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

Maximal graphs. Notation M = ( M , �· , ·� M ) ≡ compact n -dimensional Riemannian manifold without boundary, n ∈ N , n ≥ 2 . M × R 1 ≡ the Lorentzian product space M × R endowed with the Lorentzian metric �· , ·� = π ∗ M ( �· , ·� M ) − π ∗ R ( dt 2 ) = �· , ·� M − dt 2 . Ω ⊂ M ≡ connected domain. u : Ω → R ≡ smooth function. X u : Ω → M × R 1 , X u ( p ) = ( p , u ( p )) . �· , ·� u := ( X u ) ∗ ( �· , ·� ) = �· , ·� M − du 2 ≡ 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 × R 1 ≡ the Lorentzian product space M × R endowed with the Lorentzian metric �· , ·� = π ∗ M ( �· , ·� M ) − π ∗ R ( dt 2 ) = �· , ·� M − dt 2 . Ω ⊂ M ≡ connected domain. u : Ω → R ≡ smooth function. X u : Ω → M × R 1 , X u ( p ) = ( p , u ( p )) . �· , ·� u := ( X u ) ∗ ( �· , ·� ) = �· , ·� M − du 2 ≡ 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 ∇ u n div � 1 −| ∇ u | 2

Maximal graphs and harmonic diffeomorphisms Mean curvature of X u : � � H ( u ) = 1 ∇ u n div � 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 ∇ u n div � 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 × R 1 , �· , ·� ) 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 . { p 1 ,..., p m } ⊂ M . Ω = M −{ p 1 ,..., p m } .

Maximal graphs and harmonic diffeomorphisms m ∈ N , m ≥ 2 . { p 1 ,..., p m } ⊂ M . Ω = M −{ p 1 ,..., p m } . Is there a maximal graph over Ω in M × R 1 having prescribed values at p 1 ,..., p m ?

Maximal graphs and harmonic diffeomorphisms m ∈ N , m ≥ 2 . { p 1 ,..., p m } ⊂ M . Ω = M −{ p 1 ,..., p m } . Is there a maximal graph over Ω in M × R 1 having prescribed values at p 1 ,..., p m ? If M = S 2 , 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:  � �  ∇ u √ div = 0 1 −| ∇ u | 2  u | ∂ Ω = ϕ | ∂ Ω admits a solution.

