Type changes of spacelike maximal surfaces in Minkowski 3-space to - PowerPoint PPT Presentation

Type changes of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces 8th International Meeting on Lorentzian Geometry, M´ alaga Kotaro Yamada Tokyo Institute of Technology kotaro@math.titech.ac.jp 23. Sept. 2016 K. Yamada GELOMA 2016 23. Sept. 2016 1 / 50

This talk is based on the Joint works. . . S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, KY Entire zero mean curvature graphs of mixed type in Lorentz- Minkowski 3 -space preprint, arXiv:1511.07954, to appear. A. Honda, M. Koiso, M. Kokubu, M. Umehara and KY Mixed type surfaces with bounded mean curvature in 3 -dimensional space-times preprint, arXiv:1508.02514. S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara, S.-D. Yang and KY Zero mean curvature surfaces in Lorentz-Minkowski 3 -space and 2 -dimensional fluid mechanics Math. J. Okayama Univ., vol. 57 (2015). K. Yamada GELOMA 2016 23. Sept. 2016 2 / 50

Surfaces in L 3 A surface in the Lorentz-Minkowki 3-space ( ) L 3 := ( R 3 ; t, x, y ) , ⟨ , ⟩ = − dt 2 + dx 2 + dy 2 is said to be Spacelike if all tangent planes are spacelike. Timelike if all tangent planes are timelike. spacelike timelike type change K. Yamada GELOMA 2016 23. Sept. 2016 4 / 50

What kind of surfaces can change types? Theorem (Honda-Koiso-Kokubu-Umehara-Y) f : R 2 ⊂ D → L 3 : an immersion such that U + := { spacelike points } ̸ = ∅ , U − := { timelike points } ̸ = ∅ , and the mean curvature function H is bounded on U + ∪ U − . Then for ∀ p ∈ U + ∩ U − , ∃ a sequence { p n } ⊂ U + ∪ U − such that p n → p ( n → ∞ ) , and H ( p n ) → 0 ( n → ∞ ) . Corollary Surfaces of non-zero constant mean curvature cannot change causal types. K. Yamada GELOMA 2016 23. Sept. 2016 5 / 50

ZMC surfaces Question Can zero mean curvature (ZMC) surfaces change types? K. Yamada GELOMA 2016 23. Sept. 2016 6 / 50

ZMC surfaces A surface in L 3 is said to be Zero Mean Curvature (ZMC) if the union of the spacelike part and the timelike part is dense on the surface, and the mean curvature vanishes identically on spacelike/timelike part. a spacelike ZMC surface is called a (spacelike) maximal surface. a timelike ZMC surface is called a (timelike) minimal surface. This talk deals Description of type-change phenomenon for ZMC surfaces from the viewpoint of maximal surfaces. Examples of embedded ZMC surfaces. Examples of entire ZMC graphs. K. Yamada GELOMA 2016 23. Sept. 2016 7 / 50

Maximal surfaces A maximal surface is A spacelike surface in L 3 with vanishing mean curvature. A critical point of the area functional. A Weierstrass-type representation formula (Osamu Kobayashi, 1983) Written in terms of holomorphic data on the surface. cf. The Weierstrass representation for minimal surfaces in R 3 . A complete maximal surface in L 3 is a spacelike plane (Calabi, cf. Rubio’s talk) It is natural to consider maximal surfaces with singularities . O. Kobayashi Maximal surfaces in the 3-dimensional Minkowski space L 3 Tokyo J. Math., 6 (1983), 297–309. O. Kobayashi Maximal surfaces with conelike singularities J. Math. Soc. Japan, 36 (1984), 609–617. K. Yamada GELOMA 2016 23. Sept. 2016 8 / 50

The Weierstrass-type representation O. Kobayashi (1983): A spacelike maximal immersion f : M 2 → L 3 is expressed as √ ∫ ( ) − 2 g, 1 + g 2 , − 1(1 − g 2 ) f = Re F ; F = ω where ( ) ( g, ω ) = a meromorphic fct. , a holomorphic 1 -form on M 2 (with complex structure induced by the induced metric). The induced metric is ds 2 = (1 − | g | 2 ) 2 | ω | 2 ( | g | = 1 : singularity) Remark: f ∗ = Im F is also a maximal surface called the conjugate of f . cf. Yasumoto’s talk K. Yamada GELOMA 2016 23. Sept. 2016 9 / 50

The Weierstrass-type representation: Gauss maps The unit normal vector of f is expressed by the Weierstrass data ( g, ω ) as ( | g | 2 + 1 | g | 2 − 1 , 2 Re g 1 − | g | 2 , 2 Im g ) : M 2 → H 2 ± ⊂ L 3 , ν = 1 − | g | 2 that is, g is the unit normal vector ν composed with the stereographic projection (the Gauss map). K. Yamada GELOMA 2016 23. Sept. 2016 10 / 50

Maxfaces—Maximal surfaces with singularities Definition (Umehara-Y, 2006) A map f : M 2 → L 3 is a maxface ( M 2 : a Riemann surface) ⇔ ∃ W ⊂ M 2 : open dense, such that f W is a conformal spacelike maximal immersion, and f ( p ) ̸ = 0 ( ∀ p ∈ M 2 ) . d Estudillo and Romero (1992) introduced a notion of and investigated global properties of them. Our notion of maxfaces is equivalent to non-branched generalized maximal surfaces in the sense of Estudillo-Romero. K. Yamada GELOMA 2016 23. Sept. 2016 11 / 50

