Integral transforms and the twistor theory for indefinite metrics Fuminori NAKATA Tokyo University of Science Dec. 5, 2011, The 10th Pacific Rim Geometry Conference F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 1 / 36
Introduction Twistor correspondence is a correspondence between complex manifolds with a family of CP 1 or holomorphic disks, and manifolds equipped with a certain integrable structure. self-dual conformal 4-mfd Einstein-Weyl 3-mfd complex Penrose (1976) Hitchin (1982) Hitchin (1982) Riemannian Atiyah-Hitchin-Singer (1978) Pedersen-Tod (1993) indefinite LeBrun-Mason (2007) LeBrun-Mason (2009) Results in hyperbolic PDE and integral transforms are obtained in the way of constructing explicit examples of twistor correspondences. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 2 / 36
Introduction Twistor correspondence is a correspondence between complex manifolds with a family of CP 1 or holomorphic disks, and manifolds equipped with a certain integrable structure. self-dual conformal 4-mfd Einstein-Weyl 3-mfd complex Penrose (1976) Hitchin (1982) Hitchin (1982) Riemannian Atiyah-Hitchin-Singer (1978) Pedersen-Tod (1993) indefinite LeBrun-Mason (2007) LeBrun-Mason (2009) Results in hyperbolic PDE and integral transforms are obtained in the way of constructing explicit examples of twistor correspondences. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 2 / 36
Introduction Twistor correspondence is a correspondence between complex manifolds with a family of CP 1 or holomorphic disks, and manifolds equipped with a certain integrable structure. self-dual conformal 4-mfd Einstein-Weyl 3-mfd complex Penrose (1976) Hitchin (1982) Hitchin (1982) Riemannian Atiyah-Hitchin-Singer (1978) Pedersen-Tod (1993) indefinite LeBrun-Mason (2007) LeBrun-Mason (2009) Results in hyperbolic PDE and integral transforms are obtained in the way of constructing explicit examples of twistor correspondences. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 2 / 36
Integral transforms on 2-sphere 1 Integral transforms on a cylinder 2 Minitwistor theory 3 LeBrun-Mason twistor theory 4 general theory S 1 -invariant case R -invariant case F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 3 / 36
1. Integral transforms on 2-sphere F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 4 / 36
S Ω (t,y) y 2 C (t,y) Small circles Let us define M = { oriented small circles on S 2 } � � domain on S 2 ∼ = bouded by a small circle tanh t Each domain is described as Ω ( t,y ) = { u ∈ S 2 | u · y > tanh t } by using ( t, y ) ∈ R × S 2 . Hence M ∼ = R × S 2 . F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 5 / 36
S Ω (t,y) y 2 C (t,y) Small circles Let us define M = { oriented small circles on S 2 } � � domain on S 2 ∼ = bouded by a small circle tanh t Each domain is described as Ω ( t,y ) = { u ∈ S 2 | u · y > tanh t } by using ( t, y ) ∈ R × S 2 . Hence M ∼ = R × S 2 . F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 5 / 36
S Ω (t,y) y 2 C (t,y) Small circles Let us define M = { oriented small circles on S 2 } � � domain on S 2 ∼ = bouded by a small circle tanh t Each domain is described as Ω ( t,y ) = { u ∈ S 2 | u · y > tanh t } by using ( t, y ) ∈ R × S 2 . Hence M ∼ = R × S 2 . F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 5 / 36
2 y S Ω (t,y) C (t,y) Small circles M = { ( t, y ) ∈ R × S 2 } Let us introduce an indefinite metric on M by g = − dt 2 + cosh 2 t · g S 2 . tanh t ( M, g ) is identified with the de Sitter 3-space ( S 3 1 , g S 3 1 ) This identification M ∼ = S 3 1 arises from minitwistor correspondence. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 6 / 36
2 y S Ω (t,y) C (t,y) Small circles M = { ( t, y ) ∈ R × S 2 } Let us introduce an indefinite metric on M by g = − dt 2 + cosh 2 t · g S 2 . tanh t ( M, g ) is identified with the de Sitter 3-space ( S 3 1 , g S 3 1 ) This identification M ∼ = S 3 1 arises from minitwistor correspondence. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 6 / 36
2 y S Ω (t,y) C (t,y) Small circles M = { ( t, y ) ∈ R × S 2 } Let us introduce an indefinite metric on M by g = − dt 2 + cosh 2 t · g S 2 . tanh t ( M, g ) is identified with the de Sitter 3-space ( S 3 1 , g S 3 1 ) This identification M ∼ = S 3 1 arises from minitwistor correspondence. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 6 / 36
