Surfaces in the Euclidean 3-space E³ with constant mean curvature H which are often called H-surfaces and those with constant Gauss curvature K are called K-surfaces.

As Bonnet pointed out that K-surfaces and H-surfaces are big classes hard to classify, the so-called Weingarten condition is considered. We call a surface S in E³linear Weingarten if a linear combination of the mean curvature H and the Gauss curvature K is a constant, i.e.,

for some real numbers a, b and c which are not all zero. If a² + bc > 0, the local graph of the surface satisfies the elliptic condition for differential equation relative to the principal curvatures [3, 4, 5].

On the other hand, the eigenvalue problem of an isometric immersion x: M → E^m of a Riemannian manifold M into a Euclidean space E^m is a nice tool to determine a geometric character for a sphere, i.e., if Δx = kx is satisfied for a non-zero real number k, then M is part of a sphere [6]. Generalizing this notion, B.-Y. Chen defined the notion of order and type for the immersion of M into E^m. By definition, a finite-type immersion x: M → E^m of a submanifold M into a Euclidean space E^m means x is decomposed as a finite sum of the eigenvectors of the Laplace operator Δ of M in the following

where x₀ is a constant vector and x₁, …, x_k are non-constant vectors satisfying Δx_i = λ_ix_i, i = 1, 2, …, k. In particular, if all of λ₁, …, λ_k are different, it is called k-type or the submanifold M is said to be of k-type (cf. [1, 2]). Thus, if a submanifold M of E^m is 1-type, then its immersion x satisfies

for some non-zero real number k and a constant vector C.

All surfaces under consideration is smooth and connected unless otherwise stated.

Let S be an oriented surface in the 3-dimensional Euclidean space E³ and x: S → E³ an isometric immersion. Then a unit vector field N called the Gauss map is well-defined on S.

We now assume that the immersion x satisfies (1.1) with a² + bc > 0. In this case, the surface S is called the elliptic linear Weingarten or shortly ELW surface. An ELW surface with b = 0 has constant mean curvature and that with a = 0 has constant Gauss curvature.

We put E₁ = 〈x_s, x_s〉, F₁ = 〈x_s, x_t〉, G₁ = 〈x_t, x_t〉, E₂ = 〈x_ss, N〉, F₂ = 〈x_st, N〉, and G₂ = 〈x_tt, N〉, where x = x(s, t) for some coordinate system (s, t) of S. We then have the first and second fundamental forms, respectively,

Then, similarly to Lemma 1 in [3], we have

Lemma 2.1

Let x: M → E³be an ELW immersion of a surface S in E³satisfying (1.1). Then,

defines a Riemannian metric on M.

The Riemannian metric σ defined in Lemma 2.1 is called the ELW metric. Then we have the Gauss map η relative to the Riemannian metric σ, which is called the associated Gauss map.

Let (u, v) be the isothermal coordinates for σ. If we adopt the same notations by E₁ = 〈x_u, x_u〉, F₁ = 〈x_u, x_v〉, G₁ = 〈x_v, x_v〉, E₂ = 〈x_uu, N〉, F₂ = 〈x_uv, N〉, and G₂ = 〈x_vv, N〉 as above relative to the isothermal coordinates (u, v), we have

for some positive function λ. Without loss of generality, we may assume that

by taking the appropriate direction for the Gauss map if necessary. In particular, (2.3) shows that if b = 0, then we may assume that a = 1. In this case, the ELW metric σ is nothing but the first fundamental form I. The ELW metric σ is said to be non-trivial if b ≠ 0.

We then have the Laplacian Δ^σ with respect to the Riemannian metric σ by

If we compute λ² by using (2.2), we have

from which,

Since 2aH + bK = c ≥ 0, we get

or, equivalently

Then, we get

Lemma 2.2

Let S be an ELW surface in E³satisfying (1.1) with a² + bc = 1. Then, the associated Gauss map η and the Gauss map N are the same.

From the first and second fundamental forms I and II, we have the shape operator A of the form

where

As is given in [3], we have

Theorem 2.3

Let x: S → E³be an ELW immersion satisfying (1.1) with a²+bc = 1. Then, we have

In this section, we characterize harmonic and bi-harmonic ELW surfaces in E³ with respect to the ELW metric σ.

Let S be an ELW surface with the metric σ defined by (2.1) satisfying a²+bc = 1.

Definition 3.1

An ELW surface S in E³ is said to be σ-harmonic or ELW harmonic if its immersion x satisfies Δ^σx = 0. It is said to be σ-biharmonic or ELW biharmonic if its immersion x satisfies (Δ^σ)²x = 0.

First of all, we prove

Theorem 3.2

Let S be an ELW surface in E³with the ELW metric σ. Then, S is σ-harmonic if and only if S is minimal or part of a plane.

Theorem 3.3

Let S be an ELW surface in E³with the ELW metric σ. Then, if S is σ-biharmonic, then S is part of either a minimal surface or an isoparametric surface in E³, i.e., S is part of a sphere, a plane or a circular cylinder. Conversely, if we take appropriate real numbers a, b and c, then a sphere, a plane or a minimal surface is σ-biharmonic.

Let S be a surface in E³ with non-degenerate second fundamental form via an isometric immersion x: S → E³.

