, z(θ) + ry(θ) cos θ − rx(θ) sin θ).
We have
Yr = (cos θ, sin θ, y(θ) cos θ − x(θ) sin θ), Yθ =
(
x′(θ)− r sin θ, y′(θ) + r cos θ, ∗
)
,
where
∗ = z′(θ) + ry′(θ) cos θ − y(θ) sin θ − x′(θ) sin θ − x(θ) cos θ.
Then
Yr × Yθ =
∣∣∣∣∣∣
i j k
cos θ sin θ y(θ) cos θ − x(θ) sin θ
x′(θ)− r sin θ y′(θ) + r cos θ ∗
∣∣∣∣∣∣
= ρ
(
sin θ
(
y′(θ) cos θ − x′(θ) sin θ
)
− y(θ),
− cos θ
(
y′(θ) cos θ − x′(θ) sin θ
)
+ x(θ), 1
)
,
where ρ = r +
(
y′(θ) cos θ − x′(θ) sin θ
)
. For p-minimal surfaces of special type I, we have
α =
1
r +
(
y′(θ) cos θ − x′(θ) sin θ
) .
Therefore, Yr and Yθ are linearly dependent along the curve when α blows up.
For constant p-mean curvature surfaces of special type I, (4.2) and (4.3) immediately imply
that Ys ∧ Yθ = 0 if and only if
0 = A(θ) cos 2λs+
(
1
2λ
−B(θ)
)
sin 2λs,
−D(θ) = −
(
1
2λ
−B(θ)
)
cos 2λs+A(θ) sin 2λs,
that is,(
cos 2λs sin 2λs
sin 2λs − cos 2λs
)(
A(θ)/∥V ∥(
1
2λ −B(θ)
)
/∥V ∥
)
=
(
0
−G(θ)
)
.
This implies that Ys ∧ Yθ = 0 holds if and only if G(θ) = ±1, namely, it happens only on the
surface Y of special type I at points where the function α blows up.
5.3 Examples of p-minimal surfaces
In what follows, we give some p-minimal surfaces of special type II (i.e., ζ2 < 0 and linear ζ1).
We first recall in [5] that C(θ) = (x(θ), y(θ), z(θ)) satisfying
ζ1(θ) = −Γ(θ) + y′(θ) cos θ − x′ sin θ,
ζ2(θ) = z′(θ) + x(θ)y′(θ)− y(θ)x′(θ)−
(
y′(θ) cos θ − x′(θ) sin θ
)2
,
where Γ(θ) =
∫
x′(θ) cos θ + y′(θ) sin θdθ, will result in a p-minimal surface. For any nonzero
r ∈ R, if x(θ) = r sin θ and y(θ) = −r cos θ, then Γ(θ) = rθ (up to a constant), ζ1(θ) = −rθ,
and ζ2(θ) = z′(θ) + r2. We choose z(θ) such that z′(θ) + r2 < 0 to have negative ζ2.
On Invariants of Constant p-Mean Curvature Surfaces in the Heisenberg Group H1 25
Acknowledgements
The first author’s research was supported in part by NSTC 112-2115-M-007-009-MY3. The
second author’s research was supported in part by NSTC 110-2115-M-167-002-MY2 and NSTC
112-2115-M-167-002-MY2. The third author’s research was supported in part by NSTC 112-
2628-M-032-001-MY4. We all thank the anonymous referees for carefully reading our manuscript
and their insightful comments and suggestions for improving the article.
References
[1] Cheng J.-H., Chiu H.-L., Hwang J.-F., Yang P., Umbilicity and characterization of Pansu spheres in the
Heisenberg group, J. Reine Angew. Math. 738 (2018), 203–235, arXiv:1406.2444.
[2] Cheng J.-H., Hwang J.-F., Malchiodi A., Yang P., Minimal surfaces in pseudohermitian geometry, Ann. Sc.
Norm. Super. Pisa Cl. Sci. 4 (2005), 129–177, arXiv:math.DG/0401136.
[3] Chiu H.-L., Huang Y.-C., Lai S.-H., An application of the moving frame method to integral geometry in the
Heisenberg group, SIGMA 13 (2017), 097, 27 pages, arXiv:1509.00950.
[4] Chiu H.-L., Lai S.-H., The fundamental theorem for hypersurfaces in Heisenberg groups, Calc. Var. Partial
Differential Equations 54 (2015), 1091–1118.