Conjugate Maxfaces Weierstrass representation √ ∫ ( ) − 2 g, 1 + g 2 , − 1(1 − g 2 ) f = Re F ; F = ω gives a maxface # := (1 + | g | 2 ) 2 | ω | 2 is positive definite. ⇔ ds 2 cf. the induced metric: ds 2 = (1 − | g | 2 ) 2 | ω | 2 The singular set of f is { p ∈ M 2 ; | g | = 1 } . The conjugate f ∗ = Im F corresponds to ( g, −√− 1 ω ) . The singular set of f ∗ coincides with that of f . K. Yamada GELOMA 2016 23. Sept. 2016 12 / 50

Singularities of Maxfaces Theorem (Fujimori-Saji-Umehara-Y, 2008) Generic singular points of maxfaces are cuspidal edges, swallowtails, and cuspidal crosscaps. The conjugate f ∗ The duality Maxface f cuspidal edge cuspidal edge swallowtail cuspidal crosscap cuspidal crosscap swallowtail Cuspidal edge swallowtail cuspidal crosscap Note: Similar duality properties hold for timelike minimal surfaces. (cf. H. Takahashi, Master Thesis, 2012); K. Yamada GELOMA 2016 23. Sept. 2016 13 / 50

Example (the catenoid) M 2 = C \ { 0 } , ( g, ω ) = M 2 = C \ { 0 } , ( g, ω ) = ( z, dz ) ( z, dz ) z 2 z 2 1 1 Lorentzian-Catenoid in L 3 Euclidean-Catenoid in R 3 | z | = 1 : cone-like singularities non-generic K. Yamada GELOMA 2016 23. Sept. 2016 14 / 50

Example (the helicoid) The conjugate surface of the catenoid: Helicoid in R 3 Lorentz-Helicoid in L 3 | z | = 1 corresponds to the fold singularities. the image of fold singularities consists of a null (light-like) curve. K. Yamada GELOMA 2016 23. Sept. 2016 15 / 50

Fold singularities − → − → Fact (cf. Fujimori-Kim-Koh-Rossman-Shin-Umehara-Yang-Y, 2015) The conjugate of a cone-like singularity is a fold singularity. For a maxface obtained by the Weierstrass data ( g, ω ) , the singular curve γ is a fold singularity Re dg ⇔ | g | = 1 , dg ̸ = 0 , and g 2 ω = 0 hold on γ . K. Yamada GELOMA 2016 23. Sept. 2016 16 / 50

The analytic extension of the helicoid The image of the Lorentz helicoid can be extended across the singular curve: Lorentz helicoid in L 3 The analytic extension The dark part is a timelike minimal surface. Remark: The image of the right-hand figure coincides with the Euclidean helicoid (cf. Albujer’s talk). K. Yamada GELOMA 2016 23. Sept. 2016 17 / 50

Analytic extensions of along fold singularities Fact f : M 2 → L 3 : a maxface with fold singularities γ ( t ) . ⇒ γ ( t ) = f ◦ γ ( t ) is a null curve in L 3 which is The image ˆ non-degenerate ( i.e. ˙ γ ( t ) and ¨ γ ( t ) are linearly independent for all t ) . The map γ ( u − v ) f ( u, v ) := ˆ γ ( u + v ) + ˆ ˜ 2 gives a timelike minimal surface. The image of ˜ f gives the analytic extension of the image of f along ˆ γ . The union of the images of f and ˜ f is an immersed surface near ˆ γ . C. Gu (1985); V. A. Klyachin (2003); Kim-Koh-Shin-Yang (2006). (cf. [3]) K. Yamada GELOMA 2016 23. Sept. 2016 18 / 50

The analytic extension Take ˆ γ ( t ) : a null curve, which is nondegenerae, i.e. ¨ γ ( t ) is not proportional to ˙ ˆ γ ( t ) . ˆ Take the midpoint of two points: f ( s, t ) := 1 ( ) ˆ γ ( s ) + ˆ γ ( t ) 2 f gives a timelike minimal surface. K. Yamada GELOMA 2016 23. Sept. 2016 19 / 50

Example (Scherk-type surface) M 2 = C \ {± 1 , ±√− 1 } M 2 = C \ {± 1 , ±√− 1 } √− 1 dz ( ) ( ) dz ( g, ω ) = z, ( g, ω ) = z, 1 − z 4 1 − z 4 Lorentz-Scherk in L 3 Scherk R 3 The graph x 3 = log cos x 1 . cos x 2 K. Yamada GELOMA 2016 23. Sept. 2016 20 / 50

The analytic extension of Scherk-type surface t = log cosh x cosh y Entire ZMC graph (O. Kobayashi (1983)) K. Yamada GELOMA 2016 23. Sept. 2016 21 / 50

Embedded Examples: Schwarz type (Euclidean case) ( z, w ) ∈ ( C ∪ {∞} ) 2 ; w 2 = z 8 + ( a 4 + a − 4 ) z 4 + 1 { } M a := , ( 0 < a < 1 ) Schwarz P g = z ω = dz w Schwarz D g = z ω = √− 1 dz w Figure: Shoichi Fujimori (Okayama Univ.) K. Yamada GELOMA 2016 23. Sept. 2016 22 / 50

Embedded Examples: Schwarz type (Lorentzian case) ( z, w ) ∈ ( C ∪ {∞} ) 2 ; w 2 = z 8 + ( a 4 + a − 4 ) z 4 + 1 { } M a := , ( 0 < a < 1 ) Figure: Fujimori Schwarz P g = z ω = dz w Schwarz D g = z ω = √− 1 dz w K. Yamada GELOMA 2016 23. Sept. 2016 23 / 50

The analytic extension of the Schwarz D-type maxface → Embedded (Fujimori-Rossman-Umehara-Yang-Y (2014)) Figure: Fujimori K. Yamada GELOMA 2016 23. Sept. 2016 24 / 50