Let ∇̃be the Levi-Civita connection on E³ and ∇ the induced connection on S. Then, the Gauss and Codazzi equations of S in E³ are respectively given by

where A is the shape operator of S and X, Y and Z are the vector fields tangent to S.

Since A is non-degenerate, we can choose a coordinate patch x(u, v) on a neighborhood around p such that x_u, x_v are in the principal directions. Then, we have

with respect to the coordinate frame {x_u, x_v} so that the mean curvature and the Gaussian curvature are respectively given by H = (κ₁ + κ₂)/2 and K = κ₁κ₂. Define a symmetric tensor h by

for tangent vector fields X and Y to S.

Since h is non-degenerate, h is regarded as a non-degenerate metric on M, which is called the II-metric with representation given by

On the other hand, it is easy to derive

Without loss of generality, we may regard as κ₁ > 0. We put

for some positive functions a and b, where ɛ = ±1 depending upon the signature of h₂₂. Then, we have the equations of Gauss

We then define the II-Laplace operator Δ^II with respect to the metric h by

If we put f = b/a, then (4.12) can be written as

Using (4.9), (4.10) and (4.11), we have

Lemma 4.1

Let M be a surface of S³(1) with non-degenerate second fundamental form. Then, we have

We then have immediately from Lemma 4.1

Proposition 4.2

There do not exist II-harmonic surfaces of S³(1) with non-degenerate second fundamental form satisfying Δ^IIx = 0.

Suppose that the surface S satisfies Δ^IIx = kx+C for some real number k ≠ 0 and a constant vector C, that is, S is of 1-type with respect to II-metric. From equation (4.14), we see that kx + C is in the normal direction, i.e., kx + C = ρN for some function ρ. It follows that

from which, we get

and ρ is a constant. Thus, the surface S is part of a sphere.

Conversely, suppose that the surface S is part of sphere with radius r. Without loss of generality, we may assume that the center of S is the origin. It is straightforward to compute

Therefore, S is of 1-type with respect to II-metric. Therefore, we have

Theorem 4.3

Let S be a surface of E³with non-degenerate fundamental form. Then, S is of 1-type with respect to II-metric if and only if S is part of a sphere.

In this section, we discuss about the geometric meaning of the Gauss curvature K^σ on the ELW surface M defined by the Riemannian metric σ. We call K^σ the ELW-Gauss curvature. Let ∇^σ be the Levi-Civita connection compatible with the Riemannian metric σ on M.

By straightforward computation, we have the following

Lemma 5.1

Let M be an ELW surface with the metric σ. Then, the Christoffel symbolsΓ¯jihare given by

Making use of Lemma 5.1, we have the ELW-Gauss curvature K^σ

where R^σ is the curvature tensor defined by ∇^σ and we have put

for tangent vector field X, Y, Z, W on M.

We now consider the non-trivial ELW metric σ with a constant λ, i.e., S satisfies a² + bc = 1 with b ≠ 0.

Theorem 5.2

Let S be a non-trivial ELW surface of E³with the ELW metric σ. Then, S is flat if and only if λ is constant.

Corollary 5.3

Let S be a non-trivial ELW surface with the ELW metric σ. If S is flat, the ELW-Gauss curvature K^σ of S vanishes.

A complete Riemannian manifold (M, g) is a Ricci soliton if there exists a smooth function f on M satisfying

for some constant ρ, where ∇² is the Hessian defined by (∇²f)(X, Y) = ∇_X∇_Yf − (∇_XY)f for vector fields X, Y on M. In this case, f is called a potential function of the Ricci soliton. The Ricci soliton is called steady if ρ = 0, shrinking if ρ > 0 and expanding if ρ < 0. If f is constant, (M, g) is Einstein. Thus, a Ricci soliton is a natural extension of Einstein manifolds.

Let (S, σ) be a Ricci soliton and ELW surface in E³ with λ as a potential function, i.e., M satisfies

for a constant ρ, where Ric^σ is the Ricci tensor associated with the metric σ and ∇^σ is the Levi-Civita connection on S compatible with σ. The Ricci tensor is given by

where e₁, e₂ are orthonormal frame along S with respect to the metric σ and R^σ is the curvature tensor defined by (5.4).

Let x: S → E³ be an immersion of a Ricci soliton and ELW surface S into E³ with the isothermal coordinate system (u, v) with respect to σ. Then, we have a natural orthonormal frame e₁ = x_u/λ and e₂ = x_v/λ. Using (6.2), we get

Therefore, equation (6.1) gives

from which, we get

for some non-zero functions Φ = Φ(u) and Ψ = Ψ(v). Therefore, we have

Suppose λ_u ≠ 0 on an open subset U on S. Then, λ_u = λ_v and the functions Φ and Ψ are constant on U. Thus, on U, λ_v = λ_u = λC for some non-zero constant C. It follows u = v, which is a contradiction. Hence, λ_u = 0 on S. Similarly, we can have λ_v = 0 on S. Thus, λ is a constant. According to Corollary 5.3, S is flat and its ELW Gauss curvature K^σ is also vanishing.

Conversely, if the function λ is constant, it is trivial that S is a Ricci soliton satisfying (6.1).

Thus, we have

Theorem 6.1

Let S be an ELW surface in E³with the ELW metric σ. Then, S is a Ricci soliton with respect to σ with λ as a potential function if and only if S is part of a plane or a circular cylinder.