Harmonic diffeomorphisms and maximal surfaces Rabah Souam CNRS, - - PowerPoint PPT Presentation

Harmonic diffeomorphisms and maximal surfaces

Rabah Souam CNRS, Institut de Math´ ematiques de Jussieu, Paris Joint work with A. Alarc´

• n

PADGE 2012 Leuven, August 2012

Harmonic maps between Riemannian manifolds

Let M = (Mm,g) and N = (Nn,h) be smooth Riemannian manifolds. Given a smooth map f : M → N and a domain Ω ⊂ M, the quantity (possibly infinite) EΩ(f) = 1 2

• Ω |df|2dVg

is said to be the energy of f over Ω. Energy density: |df|2 = trg(f∗h). A smooth map f : M → N is said to be harmonic if it is a critical point of the energy functional for variations with compact support.

Euler-Lagrange equation

Let (x1,...,xm) be local coordinates in M and (y1,...,yn) local coordinates in N and set f = (f1,...,fn), the map f is harmonic iff it satisfies the quasi-linear elliptic system of second order: ∆Mfi + ∑

j,l,α,β

gαβ(x) Γi

jl(f(x)) ∂φj

∂xα (x) ∂φl ∂xβ (x) = 0, i = 1,...,n. ∆M denotes the Laplace-Beltrami operator on M.

Well known facts

An isometric immersion f : M → N is harmonic if and only if f(M) is a minimal submanifold of N.

Well known facts

An isometric immersion f : M → N is harmonic if and only if f(M) is a minimal submanifold of N. A map f = (f1,f2) : (M,g) → (N1 x N2,h1 ⊕h2) is harmonic if and only if f1 and f2 are harmonic.

Well known facts

An isometric immersion f : M → N is harmonic if and only if f(M) is a minimal submanifold of N. A map f = (f1,f2) : (M,g) → (N1 x N2,h1 ⊕h2) is harmonic if and only if f1 and f2 are harmonic. Harmonicity of a map from a Riemann surface is well defined. If M = (M2,g) is a surface, then the energy integral of a smooth map f : M → N is invariant under conformal changes

• f the metric g, and thus so is the harmonicity of f.

Well known facts

An isometric immersion f : M → N is harmonic if and only if f(M) is a minimal submanifold of N. A map f = (f1,f2) : (M,g) → (N1 x N2,h1 ⊕h2) is harmonic if and only if f1 and f2 are harmonic. Harmonicity of a map from a Riemann surface is well defined. If M = (M2,g) is a surface, then the energy integral of a smooth map f : M → N is invariant under conformal changes

• f the metric g, and thus so is the harmonicity of f.

Harmonicity of a map into a Riemannian surface N = (N2,h) is highly sensitive to conformal changes of the metric h.

Well known facts

An isometric immersion f : M → N is harmonic if and only if f(M) is a minimal submanifold of N. A map f = (f1,f2) : (M,g) → (N1 x N2,h1 ⊕h2) is harmonic if and only if f1 and f2 are harmonic. Harmonicity of a map from a Riemann surface is well defined. If M = (M2,g) is a surface, then the energy integral of a smooth map f : M → N is invariant under conformal changes

• f the metric g, and thus so is the harmonicity of f.

Harmonicity of a map into a Riemannian surface N = (N2,h) is highly sensitive to conformal changes of the metric h. If M = (M2,g) and N = (N2,h) are two oriented Riemannian surfaces then a map f : M → N which is holomorphic or antiholomorphic (for the underlying complex structures) is harmonic.

Existence or not of harmonic diffeomorphisms

[Liouville] There is no non-constant harmonic map C → D, with the euclidean metric.

Existence or not of harmonic diffeomorphisms

[Liouville] There is no non-constant harmonic map C → D, with the euclidean metric. [Heinz 1952] There is no harmonic diffeomorphism D → C with the euclidean metric.

Existence or not of harmonic diffeomorphisms

[Liouville] There is no non-constant harmonic map C → D, with the euclidean metric. [Heinz 1952] There is no harmonic diffeomorphism D → C with the euclidean metric. Consequence: Bernstein’s theorem: An entire minimal graph

• ver the euclidean plane in R3 is a plane.

Existence or not of harmonic diffeomorphisms

[Liouville] There is no non-constant harmonic map C → D, with the euclidean metric. [Heinz 1952] There is no harmonic diffeomorphism D → C with the euclidean metric. Consequence: Bernstein’s theorem: An entire minimal graph

• ver the euclidean plane in R3 is a plane.

Question (Schoen-Yau 1985)

Are Riemannian surfaces which are related by a harmonic diffeomorphism quasiconformally related?

Existence or not of harmonic diffeomorphisms

[Liouville] There is no non-constant harmonic map C → D, with the euclidean metric. [Heinz 1952] There is no harmonic diffeomorphism D → C with the euclidean metric. Consequence: Bernstein’s theorem: An entire minimal graph

• ver the euclidean plane in R3 is a plane.

Question (Schoen-Yau 1985)

Are Riemannian surfaces which are related by a harmonic diffeomorphism quasiconformally related? No!

