Rotational symmetry of Weingarten spheres in homogeneous three-manifolds

A Statement of the results in ℝ 3 and 𝕊 3

As explained in Remark 2.1, the Euclidean space ℝ 3 can be seen as a degenerate case of the spaces 𝔼 3 ⁢ ( κ , τ ) , obtained by choosing κ = τ = 0 . A similar situation happens in the round sphere 𝕊 3 ⁢ ( c ) , which can be recovered as the degenerate case of the spaces 𝔼 3 ⁢ ( κ , τ ) for κ = 4 ⁢ τ 2 = 4 ⁢ c > 0 . Note that 𝕊 3 ⁢ ( c ) can be viewed as ( 𝕊 3 , g ) with the metric g in (B.1) for κ = 4 ⁢ τ 2 .

It then turns out that the results that we have obtained here for M = 𝔼 3 ⁢ ( κ , τ ) also work for M = ℝ 3 and 𝕊 3 ⁢ ( c ) , in general with simpler arguments and computations. Next, we state some of these results explicitly.

Let M be ℝ 3 or 𝕊 3 ⁢ ( c ) , and let ξ denote a unit Killing field on M. Then, for any immersed oriented surface Σ in M we can define its angle function in the direction ξ as ν := 〈 η , ξ 〉 ∈ C ∞ ⁢ ( Σ ) , where η is the unit normal of Σ in M.

In these conditions, the definition of a general (elliptic) Weingarten surface given in Definition 3.2 also makes sense when M = ℝ 3 , 𝕊 3 ⁢ ( c ) .

As proved for 𝔼 3 ⁢ ( κ , τ ) -spaces, given a general Weingarten class 𝒲 in M, there is an inextendible rotational surface S of the class 𝒲 with rotation axis L tangent to ξ, and such that S touches L orthogonally at some point. Then the following result holds, which corresponds to Theorem 1.6 for our situation.

The statement of Theorem 6.1 also holds if M = ℝ 3 or 𝕊 3 ⁢ ( c ) . Thus, its Corollary 6.2 also holds in these cases. We must note, however, that Corollary 6.2 was already known if M = ℝ 3 , as a consequence of previous results by the authors; see [5, 13, 14].

Similarly, the Minkowski-type result in Theorem 7.3 also holds for M = ℝ 3 or 𝕊 3 ⁢ ( c ) , but for ℝ 3 it is an immediate consequence of the classical solution to the Minkowski problem. For M = 𝕊 3 ⁢ ( c ) , Theorem 7.3 seems new, and is motivated by the following natural extension of the classical Minkowski problem (which Theorem 7.3 solves in the case that 𝒦 is rotationally symmetric in some direction).

Proposition 8.4 also holds when the ambient space is ℝ 3 instead of ℍ 2 × ℝ . In ℝ 3 one can also create, similarly to Proposition 8.3, examples of general Weingarten classes 𝒲 for which the canonical rotational example S is a non-complete graph of unbounded second fundamental form.

B On the geometry of Berger spheres

In this Appendix we describe in more detail the geometry of Berger spheres, i.e. of the 𝔼 3 ⁢ ( κ , τ ) spaces with κ > 0 and τ ≠ 0 , all of which are diffeomorphic to 𝕊 3 .

Consider first of all 𝕊 3 = { ( z , w ) ∈ ℂ 2 : | z | 2 + | w | 2 = 1 } . A basis of the tangent bundle of 𝕊 3 is given by the vector fields

The Berger sphere M = 𝔼 3 ⁢ ( κ , τ ) can be seen then as ( 𝕊 3 , g ) , where g is the Riemannian metric on 𝕊 3 given for any X , Y ∈ T ⁢ 𝕊 3 by

where 〈 ⋅ , ⋅ 〉 denotes the Euclidean metric in ℝ 4 ≡ ℂ 2 .

