The Global Geometry of Stationary Surfaces in 4-dimensional Lorentz space Xiang Ma (Joint with Zhiyu Liu, Changping Wang, Peng Wang) Peking University the 10th Pacific Rim Geometry Conference 3 December, 2011, Osaka Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction Total curvature and singularities Constructing embedded examples 1 Introduction What is a stationary surface Main results The Weierstrass representation 2 Total curvature and singularities The failure of Osserman’s theorem Singular ends Gauss-Bonnet type theorems 3 Constructing embedded examples Generalized catenoid and k-noids Generalized helicoid and Enneper surface Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation 1 Introduction What is a stationary surface Main results The Weierstrass representation 2 Total curvature and singularities The failure of Osserman’s theorem Singular ends Gauss-Bonnet type theorems 3 Constructing embedded examples Generalized catenoid and k-noids Generalized helicoid and Enneper surface Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Freshman attempting to break a soap film Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Sharing soap films with kids Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Stationary surfaces = spacelike surfaces with H = 0 In R 4 1 : � X , X � := X 2 1 + X 2 2 + X 2 3 − X 2 4 . H = 0 ⇔ X : M → R 4 1 is harmonic ( for induced metric ). Special cases: In R 3 : Minimizer of the surface area. In R 3 1 : Maximizer of the surface area. In R 4 1 : Not local minimizer or maximizer. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Stationary surfaces = spacelike surfaces with H = 0 In R 4 1 : � X , X � := X 2 1 + X 2 2 + X 2 3 − X 2 4 . H = 0 ⇔ X : M → R 4 1 is harmonic ( for induced metric ). Special cases: In R 3 : Minimizer of the surface area. In R 3 1 : Maximizer of the surface area. In R 4 1 : Not local minimizer or maximizer. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Motivation Stationary surfaces in R 4 1 are: special examples of Willmore surfaces � ( H 2 − K ) d M ). (critical points for corresponding to Laguerre minimal surfaces � H 2 − K (critical points for d M ). K A natural generalization of classical minimal surfaces in R 3 , yet receiving little attention. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Motivation Stationary surfaces in R 4 1 are: special examples of Willmore surfaces � ( H 2 − K ) d M ). (critical points for corresponding to Laguerre minimal surfaces � H 2 − K (critical points for d M ). K A natural generalization of classical minimal surfaces in R 3 , yet receiving little attention. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Motivation Stationary surfaces in R 4 1 are: special examples of Willmore surfaces � ( H 2 − K ) d M ). (critical points for corresponding to Laguerre minimal surfaces � H 2 − K (critical points for d M ). K A natural generalization of classical minimal surfaces in R 3 , yet receiving little attention. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Motivation Stationary surfaces in R 4 1 are: special examples of Willmore surfaces � ( H 2 − K ) d M ). (critical points for corresponding to Laguerre minimal surfaces � H 2 − K (critical points for d M ). K A natural generalization of classical minimal surfaces in R 3 , yet receiving little attention. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Main Results Osserman’s theorem fails. � We construct examples with | K | < ∞ whose Gauss maps could not extend to the ends. Singular ends. We divide them into two types; define index for good type. Gauss-Bonnet type result: � � � K d M = 2 π (2 − 2 g − m − d j ) . M We construct many embedded examples (in contrast to uniqueness results in R 3 ). Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Main Results Osserman’s theorem fails. � We construct examples with | K | < ∞ whose Gauss maps could not extend to the ends. Singular ends. We divide them into two types; define index for good type. Gauss-Bonnet type result: � � � K d M = 2 π (2 − 2 g − m − d j ) . M We construct many embedded examples (in contrast to uniqueness results in R 3 ). Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Main Results Osserman’s theorem fails. � We construct examples with | K | < ∞ whose Gauss maps could not extend to the ends. Singular ends. We divide them into two types; define index for good type. Gauss-Bonnet type result: � � � K d M = 2 π (2 − 2 g − m − d j ) . M We construct many embedded examples (in contrast to uniqueness results in R 3 ). Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation Main Results Osserman’s theorem fails. � We construct examples with | K | < ∞ whose Gauss maps could not extend to the ends. Singular ends. We divide them into two types; define index for good type. Gauss-Bonnet type result: � � � K d M = 2 π (2 − 2 g − m − d j ) . M We construct many embedded examples (in contrast to uniqueness results in R 3 ). Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation The Gauss Map in R 3 Minimal ⇔ N : M → S 2 anti-conformal. ⇔ G = p ◦ N meromorphic. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
� � � Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation The Gauss Maps in R 4 1 Space-like X : M 2 → R 4 1 : normal plane ( TM ) ⊥ is a Lorentz plane; splits into light-like lines ( TM ) ⊥ = Span { Y , Y ∗ } . [ Y ] , [ Y ∗ ] ∼ ( M 2 , z ) Q 2 = S 2 � Y , Y � = � Y ∗ , Y ∗ � = 0 , � � Y , Y ∗ � = 1 . � � p � � � Q 2 = { [ v ] ∈ R P 3 |� v , v � = 0 } . φ,ψ � � � � C Stationary ⇔ [ Y ] conformal, [ Y ∗ ] anti-conformal. ⇔ φ, ψ : M → C meromorphic. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
� � � Introduction What is a stationary surface Total curvature and singularities Main results Constructing embedded examples The Weierstrass representation The Gauss Maps in R 4 1 Space-like X : M 2 → R 4 1 : normal plane ( TM ) ⊥ is a Lorentz plane; splits into light-like lines ( TM ) ⊥ = Span { Y , Y ∗ } . [ Y ] , [ Y ∗ ] ∼ ( M 2 , z ) Q 2 = S 2 � Y , Y � = � Y ∗ , Y ∗ � = 0 , � � Y , Y ∗ � = 1 . � � p � � � Q 2 = { [ v ] ∈ R P 3 |� v , v � = 0 } . φ,ψ � � � � C Stationary ⇔ [ Y ] conformal, [ Y ∗ ] anti-conformal. ⇔ φ, ψ : M → C meromorphic. Global Geometry of Stationary Surfaces in R 4 Z. Liu, X. Ma, C. Wang, P. Wang 1
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