Theorem (Markovic 2002)

There is a pair of Riemannian surfaces of infinite topology which are related by a harmonic diffeomorphism but not by a quasiconformal diffeomorphism.

Existence or not of harmonic diffeomorphisms

But...

Theorem (Markovic 2002)

The answer to the question by Schoen and Yau is positive in the finite topology case, under some additional assumptions.

Existence or not of harmonic diffeomorphisms

Conjecture (Schoen-Yau 1985)

There is no harmonic diffeomorphism from C onto the hyperbolic plane H.

Existence or not of harmonic diffeomorphisms

Conjecture (Schoen-Yau 1985)

There is no harmonic diffeomorphism from C onto the hyperbolic plane H. [Collin-Rosenberg 2010] There exists an entire minimal graph Σ over H in the Riemannian product H×R with the conformal type of C. In particular, the vertical projection Σ → H is a harmonic diffeomorphism from C into H.

Existence or not of harmonic diffeomorphisms

Problem: Consider the sphere S2 endowed with its canonical

• metric. Take p1,...,pm, m distinct points on S2 and

D1,...,Dm, m pairwise disjoint closed circular discs in S2. Study the existence or not of harmonic diffeomorphisms between S2 −{p1,...,pm} and S2 −∪m

j=1Dj.

Existence or not of harmonic diffeomorphisms

An open domain in the Riemann sphere C is said to be a circular domain if every connected component of its boundary is a circle.

Theorem (Alarc´

• n-S. 2011)

(i) For any m ∈ N, m ≥ 2, and any subset {p1,...,pm} ∈ S2 there exist a circular domain U ⊂ C and a harmonic diffeomorphism U → S2 −{p1,...,pm}. (ii) There exists no harmonic diffeomorphism D → S2 −{p}, p ∈ S2. (iii) For any m ∈ N, any subset {z1,...,zm} ⊂ C and any pairwise disjoint closed discs D1,...,Dm in S2, there exists no harmonic diffeomorphism C−{z1,...,zm} → S2 −∪m

j=1Dj.

(i) Existence of harmonic diffeomorphisms U → S2 −{p1,...,pm}.

Strategy

Similarly to Collin-Rosenberg,

• ur strategy to show the harmonic diffeomorphism of Item (i)

consists of constructing a maximal graph Σ over S2 −{p1,...,pm} in the Lorentzian manifold S2 ×R1, with the conformal structure of a circular domain. Then, the projection Σ → S2 −{p1,...,pm} is a surjective harmonic diffeomorphism,

Strategy

Similarly to Collin-Rosenberg,

• ur strategy to show the harmonic diffeomorphism of Item (i)

consists of constructing a maximal graph Σ over S2 −{p1,...,pm} in the Lorentzian manifold S2 ×R1, with the conformal structure of a circular domain. Then, the projection Σ → S2 −{p1,...,pm} is a surjective harmonic diffeomorphism, but...

• ur construction method is completely different and relies on

the theory of maximal hypersurfaces in Lorentzian manifolds. More precisely, we proceed by solving Dirichlet problems.

Maximal graphs. Notation

M = (M,·,·M) ≡ compact n-dimensional Riemannian manifold without boundary, n ∈ N, n ≥ 2.

Maximal graphs. Notation

M = (M,·,·M) ≡ compact n-dimensional Riemannian manifold without boundary, n ∈ N, n ≥ 2. M×R1 ≡ the Lorentzian product space M×R endowed with the Lorentzian metric ·,· = π∗

M(·,·M)−π∗ R(dt2) = ·,·M −dt2.

Maximal graphs. Notation

M = (M,·,·M) ≡ compact n-dimensional Riemannian manifold without boundary, n ∈ N, n ≥ 2. M×R1 ≡ the Lorentzian product space M×R endowed with the Lorentzian metric ·,· = π∗

M(·,·M)−π∗ R(dt2) = ·,·M −dt2.

Ω ⊂ M ≡ connected domain.

Maximal graphs. Notation

M = (M,·,·M) ≡ compact n-dimensional Riemannian manifold without boundary, n ∈ N, n ≥ 2. M×R1 ≡ the Lorentzian product space M×R endowed with the Lorentzian metric ·,· = π∗

M(·,·M)−π∗ R(dt2) = ·,·M −dt2.

Ω ⊂ M ≡ connected domain. u : Ω → R ≡ smooth function.

Maximal graphs. Notation

M = (M,·,·M) ≡ compact n-dimensional Riemannian manifold without boundary, n ∈ N, n ≥ 2. M×R1 ≡ the Lorentzian product space M×R endowed with the Lorentzian metric ·,· = π∗

M(·,·M)−π∗ R(dt2) = ·,·M −dt2.

Ω ⊂ M ≡ connected domain. u : Ω → R ≡ smooth function. X u : Ω → M×R1, X u(p) = (p,u(p)).

Maximal graphs. Notation

M = (M,·,·M) ≡ compact n-dimensional Riemannian manifold without boundary, n ∈ N, n ≥ 2. M×R1 ≡ the Lorentzian product space M×R endowed with the Lorentzian metric ·,· = π∗

M(·,·M)−π∗ R(dt2) = ·,·M −dt2.