Let 𝕊 2 ⁢ ( κ ) denote the two-dimensional sphere of constant curvature κ > 0 , which we will view as 𝕊 2 ⁢ ( κ ) = { x ∈ ℝ 3 : 〈 x , x 〉 = 1 κ } . Then the Hopf fibration π : ( 𝕊 3 , g ) → 𝕊 2 ⁢ ( κ ) , given by

is a Riemannian submersion, with kernel generated by the vector field ξ ^ , which is a Killing field of ( 𝕊 3 , g ) of constant length. Thus, π corresponds to the canonical submersion of 𝔼 3 ⁢ ( κ , τ ) , and ξ = ξ ^ / | ξ ^ | is the vertical unit Killing field of 𝔼 3 ⁢ ( κ , τ ) .

In order to construct a canonical coordinate model ℛ 3 ⁢ ( κ , τ ) associated to the space, we first parametrize 𝕊 2 ⁢ ( κ ) ∖ { ( 0 , 0 , - 1 / κ ) } by inverse stereographic projection:

where

This provides the ( x 1 , x 2 ) -coordinates in ℛ 3 ⁢ ( κ , τ ) . We let the x 3 -coordinate of ℛ 3 ⁢ ( κ , τ ) be the unit speed parametrization of the fiber π - 1 ⁢ ( φ ⁢ ( x 1 , x 2 ) ) . In this way, we obtain that the coordinates ( x 1 , x 2 , x 3 ) ∈ ℝ 3 cover 𝕊 3 minus the fiber π - 1 ⁢ ( ( 0 , 0 , - 1 κ ) ) , which equals

More specifically, this ℛ 3 ⁢ ( κ , τ ) model corresponds then to the universal cover of 𝕊 3 ∖ L * , and two points ( x 1 , x 2 , x 3 ) and ( x 1 , x 2 , x 3 + 8 ⁢ τ ⁢ π κ ) correspond to the same point of 𝕊 3 .

Explicitly, we have the isometric immersion Ψ : ( ℛ 3 ⁢ ( κ , τ ) , d ⁢ s 2 ) → ( 𝕊 3 ∖ L * , g ) ,

where λ is given by (B.2) and d ⁢ s 2 is the metric (2.1).

Note that the x 3 -axis in ℛ 3 ⁢ ( κ , τ ) corresponds to the universal cover of the fiber

of 𝕊 3 , and that the fiber L * does not appear in this ℛ 3 ⁢ ( κ , τ ) model. In order to have a model for 𝕊 3 , where both L , L * appear, we consider the stereographic projection of 𝕊 3 ∖ { p N } ⊂ ℝ 4 into ℝ 3 , p N := ( 0 , 0 , 0 , 1 ) , given by

In these ( y 1 , y 2 , y 3 ) -coordinates, L ∖ { p N } corresponds to the y 3 -axis, while the fiber L * corresponds diffeomorphically to the circle { y 1 2 + y 2 2 = 1 , y 3 = 0 } .

With these different models in hand, we turn now our attention to rotational surfaces S in ( 𝕊 3 , g ) . Assume that the rotation axis of S is the fiber L, and that S remains away from its antipodal fiber L * . Then, in the ℛ 3 ⁢ ( κ , τ ) model, S defines a rotational surface S ^ around the x 3 -axis, that can be parametrized as

In the ( y 1 , y 2 , y 3 ) -coordinates for 𝕊 3 ∖ { p N } , S is given by

where σ := κ 4 ⁢ τ and we are denoting ρ = ρ ⁢ ( u ) , h = h ⁢ ( u ) . We note that the surface in ℝ 3 given by (B.5) is a rotational surface around the y 3 -axis, with profile curve in the ( y 1 , y 3 ) -plane given by

When the rotational surface S approaches its opposite axis L * in 𝕊 3 , the radius ρ of its associated surface S ^ in the ℛ 3 ⁢ ( κ , τ ) model blows up, while in the ( y 1 , y 2 , y 3 ) -coordinates, the profile curve (B.6) converges to the point ( 1 , 0 ) .