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 A = { ( p i , t i ) } m i =1 ⊂ M × R such that | t i − t j | < dist M ( p i , p j ) ∀ i , j ∈ { 1 ,..., m } , i � = j (spacelike condition) .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 A = { ( p i , t i ) } m i =1 ⊂ M × R such that | t i − t j | < dist M ( p i , p j ) ∀ i , j ∈ { 1 ,..., m } , i � = j (spacelike condition) . B n i , ( i , n ) ∈ { 1 ,..., m }× N , open disc in M ∂ B n i smooth Jordan curve, B n i ∩ B n j = / 0 if i � = j , B n +1 ⊂ B n i , i { p i } = ∩ n ∈ N B n i .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 A = { ( p i , t i ) } m i =1 ⊂ M × R such that | t i − t j | < dist M ( p i , p j ) ∀ i , j ∈ { 1 ,..., m } , i � = j (spacelike condition) . B n i , ( i , n ) ∈ { 1 ,..., m }× N , open disc in M ∂ B n i smooth Jordan curve, B n i ∩ B n j = / 0 if i � = j , B n +1 ⊂ B n i , i { p i } = ∩ n ∈ N B n i . i =1 B n ∆ n = M −∪ m i , n ∈ N .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 A = { ( p i , t i ) } m i =1 ⊂ M × R such that | t i − t j | < dist M ( p i , p j ) ∀ i , j ∈ { 1 ,..., m } , i � = j (spacelike condition) . B n i , ( i , n ) ∈ { 1 ,..., m }× N , open disc in M ∂ B n i smooth Jordan curve, B n i ∩ B n j = / 0 if i � = j , B n +1 ⊂ B n i , i { p i } = ∩ n ∈ N B n i . i =1 B n ∆ n = M −∪ m i , n ∈ N . t n i ∈ R , { t n i } n ∈ N → t i , i = 1 ,..., m .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 ϕ n : ∂ ∆ n → R i = t n ϕ n | ∂ B n i , i = 1 ,..., m , is ε n -Lipschitz, ε n ∈ (0 , 1) .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 ϕ n : ∂ ∆ n → R i = t n ϕ n | ∂ B n i , i = 1 ,..., m , is ε n -Lipschitz, ε n ∈ (0 , 1) . [Federer 1969] ϕ n extends to ∆ n as an ε n -Lipschitz function � ϕ n : ∆ n → R .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 ϕ n : ∂ ∆ n → R i = t n ϕ n | ∂ B n i , i = 1 ,..., m , is ε n -Lipschitz, ε n ∈ (0 , 1) . [Federer 1969] ϕ n extends to ∆ n as an ε n -Lipschitz function � ϕ n : ∆ n → R . Smoothing � ϕ n , there exists a smooth spacelike function ϕ n : ∆ n → R such that i = t n ϕ n | ∂ B n i , i = 1 ,..., m .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 By Gerhardt’s result there exists a maximal function u n : ∆ n → R i = t n such that u n | ∂ B n i = ϕ n | ∂ B n i .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 By Gerhardt’s result there exists a maximal function u n : ∆ n → R i = t n such that u n | ∂ B n i = ϕ n | ∂ B n i . � { u n } n ∈ N uniformly bounded (Ascoli + a diagonal argument) = ⇒ | ∇ u n | < 1 on ∆ n up to taking a subsequence, { u n } n ∈ N uniformly converges on compact sets of M −{ p i } m i =1 = ∪ n ∈ N ∆ n to a Lipschitz function u : M −{ p i } m ˆ i =1 → R u | ≤ 1 a.e. in M −{ p i } m with | ∇ ˆ i =1 .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 By Gerhardt’s result there exists a maximal function u n : ∆ n → R i = t n such that u n | ∂ B n i = ϕ n | ∂ B n i . � { u n } n ∈ N uniformly bounded (Ascoli + a diagonal argument) = ⇒ | ∇ u n | < 1 on ∆ n up to taking a subsequence, { u n } n ∈ N uniformly converges on compact sets of M −{ p i } m i =1 = ∪ n ∈ N ∆ n to a Lipschitz function u : M −{ p i } m ˆ i =1 → R u | ≤ 1 a.e. in M −{ p i } m with | ∇ ˆ i =1 . ˆ u extends to a Lipschitz function u : M → R | ∇ u | ≤ 1 a.e. in M −{ p i } m with i =1 and u ( p i ) = t i ∀ i = 1 ,..., m .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 [Bartnik 1988] ˆ u is a smooth maximal function except for a set of points Λ ⊂ M −{ p i } m i =1 , � p ∈ M −{ p i } m Λ := i =1 | ( p , ˆ u ( p )) = γ ( s 0 ) for some 0 < s 0 < 1 , where γ : [0 , 1] → M × R 1 is a lightlike geodesic such that γ ((0 , 1)) ⊂ X ˆ u ( M −{ p i } m i =1 ) and π M ( { γ (0) , γ (1) } ) ⊂ { p i } m i =1 .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 [Bartnik 1988] ˆ u is a smooth maximal function except for a set of points Λ ⊂ M −{ p i } m i =1 , � p ∈ M −{ p i } m Λ := i =1 | ( p , ˆ u ( p )) = γ ( s 0 ) for some 0 < s 0 < 1 , where γ : [0 , 1] → M × R 1 is a lightlike geodesic such that γ ((0 , 1)) ⊂ X ˆ u ( M −{ p i } m i =1 ) and π M ( { γ (0) , γ (1) } ) ⊂ { p i } m i =1 . Since A satisfies the spacelike condition then Λ = / 0 and ˆ u : M −{ p 1 ,..., p m } → R determines a maximal graph over M −{ p 1 ,..., p m } in M × R 1 .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 Theorem Let M be a compact Riemannian manifold, let m ∈ N , m ≥ 2 , and let A = { ( p i , t i ) } m i =1 ⊂ M × R satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M × R 1 such that A ⊂ Σ and Σ − A is a spacelike maximal graph over M −{ p i } i =1 ,..., m .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 Theorem Let M be a compact Riemannian manifold, let m ∈ N , m ≥ 2 , and let A = { ( p i , t i ) } m i =1 ⊂ M × R satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M × R 1 such that A ⊂ Σ and Σ − A is a spacelike maximal graph over M −{ p i } i =1 ,..., m .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 Theorem Let M be a compact Riemannian manifold, let m ∈ N , m ≥ 2 , and let A = { ( p i , t i ) } m i =1 ⊂ M × R satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M × R 1 such that A ⊂ Σ and Σ − A is a spacelike maximal graph over M −{ p i } i =1 ,..., m . Moreover the space G m of entire maximal graphs over M in M × R 1 with precisely m singularities, endowed with the topology of uniform convergence