Ω ⊂ M ≡ connected domain. u : Ω → R ≡ smooth function. X u : Ω → M×R1, X u(p) = (p,u(p)). ·,·u := (X u)∗(·,·) = ·,·M −du2 ≡ metric induced on Ω by ·,· via X u.

Maximal graphs. Notation

M = (M,·,·M) ≡ compact n-dimensional Riemannian manifold without boundary, n ∈ N, n ≥ 2. M×R1 ≡ the Lorentzian product space M×R endowed with the Lorentzian metric ·,· = π∗

M(·,·M)−π∗ R(dt2) = ·,·M −dt2.

Ω ⊂ M ≡ connected domain. u : Ω → R ≡ smooth function. X u : Ω → M×R1, X u(p) = (p,u(p)). ·,·u := (X u)∗(·,·) = ·,·M −du2 ≡ metric induced on Ω by ·,· via X u. X u, u is spacelike (i.e., induces a Riemannian metric on Ω) if and only if |∇u| < 1 on Ω.

Maximal graphs and harmonic diffeomorphisms

Mean curvature of X u : H(u) = 1 n div

• ∇u
• 1−|∇u|2

Maximal graphs and harmonic diffeomorphisms

Mean curvature of X u : H(u) = 1 n div

• ∇u
• 1−|∇u|2
• X u, u is maximal if u is spacelike and H(u) vanishes

identically on Ω.

Maximal graphs and harmonic diffeomorphisms

Mean curvature of X u : H(u) = 1 n div

• ∇u
• 1−|∇u|2
• X u, u is maximal if u is spacelike and H(u) vanishes

identically on Ω. The equation H(u) = 0 with |∇u| < 1 is elliptic.

Maximal graphs and harmonic diffeomorphisms

If u : Ω → R is maximal then X u : (Ω,·,·u) → (M×R1,·,·) is a harmonic map. In particular Id : (Ω,·,·u) → (Ω,·,·M) is a harmonic diffeomorphism, and u : (Ω,·,·u) → R is a harmonic function.

Maximal graphs and harmonic diffeomorphisms

m ∈ N, m ≥ 2. {p1,...,pm} ⊂ M. Ω = M−{p1,...,pm}.

Maximal graphs and harmonic diffeomorphisms

m ∈ N, m ≥ 2. {p1,...,pm} ⊂ M. Ω = M−{p1,...,pm}. Is there a maximal graph over Ω in M×R1 having prescribed values at p1,...,pm?

Maximal graphs and harmonic diffeomorphisms

m ∈ N, m ≥ 2. {p1,...,pm} ⊂ M. Ω = M−{p1,...,pm}. Is there a maximal graph over Ω in M×R1 having prescribed values at p1,...,pm? If M = S2, does such a graph have the conformal structure of a circular domain?

Maximal graphs and harmonic diffeomorphisms

[Gerhardt 1983] Let Ω ⊂ M be a compact domain with C 2 boundary and ϕ ∈ C 2(Ω) satisfying |∇ϕ| < 1 in Ω then the Dirichlet problem:    div

• ∇u

√

1−|∇u|2

• = 0

u|∂Ω = ϕ|∂Ω admits a solution.

Maximal graphs over M−{p1,...,pm} in M×R1

A = {(pi,ti)}m

i=1 ⊂ M×R such that

|ti −tj| < distM(pi,pj) ∀i,j ∈ {1,...,m}, i = j (spacelike condition).

Maximal graphs over M−{p1,...,pm} in M×R1

A = {(pi,ti)}m

i=1 ⊂ M×R such that

|ti −tj| < distM(pi,pj) ∀i,j ∈ {1,...,m}, i = j (spacelike condition). Bn

i , (i,n) ∈ {1,...,m}×N, open disc in M

∂Bn

i smooth Jordan curve,

Bn

i ∩Bn j = /

0 if i = j, Bn+1

i

⊂ Bn

i ,

{pi} = ∩n∈NBn

i .

Maximal graphs over M−{p1,...,pm} in M×R1

A = {(pi,ti)}m

i=1 ⊂ M×R such that

|ti −tj| < distM(pi,pj) ∀i,j ∈ {1,...,m}, i = j (spacelike condition). Bn

i , (i,n) ∈ {1,...,m}×N, open disc in M

∂Bn

i smooth Jordan curve,

Bn

i ∩Bn j = /

0 if i = j, Bn+1

i

⊂ Bn

i ,

{pi} = ∩n∈NBn

i .

∆n = M−∪m

i=1Bn i , n ∈ N.

Maximal graphs over M−{p1,...,pm} in M×R1

A = {(pi,ti)}m

i=1 ⊂ M×R such that

|ti −tj| < distM(pi,pj) ∀i,j ∈ {1,...,m}, i = j (spacelike condition). Bn

i , (i,n) ∈ {1,...,m}×N, open disc in M

∂Bn

i smooth Jordan curve,

Bn

i ∩Bn j = /

0 if i = j, Bn+1

i

⊂ Bn

i ,

{pi} = ∩n∈NBn

i .

∆n = M−∪m

i=1Bn i , n ∈ N.

tn

i ∈ R, {tn i }n∈N → ti, i = 1,...,m.

