Biharmonic maps Let ( M m , g ) and ( N n , h ) be two Riemannian manifolds. Assume M is compact and consider The bienergy functional E 2 ( φ ) = 1 � M � τ ( φ ) � 2 v g . E 2 : C ∞ ( M , N ) → R , 2 The biharmonic maps are critical points of E 2 , i.e., for any variation { φ t } t ∈ R of φ we have � d � { E 2 ( φ t ) } = 0 . � dt � t = 0 18 / 88
Biharmonic maps Theorem ([11]) A smooth map φ : ( M m , g ) → ( N n , h ) is biharmonic if and only if the bitension field associated to φ , τ 2 ( φ ) = − ∆ φ τ ( φ ) − trace g R N ( d φ , τ ( φ )) d φ , vanishes. Here, ∆ φ = − trace g ∇ φ ∇ φ − ∇ φ � � ∇ is the rough Laplacian on the sections of φ − 1 TN and R N ( X , Y ) Z = ∇ N X ∇ N Y Z − ∇ N Y ∇ N X Z − ∇ N [ X , Y ] Z . 19 / 88
The biharmonic equation (G.Y. Jiang, 1986) τ 2 ( φ ) = − ∆ φ τ ( φ ) − trace g R N ( d φ , τ ( φ )) d φ = 0 20 / 88
The biharmonic equation (G.Y. Jiang, 1986) τ 2 ( φ ) = − ∆ φ τ ( φ ) − trace g R N ( d φ , τ ( φ )) d φ = 0 is a fourth-order non-linear elliptic equation; 20 / 88
The biharmonic equation (G.Y. Jiang, 1986) τ 2 ( φ ) = − ∆ φ τ ( φ ) − trace g R N ( d φ , τ ( φ )) d φ = 0 is a fourth-order non-linear elliptic equation; any harmonic map is biharmonic; 20 / 88
The biharmonic equation (G.Y. Jiang, 1986) τ 2 ( φ ) = − ∆ φ τ ( φ ) − trace g R N ( d φ , τ ( φ )) d φ = 0 is a fourth-order non-linear elliptic equation; any harmonic map is biharmonic; a non-harmonic biharmonic map is called proper biharmonic; 20 / 88
The stress-bienergy tensor G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S 2 for the bienergy functional, and called it the stress-bienergy tensor: S 2 ( X , Y ) = 1 2 � τ ( φ ) � 2 � X , Y � + � d φ , ∇ τ ( φ ) �� X , Y � −� d φ ( X ) , ∇ Y τ ( φ ) �−� d φ ( Y ) , ∇ X τ ( φ ) � . It satisfies div S 2 = −� τ 2 ( φ ) , d φ � . 21 / 88
The stress-bienergy tensor G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S 2 for the bienergy functional, and called it the stress-bienergy tensor: S 2 ( X , Y ) = 1 2 � τ ( φ ) � 2 � X , Y � + � d φ , ∇ τ ( φ ) �� X , Y � −� d φ ( X ) , ∇ Y τ ( φ ) �−� d φ ( Y ) , ∇ X τ ( φ ) � . It satisfies div S 2 = −� τ 2 ( φ ) , d φ � . φ = biharmonic ⇒ div S 2 = 0 . 21 / 88
The stress-bienergy tensor G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S 2 for the bienergy functional, and called it the stress-bienergy tensor: S 2 ( X , Y ) = 1 2 � τ ( φ ) � 2 � X , Y � + � d φ , ∇ τ ( φ ) �� X , Y � −� d φ ( X ) , ∇ Y τ ( φ ) �−� d φ ( Y ) , ∇ X τ ( φ ) � . It satisfies div S 2 = −� τ 2 ( φ ) , d φ � . φ = biharmonic ⇒ div S 2 = 0 . If φ is a submersion, div S 2 = 0 if and only if φ is biharmonic. 21 / 88
The stress-bienergy tensor G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S 2 for the bienergy functional, and called it the stress-bienergy tensor: S 2 ( X , Y ) = 1 2 � τ ( φ ) � 2 � X , Y � + � d φ , ∇ τ ( φ ) �� X , Y � −� d φ ( X ) , ∇ Y τ ( φ ) �−� d φ ( Y ) , ∇ X τ ( φ ) � . It satisfies div S 2 = −� τ 2 ( φ ) , d φ � . φ = biharmonic ⇒ div S 2 = 0 . If φ is a submersion, div S 2 = 0 if and only if φ is biharmonic. If φ : M → N is a Riemannian immersion then ( div S 2 ) ♯ = − τ 2 ( φ ) ⊤ . In general, for a Riemannian immersion, div S 2 � = 0 . 21 / 88