[5] Chiu H.-L., Liu H.-F., A characterization of constant p-mean curvature surfaces in the Heisenberg group H1,
Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780.
[6] Dragomir S., Tomassini G., Differential geometry and analysis on CR manifolds, Progr. Math., Vol. 246,
Birkhäuser, Boston, MA, 2006.
[7] Polyanin A.D., Zaitsev V.F., Handbook of exact solutions for ordinary differential equations, 2nd ed.,
Chapman & Hall/CRC, Boca Raton, FL, 2003.
[8] Ritoré M., Rosales C., Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg
group Hn, J. Geom. Anal. 16 (2006), 703–720, arXiv:math.DG/0504439.
https://doi.org/10.1515/crelle-2015-0044
https://arxiv.org/abs/1406.2444
http://www.numdam.org/item/ASNSP_2005_5_4_1_129_0/
http://www.numdam.org/item/ASNSP_2005_5_4_1_129_0/
https://arxiv.org/abs/math.DG/0401136
https://doi.org/10.3842/SIGMA.2017.097
https://arxiv.org/abs/1509.00950
https://doi.org/10.1007/s00526-015-0818-1
https://doi.org/10.1007/s00526-015-0818-1
https://doi.org/10.1016/j.aim.2022.108514
https://arxiv.org/abs/2101.11780
https://doi.org/10.1007/0-8176-4483-0
https://doi.org/10.1007/BF02922137
https://arxiv.org/abs/math.DG/0504439
1 Introduction
2 Solutions to the Codazzi-like equation
2.1 The Pansu sphere
2.2 The normalization
2.3 The structure of the singular sets
3 Rotationally invariant surfaces in H_1
3.1 The computation of H, alpha, a and b
3.2 Another understanding of energy E
3.3 The Coddazi-like equation
3.4 The relation between k and E
3.5 Horizontal generating curves for c neq 0
3.6 The invariants zeta_1 and zeta_2 for surfaces with c neq 0
3.7 The allowed values of k and E with c=0
3.8 The invariants zeta_1 and zeta_2 for surfaces with c=0 and E neq 0
4 The construction of constant p-mean curvature surfaces
5 Examples
5.1 Examples of constant p-mean curvature surfaces
5.2 Basic properties of surfaces of special type I
5.3 Examples of p-minimal surfaces
References
|
| id | nasplib_isofts_kiev_ua-123456789-212880 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:31:18Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chiu, Hung-Lin Lai, Sin-Hua Liu, Hsiao-Fan 2026-02-13T13:50:02Z 2025 On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁. Hung-Lin Chiu, Sin-Hua Lai and Hsiao-Fan Liu. SIGMA 21 (2025), 011, 25 pages 1815-0659 2020 Mathematics Subject Classification: 53A10; 53C42; 53C22; 34A26 arXiv:2309.14697 https://nasplib.isofts.kiev.ua/handle/123456789/212880 https://doi.org/10.3842/SIGMA.2025.011 One primary objective in submanifold geometry is to discover fascinating and significant classical examples of ₁. In this paper, which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant -mean curvature surfaces, we have identified intriguing examples of such surfaces. Notably, we present a complete description of rotationally invariant surfaces of constant p-mean curvature and shed light on the geometric interpretation of the energy with a lower bound. The first author’s research was supported in part by NSTC 112-2115-M-007-009-MY3. The second author’s research was supported in part by NSTC 110-2115-M-167-002-MY2 and NSTC 112-2115-M-167-002-MY2. The third author’s research was supported in part by NSTC 1122628-M-032-001-MY4. We all thank the anonymous referees for carefully reading our manuscript and their insightful comments and suggestions for improving the article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁ Article published earlier |
| spellingShingle | On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁ Chiu, Hung-Lin Lai, Sin-Hua Liu, Hsiao-Fan |
| title | On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁ |
| title_full | On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁ |
| title_fullStr | On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁ |
| title_full_unstemmed | On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁ |
| title_short | On Invariants of Constant -Mean Curvature Surfaces in the Heisenberg Group ₁ |
| title_sort | on invariants of constant -mean curvature surfaces in the heisenberg group ₁ |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212880 |
| work_keys_str_mv | AT chiuhunglin oninvariantsofconstantmeancurvaturesurfacesintheheisenberggroup1 AT laisinhua oninvariantsofconstantmeancurvaturesurfacesintheheisenberggroup1 AT liuhsiaofan oninvariantsofconstantmeancurvaturesurfacesintheheisenberggroup1 |