Constant mean curvature surfaces inhomogeneous manifolds

Benoıt Daniel

Homogeneous manifolds

A manifold Mn is said to be homogeneous if, for every pair (x, y) ofpoints in M , there exists an isometry ϕ of M such that y = ϕ(x).

Let M3 be a simply connected homogeneous 3-manifold and G itsisometry group.

I If dimG = 6, then M has constant sectional curvature :Euclidean space R3, round sphere S3(κ), hyperbolic space H3(κ).

I If dimG = 4, then M can be of 5 different types. Together withR3 and round spheres, these manifolds belong to a 2-parameterfamily of manifolds denoted by E3(κ, τ).

I If dimG = 3, then M is one of certain Lie groups endowed with aleft-invariant metric.

There exists a Riemannian submersion π : E3(κ, τ)→M2(κ) whereM2(κ) is the simply connected surface of curvature κ. If ξ denotes aunit vertical vector field, i.e., dπ(ξ) = 0, then we have

∇Xξ = τX × ξ

for every vector field X (∇ is the Riemannian connection of E3(κ, τ)).The real number τ is called the bundle curvature.

κ < 0 κ = 0 κ > 0τ = 0 H2(κ)× R R3 S2(κ)× R

τ 6= 0 PSL2(R) Nil3round spheres,Berger spheres

Table: The different kinds of manifolds E3(κ, τ)

I PSL2(R) = UH2(κ) endowed with a Sasaki metric (U : unittangent bundle).

I Round spheres and Berger spheres = US2(κ) endowed with aSasaki metric.

I Nil3 =

1 a1 a2

0 1 a3

0 0 1

| (a1, a2, a3) ∈ R3

.

Isometries

Any isometry of E3(κ, τ) induces an isometry of M2(κ). Translationsalong fibers (vertical geodesics) are isometries called verticaltranslations. Around any vertical geodesic there exists aone-parameter group of rotations. Around any horizontal geodesic,the rotation by π is an isometry.

In H2(κ)×R and S2(κ)×R there also exist reflections with respect tothe totally geodesic surfaces H2(κ)× t or S2(κ)× t and π−1(γ)where γ is a geodesic of H2(κ) or S2(κ).

Sectional curvatures of E3(κ, τ) : τ2, τ2, κ− 3τ2.

Eight Thurston geometries : R3, S3, H3, S2 × R, H2 × R, Nil3,

PSL2(R) and Sol3.

Only H3 and Sol3 do not belong to the E3(κ, τ) family.

Uniqueness of CMC spheres

Theorem (Abresch-Rosenberg)Any CMC sphere in E3(κ, τ) is rotational.

Isometric immersions

Let M2 be an immersed surface in M3. Then for all tangent vectorfields X, Y to M we have

(Gauss) K − K = detS

(Codazzi) ∇XSY −∇Y SX − S[X,Y ] = R(X,Y )N

where

I K = intrinsic curvature of M ,

I K = (extrinsic) sectional curvature of TM in TM ,

I S = shape operator of M in M ,

I ∇ = Riemannian connection of M ,

I R = curvature tensor of M ,

I N = unit normal to M in M .

If M has constant sectional curvature c, then these equations become

K = detS + c,

∇XSY −∇Y SX − S[X,Y ] = 0.

They are well defined given the metric of M and the symmetricoperator S : TM → TM , and they are a necessary and sufficientcondition for M to be isometrically immersed into M with S as shapeoperator when M is simply connected.

When M = E3(κ, τ), these equations become

K = detS + τ2 + (κ− 4τ2)ν2,

∇XSY −∇Y SX − S[X,Y ] = (κ− 4τ2)ν(〈Y, T 〉X − 〈X,T 〉Y )

where T is the orthogonal projection of ξ on TM and ν = 〈N, ξ〉.Moreover, from ∇Xξ = τX × ξ we get

∇XT = ν(SX − τJX),

dν(X) + 〈SX − τJX, T 〉 = 0

where J is the rotation by π/2 on TM .