Maximal graphs over M−{p1,...,pm} in M×R1

ϕn : ∂∆n → R ϕn|∂Bn

i = tn

i ,

i = 1,...,m, is εn-Lipschitz, εn ∈ (0,1).

Maximal graphs over M−{p1,...,pm} in M×R1

ϕn : ∂∆n → R ϕn|∂Bn

i = tn

i ,

i = 1,...,m, is εn-Lipschitz, εn ∈ (0,1). [Federer 1969] ϕn extends to ∆n as an εn-Lipschitz function

• ϕn : ∆n → R.

Maximal graphs over M−{p1,...,pm} in M×R1

ϕn : ∂∆n → R ϕn|∂Bn

i = tn

i ,

i = 1,...,m, is εn-Lipschitz, εn ∈ (0,1). [Federer 1969] ϕn extends to ∆n as an εn-Lipschitz function

• ϕn : ∆n → R.

Smoothing ϕn, there exists a smooth spacelike function ϕn : ∆n → R such that ϕn|∂Bn

i = tn

i ,

i = 1,...,m.

Maximal graphs over M−{p1,...,pm} in M×R1

By Gerhardt’s result there exists a maximal function un : ∆n → R such that un|∂Bn

i = ϕn|∂Bn i = tn

i .

Maximal graphs over M−{p1,...,pm} in M×R1

By Gerhardt’s result there exists a maximal function un : ∆n → R such that un|∂Bn

i = ϕn|∂Bn i = tn

i .

{un}n∈N uniformly bounded |∇un| < 1 on ∆n

• (Ascoli + a diagonal argument)

= ⇒ up to taking a subsequence, {un}n∈N uniformly converges on compact sets of M−{pi}m

i=1 = ∪n∈N∆n to a Lipschitz

function ˆ u : M−{pi}m

i=1 → R

with |∇ˆ u| ≤ 1 a.e. in M−{pi}m

i=1.

Maximal graphs over M−{p1,...,pm} in M×R1

By Gerhardt’s result there exists a maximal function un : ∆n → R such that un|∂Bn

i = ϕn|∂Bn i = tn

i .

{un}n∈N uniformly bounded |∇un| < 1 on ∆n

• (Ascoli + a diagonal argument)

= ⇒ up to taking a subsequence, {un}n∈N uniformly converges on compact sets of M−{pi}m

i=1 = ∪n∈N∆n to a Lipschitz

function ˆ u : M−{pi}m

i=1 → R

with |∇ˆ u| ≤ 1 a.e. in M−{pi}m

i=1.

ˆ u extends to a Lipschitz function u : M → R with |∇u| ≤ 1 a.e. in M−{pi}m

i=1 and

u(pi) = ti ∀i = 1,...,m.

Maximal graphs over M−{p1,...,pm} in M×R1

[Bartnik 1988] ˆ u is a smooth maximal function except for a set of points Λ ⊂ M−{pi}m

i=1,

Λ :=

• p ∈ M−{pi}m

i=1 | (p,ˆ

u(p)) = γ(s0) for some 0 < s0 < 1, where γ : [0,1] → M×R1 is a lightlike geodesic such that γ((0,1)) ⊂ X ˆ

u(M−{pi}m i=1)

and πM({γ(0),γ(1)}) ⊂ {pi}m

i=1.

Maximal graphs over M−{p1,...,pm} in M×R1

[Bartnik 1988] ˆ u is a smooth maximal function except for a set of points Λ ⊂ M−{pi}m

i=1,

Λ :=

• p ∈ M−{pi}m

i=1 | (p,ˆ

u(p)) = γ(s0) for some 0 < s0 < 1, where γ : [0,1] → M×R1 is a lightlike geodesic such that γ((0,1)) ⊂ X ˆ

u(M−{pi}m i=1)

and πM({γ(0),γ(1)}) ⊂ {pi}m

i=1.

Since A satisfies the spacelike condition then Λ = / 0 and ˆ u : M−{p1,...,pm} → R determines a maximal graph over M−{p1,...,pm} in M×R1.

Maximal graphs over M−{p1,...,pm} in M×R1

Theorem

Let M be a compact Riemannian manifold, let m ∈ N, m ≥ 2, and let A = {(pi,ti)}m

i=1 ⊂ M×R

satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M×R1 such that A ⊂ Σ and Σ−A is a spacelike maximal graph over M−{pi}i=1,...,m.

Maximal graphs over M−{p1,...,pm} in M×R1

Theorem

Let M be a compact Riemannian manifold, let m ∈ N, m ≥ 2, and let A = {(pi,ti)}m

i=1 ⊂ M×R

satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M×R1 such that A ⊂ Σ and Σ−A is a spacelike maximal graph over M−{pi}i=1,...,m.