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 Theorem Let M be a compact Riemannian manifold, let m ∈ N , m ≥ 2 , and let A = { ( p i , t i ) } m i =1 ⊂ M × R satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M × R 1 such that A ⊂ Σ and Σ − A is a spacelike maximal graph over M −{ p i } i =1 ,..., m . Moreover the space G m of entire maximal graphs over M in M × R 1 with precisely m singularities, endowed with the topology of uniform convergence, is non-empty

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 Theorem Let M be a compact Riemannian manifold, let m ∈ N , m ≥ 2 , and let A = { ( p i , t i ) } m i =1 ⊂ M × R satisfying the spacelike condition. Then there exists exactly one entire graph Σ over M in M × R 1 such that A ⊂ Σ and Σ − A is a spacelike maximal graph over M −{ p i } i =1 ,..., m . Moreover the space G m of entire maximal graphs over M in M × R 1 with precisely m singularities, endowed with the topology of uniform convergence, is non-empty, and there exists a m ! -sheeted covering, G m → G m , where G m is an open subset of ( M × R ) m .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 Why is G m non-empty? Choose t 1 = ··· = t m − 1 � = t m and t m close enough to t 1 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 ( p j , t j ) , j = 1 ,..., m .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 If M is a surface X u : ( M −{ p 1 ,..., p m } , �· , ·� u ) → M × R 1 conformal harmonic map, with singularities precisely at the points { p 1 ,..., p m } .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 If M is a surface X u : ( M −{ p 1 ,..., p m } , �· , ·� u ) → M × R 1 conformal harmonic map, with singularities precisely at the points { p 1 ,..., p m } . A ≡ annular end of ( M −{ p 1 ,..., p m } , �· , ·� u ) corresponding to p i .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 If M is a surface X u : ( M −{ p 1 ,..., p m } , �· , ·� u ) → M × R 1 conformal harmonic map, with singularities precisely at the points { p 1 ,..., p m } . A ≡ annular end of ( M −{ p 1 ,..., p m } , �· , ·� u ) corresponding to p i . 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 ( p i ) .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 If M is a surface X u : ( M −{ p 1 ,..., p m } , �· , ·� u ) → M × R 1 conformal harmonic map, with singularities precisely at the points { p 1 ,..., p m } . A ≡ annular end of ( M −{ p 1 ,..., p m } , �· , ·� u ) corresponding to p i . 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 ( p i ) . [Bartnik 1989] X u ( A ) is tangent to either the upper or the lower light cone at X u ( p i ) in M × R 1 . In particular u has at p i either a strict local minimum or a strict local maximum.