Theorem (D.)Let M be a simply connected surface endowed with a metric ds2.Then M can be isometrically immersed into E3(κ, τ) with givenS : TM → TM , T ∈ X (M) and ν : M → R such that ||T ||2 + ν2 = 1if and only if the quadruple (ds2, S, T, ν) satisfies the four previousequations (compatibility equations).

Sister surfaces

Theorem (D.)There exists an isometric correspondence between CMC H1 surfacesin E3(κ1, τ1) and CMC H2 surfaces in E3(κ2, τ2) if

κ1 − 4τ21 = κ2 − 4τ2

2 , H21 + τ2

1 = H22 + τ2

2 .

Moreover, their data (ds2, S1, T1, ν1) and (ds2, S2, T2, ν2) satisfy

S2 −H2I = eθJ(S1 −H1I), T2 = eθJT1, ν2 = ν1

withτ2 + iH2 = eiθ(τ1 + iH1).

Particular cases:

I minimal surfaces in H2(κ)× R and S2(κ)× R admit an associatefamily (any θ works),

I when θ = π/2 we may expect to use this to do a “conjugatecousin Plateau construction”.

Minimal surfaces in the Heisenberg group Nil3

We view Nil3 = E3(0, 1/2) as R3 endowed with the metric

dx21 + dx2

2 + (1

2(x2dx1 − x1dx2) + dx3)2.

In Nil3, “translations” are left multiplications by elements in Nil3 forthe product coming from the Lie group structure. The group oftranslations is a normal subgroup of Isom(Nil3). They are given by

(x1, x2, x3) 7→(x1 + t, x2, x3 +

1

2tx2

),

(x1, x2, x3) 7→(x1, x2 + t, x3 −

1

2tx1

),

(x1, x2, x3) 7→ (x1, x2, x3 + t).

The Gauss map

Let X : Σ→ Nil3 be a conformal immersion, N its unit normal. Byidentifying TxNil3 with TONil3 by left multiplication, we may view Nas an application from Σ to the unit sphere of TONil3. Thencomposing with a stereographic projection we get a mapg : Σ→ C = C ∪ ∞ called the Gauss map.

Equivalently,

N =1

1 + |g|2

2 Re g2 Im g

1− |g|2

in the orthonormal frame (E1, E2, E3) defined by

E1 =∂

∂x1− x2

2

∂

∂x3, E2 =

∂

∂x2+x1

2

∂

∂x3, E3 = ξ =

∂

∂x3.

Theorem (D.)Let X : Σ→ Nil3 be a conformal minimal immersion. Then its Gaussmap g : Σ→ C satisfies

(1− |g|2)gzz + 2ggzgz = 0.

Assume now that X(Σ) is nowhere vertical, i.e., transverse to thefibers. Then up to a change oriention N points up and so |g| < 1.Then this equation means that g is harmonic in D = q ∈ C; |q| < 1endowed with the hyperbolic (Poincare) metric

4|dq|2

(1− |q|2)2.

Weierstrass-type representation

Theorem (D.)Let Σ be a simply connected Riemann surface and g : Σ→ D ≡ H2

harmonic and nowhere antiholomorphic. Then there exists a unique(up to translations) conformal minimal immersionX = (x1, x2, x3) : Σ→ Nil3 whose Gauss map is g. Moreover, settingF = x1 + ix2, X is given by

Fz = −4igz

(1− |g|2)2, Fz = −4i

g2gz(1− |g|2)2

,

(x3)z = 4iggz

(1− |g|2)2− i

4(FFz − FFz),

and the induced metric is

ds2 = 16(1 + |g|2)2

(1− |g|2)4|gz|2|dz|2. (1)

Complete graphs

A surface M ⊂ Nil3 is a

I a local graph if M is transverse to ξ, i.e., nowhere vertical,

I a graph if M is transverse to ξ and πM : M → R2 is injective,

I an entire graph if πM : M → R2 is a diffeomorphism.

Complete minimal graphs in Nil3 are rather well understood.

I Any complete local graph is an entire graph (D.-Hauswirth).

I Entire graphs are classified (Fernandez-Mira) using results ofWan-Au on harmonic maps into H2. In particular some havehyperbolic conformal type (e.g. x3 = 0) and some have parabolicconformal type (e.g. x3 = x1x2/2).