The variational meaning of S 2 If φ : M → ( N , h ) is a fixed map, then E 2 can be thought as a functional on the set of all Riemannian metrics on M . The critical points of this new functional are determined by S 2 = 0 . 22 / 88
Content The motivation of the research topic 1 General context Harmonic maps Biharmonic maps Properties of biconservative submanifolds 2 Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Complete biconservative surfaces Biconservative surfaces in N n 23 / 88
Basic facts in the submanifolds theory Let φ : M m → N n be a submanifold. 24 / 88
Basic facts in the submanifolds theory Let φ : M m → N n be a submanifold. Locally, ∇ φ X d φ ( Y ) ≡ ∇ N d φ ( X ) ≡ X ; X Y ; 24 / 88
Basic facts in the submanifolds theory Let φ : M m → N n be a submanifold. Locally, ∇ φ X d φ ( Y ) ≡ ∇ N d φ ( X ) ≡ X ; X Y ; Globally, T φ ( p ) N = d φ p ( T p M ) ⊕ d φ p ( T p M ) ⊥ ; φ − 1 ( TN ) = � T φ ( p ) N ; p ∈ M d φ p ( T p M ) ⊥ . � � TM ≡ d φ p ( T p M ) ; NM = p ∈ M p ∈ M 24 / 88
Basic facts in the submanifolds theory The Gauss equation ∇ N X Y = ∇ X Y + B ( X , Y ) ; 25 / 88
Basic facts in the submanifolds theory The Gauss equation ∇ N X Y = ∇ X Y + B ( X , Y ) ; The Weingarten equation ∇ N X η = − A η ( X )+ ∇ ⊥ X η ; 25 / 88
Basic facts in the submanifolds theory The Gauss equation ∇ N X Y = ∇ X Y + B ( X , Y ) ; The Weingarten equation ∇ N X η = − A η ( X )+ ∇ ⊥ X η ; � B ( X , Y ) , η � = � A η ( X ) , Y � ; 25 / 88
Basic facts in the submanifolds theory The Gauss equation ∇ N X Y = ∇ X Y + B ( X , Y ) ; The Weingarten equation ∇ N X η = − A η ( X )+ ∇ ⊥ X η ; � B ( X , Y ) , η � = � A η ( X ) , Y � ; H is the mean curvature vector field H = trace B ∈ C ( NM ) ; m 25 / 88
Basic facts in the submanifolds theory The Gauss equation ∇ N X Y = ∇ X Y + B ( X , Y ) ; The Weingarten equation ∇ N X η = − A η ( X )+ ∇ ⊥ X η ; � B ( X , Y ) , η � = � A η ( X ) , Y � ; H is the mean curvature vector field H = trace B ∈ C ( NM ) ; m If φ is a hypersurface, we denote f = trace A η , H = f m η , f is the ( m -times) mean curvature function. 25 / 88
Content The motivation of the research topic 1 General context Harmonic maps Biharmonic maps Properties of biconservative submanifolds 2 Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Complete biconservative surfaces Biconservative surfaces in N n 26 / 88
Biconservative submanifolds; Biharmonic submanifolds Definition A submanifold φ : M m → N n is called a biharmonic submanifold if φ is a biharmonic map, i.e., τ 2 ( φ ) = 0 . 27 / 88
Biconservative submanifolds; Biharmonic submanifolds Definition A submanifold φ : M m → N n is called a biharmonic submanifold if φ is a biharmonic map, i.e., τ 2 ( φ ) = 0 . Definition A submanifold φ : M m → N n is called a biconservative submanifold if div S 2 = 0 , i.e., τ 2 ( φ ) ⊤ = 0 . 27 / 88
M m submanifold of N n
M m submanifold of N n M m biconservative
M m submanifold of N n M m biconservative M m biharmonic
M m submanifold of N n M m biconservative M m biharmonic M m minimal 28 / 88