Geodesics There are subfamilies of small circles known as “Apollonian circles”. These families corresponde to geodesics on ( S 3 1 , g S 3 1 ) . space-like geodesic null geodesic time-like geodesic F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 7 / 36
Geodesics There are subfamilies of small circles known as “Apollonian circles”. These families corresponde to geodesics on ( S 3 1 , g S 3 1 ) . space-like geodesic null geodesic time-like geodesic F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 7 / 36
Integral transforms For given function h ∈ C ∞ ( S 2 ) , we define functions Rh, Qh ∈ C ∞ ( S 3 1 ) by Rh ( t, y ) = 1 � h dS 1 [mean value] 2 π ∂ Ω ( t,y ) Qh ( t, y ) = 1 � h dS 2 [area integral] 2 π Ω ( t,y ) where dS 1 is the standard measure on ∂ Ω ( t,y ) of total length 2 π , and dS 2 is the standard measure on S 2 . F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 8 / 36
Integral transforms For given function h ∈ C ∞ ( S 2 ) , we define functions Rh, Qh ∈ C ∞ ( S 3 1 ) by Rh ( t, y ) = 1 � h dS 1 [mean value] 2 π ∂ Ω ( t,y ) Qh ( t, y ) = 1 � h dS 2 [area integral] 2 π Ω ( t,y ) where dS 1 is the standard measure on ∂ Ω ( t,y ) of total length 2 π , and dS 2 is the standard measure on S 2 . F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 8 / 36
Wave equation on de Sitter 3-space Wave equation on ( S 3 1 , g S 3 1 ) : � V := ∗ d ∗ dV = 0 . � � � � S 2 hdS 2 = 0 ∗ ( S 2 ) = h ∈ C ∞ ( S 2 ) � Let us put C ∞ . � � Theorem (N. ’09) ∗ ( S 2 ) , the function V := Qh ∈ C ∞ ( S 3 For each function h ∈ C ∞ 1 ) satisfies (i) � V = 0 , (ii) lim t →∞ V ( t, y ) = lim t →∞ V t ( t, y ) = 0 . Conversely, if V ∈ C ∞ ( S 3 1 ) satisfies (i) and (ii), then there exists unique ∗ ( S 2 ) such that V = Qh . h ∈ C ∞ Remark A similar type theorem for the transform R is also obtained. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 9 / 36
Wave equation on de Sitter 3-space Wave equation on ( S 3 1 , g S 3 1 ) : � V := ∗ d ∗ dV = 0 . � � � � S 2 hdS 2 = 0 ∗ ( S 2 ) = h ∈ C ∞ ( S 2 ) � Let us put C ∞ . � � Theorem (N. ’09) ∗ ( S 2 ) , the function V := Qh ∈ C ∞ ( S 3 For each function h ∈ C ∞ 1 ) satisfies (i) � V = 0 , (ii) lim t →∞ V ( t, y ) = lim t →∞ V t ( t, y ) = 0 . Conversely, if V ∈ C ∞ ( S 3 1 ) satisfies (i) and (ii), then there exists unique ∗ ( S 2 ) such that V = Qh . h ∈ C ∞ Remark A similar type theorem for the transform R is also obtained. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 9 / 36
Wave equation on de Sitter 3-space Wave equation on ( S 3 1 , g S 3 1 ) : � V := ∗ d ∗ dV = 0 . � � � � S 2 hdS 2 = 0 ∗ ( S 2 ) = h ∈ C ∞ ( S 2 ) � Let us put C ∞ . � � Theorem (N. ’09) ∗ ( S 2 ) , the function V := Qh ∈ C ∞ ( S 3 For each function h ∈ C ∞ 1 ) satisfies (i) � V = 0 , (ii) lim t →∞ V ( t, y ) = lim t →∞ V t ( t, y ) = 0 . Conversely, if V ∈ C ∞ ( S 3 1 ) satisfies (i) and (ii), then there exists unique ∗ ( S 2 ) such that V = Qh . h ∈ C ∞ Remark A similar type theorem for the transform R is also obtained. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 9 / 36
Wave equation on de Sitter 3-space Wave equation on ( S 3 1 , g S 3 1 ) : � V := ∗ d ∗ dV = 0 . � � � � S 2 hdS 2 = 0 ∗ ( S 2 ) = h ∈ C ∞ ( S 2 ) � Let us put C ∞ . � � Theorem (N. ’09) ∗ ( S 2 ) , the function V := Qh ∈ C ∞ ( S 3 For each function h ∈ C ∞ 1 ) satisfies (i) � V = 0 , (ii) lim t →∞ V ( t, y ) = lim t →∞ V t ( t, y ) = 0 . Conversely, if V ∈ C ∞ ( S 3 1 ) satisfies (i) and (ii), then there exists unique ∗ ( S 2 ) such that V = Qh . h ∈ C ∞ Remark A similar type theorem for the transform R is also obtained. F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 9 / 36
2. Integral transforms on a cylinder F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 10 / 36
θ C (t,x) v Planar circles C Let C = { ( θ, v ) ∈ S 1 × R } be the 2-dimensional cylinder. Let us define M ′ = { planar circles on C } Each planar circle is described as C ( t,x ) = { ( θ, v ) | v = t + x 1 cos θ + x 2 sin θ } using ( t, x ) ∈ R × R 2 . Hence M ′ ∼ = R 3 F.Nakata (TUS) Integral transforms and the twistor theory. PRGC, Dec. 2011 11 / 36
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