I We can construct entire graphs with prescribed Gaussian imageusing results of Choi-Treibergs and Han-Tam-Treibergs-Wan onharmonic maps into H2.

Collin-Krust type theorem

Minimal graph equation:

div

(V√

1 + |V |2

)= 0, V = (ux1

+ x2/2, ux2− x1/2).

Theorem (Leandro-Rosenberg)Let Ω ⊂ R2 and u, v : Ω→ R two solutions of the minimal graphequation over Ω such that u|∂Ω ≡ v|∂Ω. Then either u ≡ v or u− v isunbounded.

Construction of properly embedded minimal annuli

Theorem (D.-Hauswirth)There exists a one parameter family (Cα)α>0 of properly embeddedminimal annuli Nil3, called “horizontal catenoid”, having thefollowing properties:

I Cα is not invariant by a one parameter group of isometries,

I the intersection of Cα and any vertical plane x2 = c is anon-empty embedded convex curve,

I Cα is invariant by rotations by π around the x1, x2 and x3 axes,and the x2 axis lies “inside” Cα,

I Cα is conformally equivalent to C \ 0.

Idea:

I start with g : C→ C given by

g(u+ iv) =sinϕ(u) + i sinh(αv)

cosϕ(u) + i cosh(αv)

where α > 0 and ϕ′(u)2

= α2 + cos2 ϕ(u) and integrate theWeierstrass-type formulas: we get a minimal “horizontalhelicoid”,

I consider the Gauss maps in the associate family (gθ)θ∈R/2πZ andfind θ so that the period closes.

These annuli Cα have∫|K| = +∞ (also K changes sign).

I Is there a geometric quantity C, replacing total curvature, suchthat if E is an embedded annular end such that C < +∞, then Eis asymptotic to an end of Cα or to a vertical plane?

I If E is a properly embedded annular end with vanishing verticalflux, then is E is asymptotic to an end of Cα or to a verticalplane?

I Construct other properly embedded annuli.

I Can we caracterize Cα among annuli?

Stability

Theorem (Manzano-Perez-Rodriguez)Let M be a complete orientable stable parabolic CMC surface inE3(κ, τ). Then one of the following statements hold:

I E3(κ, τ) = S2(κ)× R and M is a (minimal) slice S2(κ)× t,I M has mean curvature H such that H2 6 −κ/4 and M is either

a local graph or a vertical cylinder over a curve of curvature 2Hin M2(κ).

In particular a complete orientable stable parabolic minimal surfacein Nil3 is a vertical plane or an entire graph. If one had this theoremfor all complete minimal surfaces, then one would be able to prove astrong half-space theorem using the half-space theorems ofD.-Hauswirth and D.-Meeks-Rosenberg.

Construction of properly embedded CMC 1/2 annuli inH2 × R

Theorem (D.-Hauswirth)There exists a one parameter family (Cα)α>0 of properly embeddedCMC 1

2 annuli in H2 × R, called “horizontal catenoids“, such that:

I Cα is not invariant by a one parameter group of isometries,

I Cα has one horizontal and two vertical planes of symmetries, andit is a bigraph over some domain in the horizontal plane,

I Cα is conformally equivalent to C \ 0.

This annulus is explicit in terms of a solution of an autonomous order1 ODE so we know its asymptotic behaviour.

Idea: this is the sister surface of a minimal ”horizontal helicoid“ inNil3. It can be obtained explicitely using the Weierstrass-typerepresentation formula for CMC 1/2 surfaces in H2 × R byFernandez-Mira.

Gauss map in PSL2(R)

Theorem (D.-Fernandez-Mira, work in progress)For all surfaces in PSL2(R) we can define a Gauss map with values inS2 which is harmonic into a hemisphere endowed with a hyperbolicmetric for all CMC

√−κ/2 local graphs.

Remarks:

I CMC√−κ/2 surfaces in PSL2(R) are sister surfaces of minimal

surfaces in Nil3,

I this Gauss map also exists in H2 × R and extends theFernadez-Mira Gauss map (which is defined for local graphsonly),

I the Gauss map of any horocylinder is constant.