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 If M is a surface X u : ( M −{ p 1 ,..., p m } , �· , ·� u ) → M × R 1 conformal harmonic map, with singularities precisely at the points { p 1 ,..., p m } . A ≡ annular end of ( M −{ p 1 ,..., p m } , �· , ·� u ) corresponding to p i . 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 ( p i ) . [Bartnik 1989] X u ( A ) is tangent to either the upper or the lower light cone at X u ( p i ) in M × R 1 . In particular u has at p i 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 origin, a contradiction. So A has hyperbolic conformal type.

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 If M is a surface ( M −{ p 1 ,..., p m } , �· , ·� 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 { p 1 ,..., p m } ⊂ M . Then there exist an open Riemann surface R and a harmonic diffeomorphism R → M −{ p 1 ,..., p m } such that every end of R is of hyperbolic type.

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 If M is a surface ( M −{ p 1 ,..., p m } , �· , ·� 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 { p 1 ,..., p m } ⊂ M . Then there exist an open Riemann surface R and a harmonic diffeomorphism R → M −{ p 1 ,..., p m } such that every end of R is of hyperbolic type. If M = S 2 , by Koebe’s uniformization theorem, Corollary Let m ∈ N , m ≥ 2 , and let { p 1 ,..., p m } ⊂ S 2 . Then there exist a circular domain U in C and a harmonic diffeomorphism U → S 2 −{ p 1 ,..., p m } .

Maximal graphs over M −{ p 1 ,..., p m } in M × R 1 Question: Given the points p 1 ,..., p m , what are all the circular domains obtained by this construction?

(ii) Non-existence of harmonic diffeomorphisms D → S 2 −{ 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 → R 3 immersion with constant Gauss curvature K = 1 . II X positive definite metric (up to changing orientation) ⇒ II X induces on S a conformal structure, S . z = u + ı v conformal parameter on S .

Constant Gauss Curvature Surfaces S smooth simply-connected surface. X : S → R 3 immersion with constant Gauss curvature K = 1 . II X positive definite metric (up to changing orientation) ⇒ II X induces on S a conformal structure, S . z = u + ı v conformal parameter on S . ınez 2000] The Gauss map N : S → S 2 satisfies [G´ alvez-Mart´ X u = N × N v and X v = − N × N u , (1) hence it is a harmonic local diffeomorphism. Conversely, let N : S → S 2 be a harmonic local diffeomorphism. Then the map X : S → R 3 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 II X .

Constant Gauss Curvature Surfaces [Klotz 1980] There exists an immersion Y : S → R 3 of constant Gauss curvature K = 1 such that I Y = III X , II Y = II X and III Y = I X . Reason: ( III X , II X ) is a Codazzi pair.

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C .

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C . ∃ X : S → R 3 with Gauss map ϕ , constant curvature K X = 1 and such that the conformal structure of S is the one induced by II X .

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C . ∃ X : S → R 3 with Gauss map ϕ , constant curvature K X = 1 and such that the conformal structure of S is the one induced by II X . ∃ Y : S → R 3 with constant curvature K Y = 1 , I Y = III X , II Y = II X and III Y = I X .

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C . ∃ X : S → R 3 with Gauss map ϕ , constant curvature K X = 1 and such that the conformal structure of S is the one induced by II X . ∃ Y : S → R 3 with constant curvature K Y = 1 , I Y = III X , II Y = II X and III Y = I X . ϕ : S → S 2 −{ p } is a diffeomorphism and I Y = III X = � d ϕ , d ϕ � R 3 = ϕ ∗ ( �· , ·� S 2 ) , hence ϕ − 1 : S 2 −{ p } → ( S , I Y ) is an isometry.

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C . ∃ X : S → R 3 with Gauss map ϕ , constant curvature K X = 1 and such that the conformal structure of S is the one induced by II X . ∃ Y : S → R 3 with constant curvature K Y = 1 , I Y = III X , II Y = II X and III Y = I X . ϕ : S → S 2 −{ p } is a diffeomorphism and I Y = III X = � d ϕ , d ϕ � R 3 = ϕ ∗ ( �· , ·� S 2 ) , hence ϕ − 1 : S 2 −{ p } → ( S , I Y ) is an isometry. Y : ( S , I Y ) → R 3 isometric immersion.