Characterization results Theorem ([5, 13, 21]) A submanifold φ : M m → N n is biharmonic if and only if � T = 0 � R N ( · , H ) · · H ( · )+ trace ∇ A H + trace trace A ∇ ⊥ and � ⊥ = 0 . ∆ ⊥ H + trace B ( · , A H ( · ))+ trace � R N ( · , H ) · 29 / 88
Characterization results Theorem ([5, 13, 21]) A submanifold φ : M m → N n is biharmonic if and only if � T = 0 � R N ( · , H ) · · H ( · )+ trace ∇ A H + trace trace A ∇ ⊥ and � ⊥ = 0 . ∆ ⊥ H + trace B ( · , A H ( · ))+ trace � R N ( · , H ) · Proposition ([18]) Let φ : M m → N n be a submanifold. The following conditions are equivalent: M is a biconservative submanifold; 1 � T = 0 ; � R N ( · , H ) · · H ( · )+ trace ∇ A H + trace trace A ∇ ⊥ 2 � T = 0 ; m � � H � 2 � � R N ( · , H ) · + 2trace A ∇ ⊥ · H ( · )+ 2trace 2 grad 3 2trace ∇ A H − m � H � 2 � � 2 grad = 0 . 4 29 / 88
Properties of biconservative submanifolds Proposition Let φ : M m → N n be a submanifold. If ∇ A H = 0 , then M is biconservative. 30 / 88
Properties of biconservative submanifolds Proposition Let φ : M m → N n be a submanifold. If ∇ A H = 0 , then M is biconservative. Proposition Let φ : M m → N n be a submanifold. If N is a space form, i.e., has constant sectional curvature, and M is PMC , i.e., has H parallel in NM , then M is biconservative. 30 / 88
Properties of biconservative submanifolds Proposition Let φ : M m → N n be a submanifold. If ∇ A H = 0 , then M is biconservative. Proposition Let φ : M m → N n be a submanifold. If N is a space form, i.e., has constant sectional curvature, and M is PMC , i.e., has H parallel in NM , then M is biconservative. Proposition ([2]) Let φ : M m → N n be a submanifold. Assume that M is pseudoumbilical, i.e., A H = � H � 2 I , and m � = 4 . Then M is CMC . 30 / 88
Characterization theorems Theorem([2, 22]) If φ : M m → N m + 1 is a hypersurface, then M is biharmonic if and only if � T = 0 , Ricci N ( η ) � 2 A ( grad f )+ f grad f − 2 f and ∆ f + f | A | 2 − f Ricci N ( η , η ) = 0 , where η is the unit normal vector field along M in N . 31 / 88
Characterization theorems Theorem([2, 22]) If φ : M m → N m + 1 is a hypersurface, then M is biharmonic if and only if � T = 0 , Ricci N ( η ) � 2 A ( grad f )+ f grad f − 2 f and ∆ f + f | A | 2 − f Ricci N ( η , η ) = 0 , where η is the unit normal vector field along M in N . A hypersurface φ : M m → N m + 1 ( c ) is biconservative if and only if A ( grad f ) = − f 2 grad f . 31 / 88
Characterization theorems Theorem([2, 22]) If φ : M m → N m + 1 is a hypersurface, then M is biharmonic if and only if � T = 0 , Ricci N ( η ) � 2 A ( grad f )+ f grad f − 2 f and ∆ f + f | A | 2 − f Ricci N ( η , η ) = 0 , where η is the unit normal vector field along M in N . A hypersurface φ : M m → N m + 1 ( c ) is biconservative if and only if A ( grad f ) = − f 2 grad f . Every CMC hypersurface in N m + 1 ( c ) is biconservative. 31 / 88
Content The motivation of the research topic 1 General context Harmonic maps Biharmonic maps Properties of biconservative submanifolds 2 Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Complete biconservative surfaces Biconservative surfaces in N n 32 / 88