Maximal graphs over M−{p1,...,pm} in M×R1

Theorem

Let M be a compact Riemannian manifold, let m ∈ N, m ≥ 2, and let A = {(pi,ti)}m

i=1 ⊂ M×R

satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M×R1 such that A ⊂ Σ and Σ−A is a spacelike maximal graph over M−{pi}i=1,...,m. Moreover the space Gm of entire maximal graphs over M in M×R1 with precisely m singularities, endowed with the topology of uniform convergence

Maximal graphs over M−{p1,...,pm} in M×R1

Theorem

Let M be a compact Riemannian manifold, let m ∈ N, m ≥ 2, and let A = {(pi,ti)}m

i=1 ⊂ M×R

satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M×R1 such that A ⊂ Σ and Σ−A is a spacelike maximal graph over M−{pi}i=1,...,m. Moreover the space Gm of entire maximal graphs over M in M×R1 with precisely m singularities, endowed with the topology of uniform convergence, is non-empty

Maximal graphs over M−{p1,...,pm} in M×R1

Theorem

Let M be a compact Riemannian manifold, let m ∈ N, m ≥ 2, and let A = {(pi,ti)}m

i=1 ⊂ M×R

satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M×R1 such that A ⊂ Σ and Σ−A is a spacelike maximal graph over M−{pi}i=1,...,m. Moreover the space Gm of entire maximal graphs over M in M×R1 with precisely m singularities, endowed with the topology of uniform convergence, is non-empty, and there exists a m!-sheeted covering, Gm → Gm, where Gm is an open subset of (M×R)m.

Maximal graphs over M−{p1,...,pm} in M×R1

Why is Gm non-empty? Choose t1 = ··· = tm−1 = tm and tm close enough to t1 to guarantee the spacelike condition. The function ˆ u is harmonic and so the maximum principle forces the graph to have singularities at all the (pj,tj), j = 1,...,m.

Maximal graphs over M−{p1,...,pm} in M×R1

If M is a surface X u : (M−{p1,...,pm},·,·u) → M×R1 conformal harmonic map, with singularities precisely at the points {p1,...,pm}.

Maximal graphs over M−{p1,...,pm} in M×R1

If M is a surface X u : (M−{p1,...,pm},·,·u) → M×R1 conformal harmonic map, with singularities precisely at the points {p1,...,pm}. A ≡ annular end of (M−{p1,...,pm},·,·u) corresponding to pi.

Maximal graphs over M−{p1,...,pm} in M×R1

If M is a surface X u : (M−{p1,...,pm},·,·u) → M×R1 conformal harmonic map, with singularities precisely at the points {p1,...,pm}. A ≡ annular end of (M−{p1,...,pm},·,·u) corresponding to pi. A is conformally equivalent to an annulus A(r,1) := {z ∈ C | r < |z| ≤ 1} for some 0 ≤ r < 1. Identify A ≡ A(r,1) and notice that u extends continuously to S(r) = {z ∈ C | |z| = r} with u|S(r) = u(pi).

Maximal graphs over M−{p1,...,pm} in M×R1

If M is a surface X u : (M−{p1,...,pm},·,·u) → M×R1 conformal harmonic map, with singularities precisely at the points {p1,...,pm}. A ≡ annular end of (M−{p1,...,pm},·,·u) corresponding to pi. A is conformally equivalent to an annulus A(r,1) := {z ∈ C | r < |z| ≤ 1} for some 0 ≤ r < 1. Identify A ≡ A(r,1) and notice that u extends continuously to S(r) = {z ∈ C | |z| = r} with u|S(r) = u(pi). [Bartnik 1989] X u(A) is tangent to either the upper or the lower light cone at X u(pi) in M×R1. In particular u has at pi either a strict local minimum or a strict local maximum.

Maximal graphs over M−{p1,...,pm} in M×R1

If M is a surface X u : (M−{p1,...,pm},·,·u) → M×R1 conformal harmonic map, with singularities precisely at the points {p1,...,pm}. A ≡ annular end of (M−{p1,...,pm},·,·u) corresponding to pi. A is conformally equivalent to an annulus A(r,1) := {z ∈ C | r < |z| ≤ 1} for some 0 ≤ r < 1. Identify A ≡ A(r,1) and notice that u extends continuously to S(r) = {z ∈ C | |z| = r} with u|S(r) = u(pi). [Bartnik 1989] X u(A) is tangent to either the upper or the lower light cone at X u(pi) in M×R1. In particular u has at pi either a strict local minimum or a strict local maximum. u|A is harmonic. So if r = 0, i.e A conformal to D∗, u would extend as a harmonic function to D with an extremum at the

• rigin, a contradiction. So A has hyperbolic conformal type.

Maximal graphs over M−{p1,...,pm} in M×R1

If M is a surface (M−{p1,...,pm},·,·u) is conformally an open Riemann surface with the same genus as M and m hyperbolic ends.

Corollary

Let M be a compact Riemannian surface, let m ≥ 2 and let {p1,...,pm} ⊂ M. Then there exist an open Riemann surface R and a harmonic diffeomorphism R → M−{p1,...,pm} such that every end of R is of hyperbolic type.

Maximal graphs over M−{p1,...,pm} in M×R1

If M is a surface (M−{p1,...,pm},·,·u) is conformally an open Riemann surface with the same genus as M and m hyperbolic ends.