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C . ∃ X : S → R 3 with Gauss map ϕ , constant curvature K X = 1 and such that the conformal structure of S is the one induced by II X . ∃ Y : S → R 3 with constant curvature K Y = 1 , I Y = III X , II Y = II X and III Y = I X . ϕ : S → S 2 −{ p } is a diffeomorphism and I Y = III X = � d ϕ , d ϕ � R 3 = ϕ ∗ ( �· , ·� S 2 ) , hence ϕ − 1 : S 2 −{ p } → ( S , I Y ) is an isometry. Y : ( S , I Y ) → R 3 isometric immersion. Y ◦ ϕ − 1 : S 2 −{ p } → R 3 isometric immersion.

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C . ∃ X : S → R 3 with Gauss map ϕ , constant curvature K X = 1 and such that the conformal structure of S is the one induced by II X . ∃ Y : S → R 3 with constant curvature K Y = 1 , I Y = III X , II Y = II X and III Y = I X . ϕ : S → S 2 −{ p } is a diffeomorphism and I Y = III X = � d ϕ , d ϕ � R 3 = ϕ ∗ ( �· , ·� S 2 ) , hence ϕ − 1 : S 2 −{ p } → ( S , I Y ) is an isometry. Y : ( S , I Y ) → R 3 isometric immersion. Y ◦ ϕ − 1 : S 2 −{ p } → R 3 isometric immersion. [Pogorelov 1973] S 2 −{ p } is rigid in R 3 .

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C . ∃ X : S → R 3 with Gauss map ϕ , constant curvature K X = 1 and such that the conformal structure of S is the one induced by II X . ∃ Y : S → R 3 with constant curvature K Y = 1 , I Y = III X , II Y = II X and III Y = I X . ϕ : S → S 2 −{ p } is a diffeomorphism and I Y = III X = � d ϕ , d ϕ � R 3 = ϕ ∗ ( �· , ·� S 2 ) , hence ϕ − 1 : S 2 −{ p } → ( S , I Y ) is an isometry. Y : ( S , I Y ) → R 3 isometric immersion. Y ◦ ϕ − 1 : S 2 −{ p } → R 3 isometric immersion. [Pogorelov 1973] S 2 −{ p } is rigid in R 3 . Y ( S ) ⊂ R 3 is a once punctured round sphere.

Non-existence of harmonic diffeomorphisms D → S 2 −{ p } . S simply-connected Riemann surface, ϕ : S → S 2 −{ p } harmonic diffeomorphism. We will show that S is conformally equivalent to C . ∃ X : S → R 3 with Gauss map ϕ , constant curvature K X = 1 and such that the conformal structure of S is the one induced by II X . ∃ Y : S → R 3 with constant curvature K Y = 1 , I Y = III X , II Y = II X and III Y = I X . ϕ : S → S 2 −{ p } is a diffeomorphism and I Y = III X = � d ϕ , d ϕ � R 3 = ϕ ∗ ( �· , ·� S 2 ) , hence ϕ − 1 : S 2 −{ p } → ( S , I Y ) is an isometry. Y : ( S , I Y ) → R 3 isometric immersion. Y ◦ ϕ − 1 : S 2 −{ p } → R 3 isometric immersion. [Pogorelov 1973] S 2 −{ p } is rigid in R 3 . Y ( S ) ⊂ R 3 is a once punctured round sphere. The conformal structure induced on S by II Y = II X is C .

(iii) Non-existence of harmonic diffeomorphisms C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j

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, S 2 −{ p 1 ,..., p m } 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, S 2 −{ p 1 ,..., p m } 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 K N > 0 . Then φ is either holomorphic or antiholomorphic.