Biconservative surfaces Let φ : M 2 → N 3 ( c ) be a non- CMC biconservative surface. Local proprieties Global properties
Biconservative surfaces Let φ : M 2 → N 3 ( c ) be a non- CMC biconservative surface. f > 0 and grad f � = 0 on M Local proprieties Global properties
Biconservative surfaces Let φ : M 2 → N 3 ( c ) be a non- CMC biconservative surface. f > 0 on M and grad f � = 0 f > 0 and grad f � = 0 on an open and dense on M subset of M Local proprieties Global properties
Biconservative surfaces Let φ : M 2 → N 3 ( c ) be a non- CMC biconservative surface. f > 0 on M and grad f � = 0 f > 0 and grad f � = 0 on an open and dense on M subset of M Local proprieties Global properties extrinsic intrinsic
Biconservative surfaces Let φ : M 2 → N 3 ( c ) be a non- CMC biconservative surface. f > 0 on M and grad f � = 0 f > 0 and grad f � = 0 on an open and dense on M subset of M Local proprieties Global properties extrinsic intrinsic extrinsic intrinsic 33 / 88
Local extrinsic properties Theorem ([3]) Let φ : M 2 → N 3 ( c ) be a biconservative surface with f > 0 and grad f � = 0 at any point in M . Then we have f ∆ f + | grad f | 2 + 4 3 cf 2 − f 4 = 0 , (1) where ∆ is the Laplace-Beltrami operator on M . 34 / 88
Local extrinsic properties Theorem ([3]) Let φ : M 2 → N 3 ( c ) be a biconservative surface with f > 0 and grad f � = 0 at any point in M . Then we have f ∆ f + | grad f | 2 + 4 3 cf 2 − f 4 = 0 , (1) where ∆ is the Laplace-Beltrami operator on M . In fact, we proved that on a neighborhood of any point in M , there exists a local chart ( U ; u , v ) such that f = f ( u , v ) = f ( u ) and (1) is equivalent with ff ′′ − 7 f ′ � 2 − 4 3 cf 2 + f 4 = 0 , � (2) 4 i.e., f has to satisfy a second order ODE. 34 / 88
Local intrinsic properties Using the Gauss equation, K = c + det A , we get f 2 = 4 3 ( c − K ) . (3) 35 / 88
Local intrinsic properties Using the Gauss equation, K = c + det A , we get f 2 = 4 3 ( c − K ) . (3) Theorem Let φ : M 2 → N 3 ( c ) be a biconservative surface with f > 0 and grad f � = 0 at any point of M . Then we obtain ( c − K ) ∆ K −| grad K | 2 − 8 3 K ( c − K ) 2 = 0 , (4) where ∆ is Laplace-Beltrami operator on M . 35 / 88
The intrinsic problem We want to determine the necessary and sufficient conditions such that an � M 2 , g � to admit, locally, a biconservative embedding in N 3 ( c ) abstract surface with f > 0 and grad f � = 0 . 36 / 88
Ricci problem � M 2 , g � Given an abstract surface , we want to determine the necessary and sufficient conditions such that it admits, locally, a minimal embedding in N 3 ( c ) . 37 / 88
Ricci problem � M 2 , g � Given an abstract surface , we want to determine the necessary and sufficient conditions such that it admits, locally, a minimal embedding in N 3 ( c ) . M 2 , g � � It was proved (see [16, 23]) that if is an abstract surface such that c − K > 0 on M , where c ∈ R is a constant, then, locally, it admits a minimal embedding in N 3 ( c ) if and only if ( c − K ) ∆ K −| grad K | 2 − 4 K ( c − K ) 2 = 0 . (5) 37 / 88