Corollary

Let M be a compact Riemannian surface, let m ≥ 2 and let {p1,...,pm} ⊂ M. Then there exist an open Riemann surface R and a harmonic diffeomorphism R → M−{p1,...,pm} such that every end of R is of hyperbolic type. If M = S2, by Koebe’s uniformization theorem,

Corollary

Let m ∈ N, m ≥ 2, and let {p1,...,pm} ⊂ S2. Then there exist a circular domain U in C and a harmonic diffeomorphism U → S2 −{p1,...,pm}.

Maximal graphs over M−{p1,...,pm} in M×R1

Question: Given the points p1,...,pm, what are all the circular domains obtained by this construction?

(ii) Non-existence of harmonic diffeomorphisms D → S2 −{p}.

Strategy

Why does the above argument not work for m = 1?

Strategy

Why does the above argument not work for m = 1? Reason: by the maximum principle, there is no entire maximal graph with only one singularity.

Strategy

Why does the above argument not work for m = 1? Reason: by the maximum principle, there is no entire maximal graph with only one singularity. If we assume finiteness of energy then the result follows from a theorem of Lemaire.

Strategy

Why does the above argument not work for m = 1? Reason: by the maximum principle, there is no entire maximal graph with only one singularity. If we assume finiteness of energy then the result follows from a theorem of Lemaire. Our argument uses Constant Gauss Curvature Surface Theory.

Constant Gauss Curvature Surfaces

S smooth simply-connected surface. X : S → R3 immersion with constant Gauss curvature K = 1. IIX positive definite metric (up to changing orientation) ⇒ IIX induces on S a conformal structure, S . z = u +ıv conformal parameter on S .

Constant Gauss Curvature Surfaces

S smooth simply-connected surface. X : S → R3 immersion with constant Gauss curvature K = 1. IIX positive definite metric (up to changing orientation) ⇒ IIX induces on S a conformal structure, S . z = u +ıv conformal parameter on S . [G´ alvez-Mart´ ınez 2000] The Gauss map N : S → S2 satisfies Xu = N ×Nv and Xv = −N ×Nu, (1) hence it is a harmonic local diffeomorphism. Conversely, let N : S → S2 be a harmonic local

• diffeomorphism. Then the map X : S → R3 given by (1) is an

immersion with constant Gauss curvature K = 1, with Gauss map N (or −N) and the conformal structure of S is the one induced by IIX.

Constant Gauss Curvature Surfaces

[Klotz 1980] There exists an immersion Y : S → R3 of constant Gauss curvature K = 1 such that IY = IIIX, IIY = IIX and IIIY = IX. Reason: (IIIX,IIX) is a Codazzi pair.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C. ∃ X : S → R3 with Gauss map ϕ, constant curvature KX = 1 and such that the conformal structure of S is the one induced by IIX.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C. ∃ X : S → R3 with Gauss map ϕ, constant curvature KX = 1 and such that the conformal structure of S is the one induced by IIX. ∃ Y : S → R3 with constant curvature KY = 1, IY = IIIX, IIY = IIX and IIIY = IX.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C. ∃ X : S → R3 with Gauss map ϕ, constant curvature KX = 1 and such that the conformal structure of S is the one induced by IIX. ∃ Y : S → R3 with constant curvature KY = 1, IY = IIIX, IIY = IIX and IIIY = IX. ϕ : S → S2 −{p} is a diffeomorphism and IY = IIIX = dϕ,dϕR3 = ϕ∗(·,·S2), hence ϕ−1 : S2 −{p} → (S ,IY ) is an isometry.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C. ∃ X : S → R3 with Gauss map ϕ, constant curvature KX = 1 and such that the conformal structure of S is the one induced by IIX. ∃ Y : S → R3 with constant curvature KY = 1, IY = IIIX, IIY = IIX and IIIY = IX. ϕ : S → S2 −{p} is a diffeomorphism and IY = IIIX = dϕ,dϕR3 = ϕ∗(·,·S2), hence ϕ−1 : S2 −{p} → (S ,IY ) is an isometry. Y : (S ,IY ) → R3 isometric immersion.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C. ∃ X : S → R3 with Gauss map ϕ, constant curvature KX = 1 and such that the conformal structure of S is the one induced by IIX. ∃ Y : S → R3 with constant curvature KY = 1, IY = IIIX, IIY = IIX and IIIY = IX. ϕ : S → S2 −{p} is a diffeomorphism and IY = IIIX = dϕ,dϕR3 = ϕ∗(·,·S2), hence ϕ−1 : S2 −{p} → (S ,IY ) is an isometry. Y : (S ,IY ) → R3 isometric immersion. Y ◦ϕ−1 : S2 −{p} → R3 isometric immersion.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C. ∃ X : S → R3 with Gauss map ϕ, constant curvature KX = 1 and such that the conformal structure of S is the one induced by IIX. ∃ Y : S → R3 with constant curvature KY = 1, IY = IIIX, IIY = IIX and IIIY = IX. ϕ : S → S2 −{p} is a diffeomorphism and IY = IIIX = dϕ,dϕR3 = ϕ∗(·,·S2), hence ϕ−1 : S2 −{p} → (S ,IY ) is an isometry. Y : (S ,IY ) → R3 isometric immersion. Y ◦ϕ−1 : S2 −{p} → R3 isometric immersion. [Pogorelov 1973] S2 −{p} is rigid in R3.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C. ∃ X : S → R3 with Gauss map ϕ, constant curvature KX = 1 and such that the conformal structure of S is the one induced by IIX. ∃ Y : S → R3 with constant curvature KY = 1, IY = IIIX, IIY = IIX and IIIY = IX. ϕ : S → S2 −{p} is a diffeomorphism and IY = IIIX = dϕ,dϕR3 = ϕ∗(·,·S2), hence ϕ−1 : S2 −{p} → (S ,IY ) is an isometry. Y : (S ,IY ) → R3 isometric immersion. Y ◦ϕ−1 : S2 −{p} → R3 isometric immersion. [Pogorelov 1973] S2 −{p} is rigid in R3. Y (S ) ⊂ R3 is a once punctured round sphere.