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j Assume that φ preserves orientation. Is φ holomorphic?

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 ) | d φ | 2 . A conformal metric on R writes λ ( z ) | dz | 2 .

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 ) | d φ | 2 . A conformal metric on R writes λ ( z ) | dz | 2 . Partial densities: | ∂φ | 2 = ρ ( φ ( z )) ∂ z | 2 and | ∂φ | 2 = ρ ( φ ( z )) λ ( z ) | ∂φ λ ( z ) | ∂φ z | 2 . ∂ ¯

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 ) | d φ | 2 . A conformal metric on R writes λ ( z ) | dz | 2 . Partial densities: | ∂φ | 2 = ρ ( φ ( z )) ∂ z | 2 and | ∂φ | 2 = ρ ( φ ( z )) λ ( z ) | ∂φ λ ( z ) | ∂φ z | 2 . ∂ ¯ The Jacobian of φ , J ( φ ) = | ∂φ | 2 −| ∂φ | 2 > 0 .

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 ) | d φ | 2 . A conformal metric on R writes λ ( z ) | dz | 2 . Partial densities: | ∂φ | 2 = ρ ( φ ( z )) ∂ z | 2 and | ∂φ | 2 = ρ ( φ ( z )) λ ( z ) | ∂φ 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 ) = − 2 K N J ( φ )+2 K R ∆ R log( | ∂φ | 2 ) = 2 K N J ( φ )+2 K R

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 ) | d φ | 2 . A conformal metric on R writes λ ( z ) | dz | 2 . Partial densities: | ∂φ | 2 = ρ ( φ ( z )) ∂ z | 2 and | ∂φ | 2 = ρ ( φ ( z )) λ ( z ) | ∂φ 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 ) = − 2 K N J ( φ )+2 K R ∆ R log( | ∂φ | 2 ) = 2 K N J ( φ )+2 K R By contradiction: If φ is not holomorphic, then the zeros of | ∂φ | are isolated.

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j R ∗ = R −{| ∂φ | = 0 } .

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j R ∗ = R −{| ∂φ | = 0 } . log( | ∂φ | / | ∂φ | ) < 0 on R ∗ .

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j R ∗ = R −{| ∂φ | = 0 } . log( | ∂φ | / | ∂φ | ) < 0 on R ∗ . R ∗ is parabolic.

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 | ) = 2 K N J ( φ ) .

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 | ) = 2 K N J ( φ ) . Since K N > 0 , then log( | ∂φ | / | ∂φ | ) is a non-constant negative subharmonic function on the parabolic surface R ∗ , a contradiction.

Non-existence, C −{ z 1 ,..., z m } → S 2 −∪ m j =1 D j 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 | ) = 2 K N J ( φ ) . Since K N > 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 → S 2 −{ p 1 ,..., p m } ?

Harmonic diffeomorphisms and K − surfaces Answer: Theorem (Alarc´ on-S. 2012) Let { p 1 ,..., p m } ⊂ S 2 , 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 → S 2 −{ p 1 ,..., p m } . (ii) There exist positive real constants a 1 ,..., a m such that j =1 a j p j = � ∑ m 0 ∈ R 3 .

Harmonic diffeomorphisms and K − surfaces Theorem (continued) Furthermore, if S is as above and one denotes by γ j the connected component of S − S corresponding to p j via its Gauss map, then (I) γ j is a Jordan curve contained in an affine plane Π j ⊂ R 3 orthogonal to p j , and (II) K = S ∪ ( ∪ m j =1 D j ) is the boundary surface of a smooth convex body 1 in R 3 , where D j is the bounded connected component of Π j − γ j for all j ∈ { 1 ,..., m } . In addition, given { a 1 ,..., a m } satisfying (ii) , there exists a unique, up to translations, surface S satisfying (i) such that the area of D j equals a j for all j ∈ { 1 ,..., m } . 1 A convex body in R 3 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.