Ricci problem � M 2 , g � Given an abstract surface , we want to determine the necessary and sufficient conditions such that it admits, locally, a minimal embedding in N 3 ( c ) . M 2 , g � � It was proved (see [16, 23]) that if is an abstract surface such that c − K > 0 on M , where c ∈ R is a constant, then, locally, it admits a minimal embedding in N 3 ( c ) if and only if ( c − K ) ∆ K −| grad K | 2 − 4 K ( c − K ) 2 = 0 . (5) Condition (5) is called the Ricci condition with respect to c , or simple the Ricci condition. If (5) holds, then, locally, M admits a one-parametric family of minimal embeddings in N 3 ( c ) . 37 / 88
The link between the biconservativity and the Ricci condition with respect to c = 0 We can notice that relations ( 4 ) and (5) are very similar. In [7], we study the relationship between them. 38 / 88
The link between the biconservativity and the Ricci condition with respect to c = 0 We can notice that relations ( 4 ) and (5) are very similar. In [7], we study the relationship between them. � M 2 , g � � M 2 , g � bicons. in R 3 satisfies (4), K < 0
The link between the biconservativity and the Ricci condition with respect to c = 0 We can notice that relations ( 4 ) and (5) are very similar. In [7], we study the relationship between them. � M 2 , g � � M 2 , g � bicons. in R 3 satisfies (4), K < 0 M 2 , g 1 / 2 = √− Kg � � � M 2 , g � Ricci, K 1 / 2 < 0 satisfies (4), K < 0
The link between the biconservativity and the Ricci condition with respect to c = 0 We can notice that relations ( 4 ) and (5) are very similar. In [7], we study the relationship between them. � M 2 , g � � M 2 , g � bicons. in R 3 satisfies (4), K < 0 M 2 , g 1 / 2 = √− Kg � � � M 2 , g � Ricci, K 1 / 2 < 0 satisfies (4), K < 0 � M 2 , g − 1 = ( − K ) − 1 g � � M 2 , g � Ricci, K < 0 satisfies (4), K − 1 < 0 38 / 88
The link between the biconservativity and the Ricci condition with respect to c Theorem ([7]) � M 2 , g � be a biconservative surface in N 3 ( c ) . If f > 0 and grad f � = 0 on M , Let � M 2 , ( c − K ) r g � then on an open and dense subset of M , is a Ricci surface, where r is a function which locally satisfies � 1 � 4 log ( c − K r )+ r K + ∆ 2 log ( c − K ) = 0 , and K r , which denotes the Gaussian curvature on M corresponding to the metric ( c − K ) r g , is given by � 3 − 4 r � K + 1 K r = ( c − K ) − r 2 log ( c − K ) ∆ r +( c − K ) − 1 g ( grad r , grad K ) . 3 39 / 88
Content The motivation of the research topic 1 General context Harmonic maps Biharmonic maps Properties of biconservative submanifolds 2 Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Complete biconservative surfaces Biconservative surfaces in N n 40 / 88
Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Theorem ([7]) � M 2 , g � Let be an abstract surface and c ∈ R a constant. Then, locally, M can be isometrically embedded in N 3 ( c ) as a biconservative surface with f > 0 and grad f � = 0 at any point if and only if c − K > 0 , grad K � = 0 , at any point, and its level curves are circles in M with constant curvature κ = 3 | grad K | 8 ( c − K ) . 41 / 88
Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Theorem ([7]) � M 2 , g � Let be an abstract surface and c ∈ R a constant. Then, locally, M can be isometrically embedded in N 3 ( c ) as a biconservative surface with f > 0 and grad f � = 0 at any point if and only if c − K > 0 , grad K � = 0 , at any point, and its level curves are circles in M with constant curvature κ = 3 | grad K | 8 ( c − K ) . Corollary If the surface M is simply connected, then the theorem holds globally, but, in this case, instead of a local isometric embedding we have a global isometric immersion. 41 / 88
Local intrinsic characterization Theorem ([19]) M 2 , g � � Let be an abstract surface with c − K ( p ) > 0 and ( grad K )( p ) � = 0 at any point p ∈ M , where c ∈ R is a constant. Let X 1 = ( grad K ) / | grad K | and X 2 ∈ C ( TM ) be two vector fields on M such that { X 1 ( p ) , X 2 ( p ) } is a positively oriented basis at any point of p ∈ M . Then, the following conditions are equivalent: (a) the level curves of K are circles in M with constant curvature κ = 3 | grad K | 3 X 1 K 8 ( c − K ) = 8 ( c − K ) ; 42 / 88
Local intrinsic characterization Theorem ([19]) M 2 , g � � Let be an abstract surface with c − K ( p ) > 0 and ( grad K )( p ) � = 0 at any point p ∈ M , where c ∈ R is a constant. Let X 1 = ( grad K ) / | grad K | and X 2 ∈ C ( TM ) be two vector fields on M such that { X 1 ( p ) , X 2 ( p ) } is a positively oriented basis at any point of p ∈ M . Then, the following conditions are equivalent: (a) the level curves of K are circles in M with constant curvature κ = 3 | grad K | 3 X 1 K 8 ( c − K ) = 8 ( c − K ) ; (b) ∇ X 2 X 2 = − 3 X 1 K X 2 ( X 1 K ) = 0 and 8 ( c − K ) X 1 ; 42 / 88
Local intrinsic characterization Theorem ([19]) M 2 , g � � Let be an abstract surface with c − K ( p ) > 0 and ( grad K )( p ) � = 0 at any point p ∈ M , where c ∈ R is a constant. Let X 1 = ( grad K ) / | grad K | and X 2 ∈ C ( TM ) be two vector fields on M such that { X 1 ( p ) , X 2 ( p ) } is a positively oriented basis at any point of p ∈ M . Then, the following conditions are equivalent: (a) the level curves of K are circles in M with constant curvature κ = 3 | grad K | 3 X 1 K 8 ( c − K ) = 8 ( c − K ) ; (b) ∇ X 2 X 2 = − 3 X 1 K X 2 ( X 1 K ) = 0 and 8 ( c − K ) X 1 ; du 2 + dv 2 � (c) locally, the metric g can be written as g = ( c − K ) − 3 / 4 � , where ( u , v ) are local coordinates positively oriented, K = K ( u ) , and K ′ > 0 ; 42 / 88
Local intrinsic characterization du 2 + dv 2 � (d) locally, the metric g can be written as g = e 2 ϕ � , where ( u , v ) are local coordinates positively oriented, and ϕ = ϕ ( u ) satisfies the equation ϕ ′′ = e − 2 ϕ / 3 − ce 2 ϕ (6) and the condition ϕ ′ > 0 ; moreover, the solutions of the above equation, u = u ( ϕ ) , are � ϕ d τ √ u = + u 0 , − 3 e − 2 τ / 3 − ce 2 τ + a ϕ 0 where ϕ is in some open interval I and a , u 0 ∈ R are constants; 43 / 88