Non-existence of harmonic diffeomorphisms D → S2 −{p}.

S simply-connected Riemann surface, ϕ : S → S2 −{p} harmonic diffeomorphism. We will show that S is conformally equivalent to C. ∃ X : S → R3 with Gauss map ϕ, constant curvature KX = 1 and such that the conformal structure of S is the one induced by IIX. ∃ Y : S → R3 with constant curvature KY = 1, IY = IIIX, IIY = IIX and IIIY = IX. ϕ : S → S2 −{p} is a diffeomorphism and IY = IIIX = dϕ,dϕR3 = ϕ∗(·,·S2), hence ϕ−1 : S2 −{p} → (S ,IY ) is an isometry. Y : (S ,IY ) → R3 isometric immersion. Y ◦ϕ−1 : S2 −{p} → R3 isometric immersion. [Pogorelov 1973] S2 −{p} is rigid in R3. Y (S ) ⊂ R3 is a once punctured round sphere. The conformal structure induced on S by IIY = IIX is C.

(iii) Non-existence of harmonic diffeomorphisms C−{z1,...,zm} → S2 −∪m

j=1Dj

Strategy

Use the Bochner type formulas [Schoen-Yau 1997]

Strategy

Use the Bochner type formulas [Schoen-Yau 1997]

Definition

An open Riemann surface R is said to be parabolic if any negative subharmonic function on R is constant.

Strategy

Use the Bochner type formulas [Schoen-Yau 1997]

Definition

An open Riemann surface R is said to be parabolic if any negative subharmonic function on R is constant. For example, S2 −{p1,...,pm} is parabolic.

Strategy

Use the Bochner type formulas [Schoen-Yau 1997]

Definition

An open Riemann surface R is said to be parabolic if any negative subharmonic function on R is constant. For example, S2 −{p1,...,pm} is parabolic.

Proposition

Let R be a parabolic open Riemann surface, let N be an oriented Riemannian surface and let φ : R → N be a harmonic local

• diffeomorphism. Suppose that N has Gaussian curvature KN > 0.

Then φ is either holomorphic or antiholomorphic.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj Assume that φ preserves orientation. Is φ holomorphic?

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj Assume that φ preserves orientation. Is φ holomorphic? Let z (resp. φ) be a local conformal parameter in R (resp. in N). The metric on N writes ρ(φ)|dφ|2. A conformal metric

• n R writes λ(z)|dz|2.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj Assume that φ preserves orientation. Is φ holomorphic? Let z (resp. φ) be a local conformal parameter in R (resp. in N). The metric on N writes ρ(φ)|dφ|2. A conformal metric

• n R writes λ(z)|dz|2.

Partial densities: |∂φ|2 = ρ(φ(z))

λ(z) |∂φ ∂z |2 and |∂φ|2 = ρ(φ(z)) λ(z) |∂φ ∂¯ z |2.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj Assume that φ preserves orientation. Is φ holomorphic? Let z (resp. φ) be a local conformal parameter in R (resp. in N). The metric on N writes ρ(φ)|dφ|2. A conformal metric

• n R writes λ(z)|dz|2.

Partial densities: |∂φ|2 = ρ(φ(z))

λ(z) |∂φ ∂z |2 and |∂φ|2 = ρ(φ(z)) λ(z) |∂φ ∂¯ z |2.

The Jacobian of φ, J(φ) = |∂φ|2 −|∂φ|2 > 0.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj Assume that φ preserves orientation. Is φ holomorphic? Let z (resp. φ) be a local conformal parameter in R (resp. in N). The metric on N writes ρ(φ)|dφ|2. A conformal metric

• n R writes λ(z)|dz|2.

Partial densities: |∂φ|2 = ρ(φ(z))

λ(z) |∂φ ∂z |2 and |∂φ|2 = ρ(φ(z)) λ(z) |∂φ ∂¯ z |2.

The Jacobian of φ, J(φ) = |∂φ|2 −|∂φ|2 > 0. Bochner type formulas [Schoen-Yau 1997]: If |∂φ|2 (resp. |∂φ|2) is not identically zero, then its zeroes are isolated, and at points where it is nonzero, we have: ∆R log(|∂φ|2) = −2KNJ(φ)+2KR ∆R log(|∂φ|2) = 2KNJ(φ)+2KR

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj Assume that φ preserves orientation. Is φ holomorphic? Let z (resp. φ) be a local conformal parameter in R (resp. in N). The metric on N writes ρ(φ)|dφ|2. A conformal metric

• n R writes λ(z)|dz|2.

