Associate and conjugate minimal immersions in $\boldsymbol{M} \times \boldsymbol{R}$

Abstract

A beautiful phenomenon in Euclidean space is the existence of a 1-parameter family of minimal isometric surfaces connecting the catenoid and the helicoid. They are associate. A well-known fact, is that any two conformal isometric minimal surfaces in a space form are associate. What happens in other 3dimensional manifolds ?In this paper we will discuss the same phenomenon in the product space, M × R, giving a definition of associate minimal immersions. We specialize in the situations M = H 2 , the hyperbolic plane, and M = S 2 , the sphere where surprising facts occur. We will prove some existence and uniqueness results explained in the sequel. We begin with the definition.Let M be a two dimensional Riemannian manifold. Let (x, y, t) be local coordinates in M × R, where z = x + iy are conformal coordinates on M and t ∈ R. Let σ 2 | dz| 2 , be the conformal metric in M, hence ds 2 = σ 2 | dz| 2 + dt 2 is the metric in the product space M×R. Let Ω ⊂ C be a planar simply connected domain, w = u + iv ∈ Ω. We recall that if X : Ω → M × R, w → (h(w), f (w)), w ∈ Ω, is a conformal minimal immersion then h : Ω → (M, σ 2 | dz| 2 ) is a harmonic map. We recall also that for any harmonic map h : Ω ⊂ C → M there exists a related Hopf holomorphic function Q(h). Two conformal immersions X = (h, f ), X * = (h * , f * ) : Ω → H 2 × R are said associate if they are isometric and if the Hopf functions satisfy the relation Q(h * ) = e 2iθ Q(h) for a real number θ. If Q(h * ) = −Q(h) then the two immersions are said conjugate.In this paper we will show that there exist two conformal isometric minimal surfaces in H 2 ×R, with constant Gaussian curvature −1, that are non associate. We will prove also that the vertical cylinder over a planar geodesic in H 2 × R, are the only minimal surfaces with constant Gaussian curvature K ≡ 0.One of our principal results is a uniqueness theorem in H 2 × R, or S 2 × R, showing that the conformal metric and the Hopf function determine a minimal conformal immersion, up to an isometry of ambient space, see Theorem 4. We will derive the existence of the minimal associate family in H 2 × R and S 2 × R

Totally umbilic surfaces in homogeneous 3-manifolds

“…More generally many works are devoted to studying the geometry of surfaces in homogeneous 3-manifolds. See for example [14], [6], [7], [17], [15], [12], [13], [11], [9], [4], [2], [10], [5] and [8].…”

Minimal isometric immersions into S^2 x R and H^2 x R

“…The existence of the associate family was also proved by L. Hauswirth, R. Sa Earp and E. Toubiana [13] using the harmonicity of the horizontal and vertical projections of conformal minimal immersions. We also mention that L. Hauswirth and H. Rosenberg [12] developped the theory of complete finite total curvature minimal surfaces in H 2 × R using the relation between the angle function and solutions to the elliptic sinh-Gordon equation.…”

“…The systematic study of minimal surfaces in S 2 × R and H 2 × R was initiated by H. Rosenberg and W. Meeks [20,17] and has been very active since then. The existence of an associate family for simply connected minimal surfaces in S 2 × R and H 2 × R was proved independently by the author [4] and by L. Hauswirth, R. Sa Earp and E. Toubiana [13].…”

“…Remark 2.6. L. Hauswirth, R. Sa Earp and E. Toubiana [13] studied minimal immersions using the fact that the projections into H 2 and into R are harmonic. In particular they proved that two minimal isometric immersions whose height functions have the same Hopf differential are congruent.…”

“…This surface is consequently isometric to a horizontal hyperbolic plane H 2 × {a}, but these two surfaces are not associate (actually this example provides a two-parameter family of non associate minimal isometric immersions). The parabolic generalized catenoid and the helicoid of Example 18 in [13] are associate. Also, R. Sa Earp [21] gave examples of pairs of non associate isometric minimal surfaces in H 2 × R that are invariant by hyperbolic screw-motions (see Example 5.5 for details).…”