Local intrinsic characterization du 2 + dv 2 � (d) locally, the metric g can be written as g = e 2 ϕ � , where ( u , v ) are local coordinates positively oriented, and ϕ = ϕ ( u ) satisfies the equation ϕ ′′ = e − 2 ϕ / 3 − ce 2 ϕ (6) and the condition ϕ ′ > 0 ; moreover, the solutions of the above equation, u = u ( ϕ ) , are � ϕ d τ √ u = + u 0 , − 3 e − 2 τ / 3 − ce 2 τ + a ϕ 0 where ϕ is in some open interval I and a , u 0 ∈ R are constants; du 2 + dv 2 � (e) locally, the metric g can be written as g = e 2 ϕ � , where ( u , v ) are local coordinates positively oriented, and ϕ = ϕ ( u ) satisfies the equation 3 ϕ ′′′ + 2 ϕ ′ ϕ ′′ + 8 ce 2 ϕ ϕ ′ = 0 (7) and the conditions ϕ ′ > 0 and c + e − 2 ϕ ϕ ′′ > 0 ; moreover, the solutions of the above equation, u = u ( ϕ ) , are � ϕ d τ u = √ + u 0 , − 3 be − 2 τ / 3 − ce 2 τ + a ϕ 0 where ϕ is in some open interval I and a , b , u 0 ∈ R are constants, b > 0 . 43 / 88
Local intrinsic properties, c = 0 Proposition ([17]) The solutions of the equation 3 ϕ ′′′ + 2 ϕ ′ ϕ ′′ = 0 which satisfy the conditions ϕ ′ > 0 and ϕ ′′ > 0 , up to affine transformations of the parameter with α > 0 ( u = α ˜ u + β ), are given by ϕ ( u ) = 3log ( cosh u )+ constant , u > 0 . 44 / 88
Local intrinsic properties Remarks If c = 0 , we have a one-parameter family of solutions of equation (7), i.e., du 2 + dv 2 � g C 0 = C 0 ( cosh u ) 6 � , C 0 > 0 . 45 / 88
Local intrinsic properties Remarks If c = 0 , we have a one-parameter family of solutions of equation (7), i.e., du 2 + dv 2 � g C 0 = C 0 ( cosh u ) 6 � , C 0 > 0 . If c � = 0 , then we cannot determine explicitly ϕ = ϕ ( u ) , but we have u = u ( ϕ ) . Another way to see that we have only a one-parameter family of solutions of equation (7) is to rewrite the metric g in certain non-isothermal coordinates. 45 / 88
Local intrinsic properties Remarks If c = 0 , we have a one-parameter family of solutions of equation (7), i.e., du 2 + dv 2 � g C 0 = C 0 ( cosh u ) 6 � , C 0 > 0 . If c � = 0 , then we cannot determine explicitly ϕ = ϕ ( u ) , but we have u = u ( ϕ ) . Another way to see that we have only a one-parameter family of solutions of equation (7) is to rewrite the metric g in certain non-isothermal coordinates. Further, we consider only the c = 1 case. 45 / 88
Local intrinsic properties, c = 1 Proposition ([17]) be an abstract surface with g = e 2 ϕ ( u ) ( du 2 + dv 2 ) , where u = u ( ϕ ) � M 2 , g � Let satisfies � ϕ d τ u = + u 0 , � − 3 be − 2 τ / 3 − e 2 τ + a ϕ 0 where ϕ is in some open interval I , a , b ∈ R are positive constants, and u 0 ∈ R � M 2 , g � is a constant. Then is isometric to � � 3 � d ξ 2 + 1 ξ 2 d θ 2 D C 1 , g C 1 = , − ξ 8 / 3 + 3 C 1 ξ 2 − 3 ξ 2 � � � 3 3 / 2 � � where D C 1 = ( ξ 01 , ξ 02 ) × R , C 1 ∈ 4 / , ∞ is a positive constant, and ξ 01 and ξ 02 are the positive vanishing points of − ξ 8 / 3 + 3 C 1 ξ 2 − 3 , with 0 < ξ 01 < ξ 02 . 46 / 88
Local intrinsic properties, c = 1 Remark Let us consider � � 3 � d ξ 2 + 1 ξ 2 d θ 2 D C 1 , g C 1 = − ξ 8 / 3 + 3 C 1 ξ 2 − 3 ξ 2 � and 3 ξ 2 + 1 � d ˜ ξ 2 d ˜ θ 2 1 , g C ′ 1 = . D C ′ � ˜ ξ 8 / 3 + 3 C ′ ξ 2 − 3 ˜ − ˜ 1 ˜ ξ 2 � � � � are isometric if and only if C 1 = C ′ The surfaces D C 1 , g C 1 and 1 , g C ′ 1 and D C ′ 1 the isometry is Θ ( ξ , θ ) = ( ξ , ± θ + constant ) . Therefore, we have a one-parameter family of surfaces. 47 / 88
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