Partial densities: |∂φ|2 = ρ(φ(z))

λ(z) |∂φ ∂z |2 and |∂φ|2 = ρ(φ(z)) λ(z) |∂φ ∂¯ z |2.

The Jacobian of φ, J(φ) = |∂φ|2 −|∂φ|2 > 0. Bochner type formulas [Schoen-Yau 1997]: If |∂φ|2 (resp. |∂φ|2) is not identically zero, then its zeroes are isolated, and at points where it is nonzero, we have: ∆R log(|∂φ|2) = −2KNJ(φ)+2KR ∆R log(|∂φ|2) = 2KNJ(φ)+2KR By contradiction: If φ is not holomorphic, then the zeros of |∂φ| are isolated.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj R∗ = R −{|∂φ| = 0}.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj R∗ = R −{|∂φ| = 0}. log(|∂φ|/|∂φ|) < 0 on R∗.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj R∗ = R −{|∂φ| = 0}. log(|∂φ|/|∂φ|) < 0 on R∗. R∗ is parabolic.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj R∗ = R −{|∂φ| = 0}. log(|∂φ|/|∂φ|) < 0 on R∗. R∗ is parabolic. A general fact: If R is an open parabolic Riemann surface and E ⊂ R a closed subset consisting of isolated points. Then R∗ := R −E is an open parabolic Riemann surface.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj R∗ = R −{|∂φ| = 0}. log(|∂φ|/|∂φ|) < 0 on R∗. R∗ is parabolic. A general fact: If R is an open parabolic Riemann surface and E ⊂ R a closed subset consisting of isolated points. Then R∗ := R −E is an open parabolic Riemann surface. By Bochner formulas: ∆R log(|∂φ|/|∂φ|) = 2KNJ(φ).

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj R∗ = R −{|∂φ| = 0}. log(|∂φ|/|∂φ|) < 0 on R∗. R∗ is parabolic. A general fact: If R is an open parabolic Riemann surface and E ⊂ R a closed subset consisting of isolated points. Then R∗ := R −E is an open parabolic Riemann surface. By Bochner formulas: ∆R log(|∂φ|/|∂φ|) = 2KNJ(φ). Since KN > 0, then log(|∂φ|/|∂φ|) is a non-constant negative subharmonic function on the parabolic surface R∗, a contradiction.

Non-existence, C−{z1,...,zm} → S2 −∪m

j=1Dj R∗ = R −{|∂φ| = 0}. log(|∂φ|/|∂φ|) < 0 on R∗. R∗ is parabolic. A general fact: If R is an open parabolic Riemann surface and E ⊂ R a closed subset consisting of isolated points. Then R∗ := R −E is an open parabolic Riemann surface. By Bochner formulas: ∆R log(|∂φ|/|∂φ|) = 2KNJ(φ). Since KN > 0, then log(|∂φ|/|∂φ|) is a non-constant negative subharmonic function on the parabolic surface R∗, a contradiction. In the case when φ reverses orientation then a parallel argument gives that φ is antiholomorphic.

Harmonic diffeomorphisms and K−surfaces

Question: Do there exist K-surfaces (i.e surfaces with constant curvature K) whose extrinsic conformal structures are circular domains U and their Gauss maps harmonic diffeomorphisms U → S2 −{p1,...,pm}?

Harmonic diffeomorphisms and K−surfaces

Answer:

Theorem (Alarc´

• n-S. 2012)

Let {p1,...,pm} ⊂ S2,m ∈ N. The following statements are equivalent: (i) There exists a K-surface S with K = 1 such that the extrinsic conformal structure of S is a circular domain U ⊂ C, and the Gauss map of S is a harmonic diffeomorphism U → S2 −{p1,...,pm}. (ii) There exist positive real constants a1,...,am such that ∑m

j=1 ajpj =

0 ∈ R3.

Harmonic diffeomorphisms and K−surfaces

Theorem (continued)

Furthermore, if S is as above and one denotes by γj the connected component of S −S corresponding to pj via its Gauss map, then (I) γj is a Jordan curve contained in an affine plane Πj ⊂ R3

• rthogonal to pj, and

(II) K = S ∪(∪m

j=1Dj) is the boundary surface of a smooth

convex body1 in R3, where Dj is the bounded connected component of Πj −γj for all j ∈ {1,...,m}. In addition, given {a1,...,am} satisfying (ii), there exists a unique, up to translations, surface S satisfying (i) such that the area of Dj equals aj for all j ∈ {1,...,m}.

1 A convex body in R3 is said smooth if it has a unique supporting plane

at each boundary point. This is the same as saying that its boundary is a C 1 surface.

Harmonic diffeomorphisms and maximal surfaces

Rabah Souam CNRS, Institut de Math´ ematiques de Jussieu, Paris Joint work with A. Alarc´

• n

PADGE 2012 Leuven, August 2012