Information Intrinsic Geometric Flows Frédéric BARBARESCO Thales Air Defence, Surface Radar Business Line, 7/9 rue des Mathurins F-92223 Bagneux, France E-mail : [email protected] , Phone : 33.(0)1.40.84.20.04 Abstract. Geometric Flow Theory is cross fertilized by diverse elements coming from Pure Mathematic and Mathematical Physic, but its foundation is mainly based on Riemannian Geometry, as explained by M. Berger in a recent panoramic view of this discipline [4], its extension to complex manifolds, the Erich Kähler’s Geometry, vaunted for its unabated vitality by J.P. Bourguignon [6], and Minimal Surface Theory [8,9]. This paper would like to initiate seminal studies for applying intrinsic geometric flows in the framework of information geometry theory. More specifically, after having introduced Information metric deduced for Complex Auto-Regressive (CAR) models from Fisher Matrix (Siegel Metric and Hyper-Abelian Metric from Entropic Kähler Potential), we study asymptotic behavior of reflection coefficients of CAR models driven by intrinsic Information geometric Kähler-Ricci and Calabi flows. These Information geometric flows can be used in different contexts to define distance between CAR models interpreted as geodesics of Entropy Manifold. We conclude with potential application of Intrinsic Geometric Flow on Gauss Map to transform Manifold of any dimension by mean of Generalized Weierstrass Formula introduced by Kenmotsu [8] that can represent arbitrary surfaces with non-vanishing mean curvature in terms of the mean curvature function and the Gauss map. One of the advantages of the generalized formulae is that they allow to construct a new class of deformations of surfaces by use of Intrinsic Geometric Flow on Gauss Map. We conclude with the Heat equation interpretation in the framework of Information Geometry. Keywords: Chentsov Information Geometry, Siegel Metric, Hyper-Abelian Metric, Entropic Kähler Potential, Intrinsic Geometric Flow, Kähler-Ricci Flow, Calabi Flow, Gauss map, PACS: Geometric Flow Theory.

SIEGEL METRIC FOR COMPLEX AUTOREGRESSIVE MODEL In Chentsov Information geometry theory, we consider families of parametric density T functions G Θ = { p(. / θ ) : θ ∈ Θ} with Θ = [θ 1 L θ n ] , from which we can define a Riemannian Manifold Structure by meam of Fisher Information matrix : n ⎡ ∂ ln p(. / θ ) ∂ ln p(. / θ ) ⎤ 2 g ij (θ ) = Eθ ⎢ with metric ds = g ij (θ ).dθ i .dθ *j . ⎥ ∑ * ∂θ i ∂θ j i , j =1 ⎢⎣ ⎥⎦ We demonstrate easily that this Fisher metric is equivalent to the Siegel metric, introduced by Siegel in the 60’s in the framework of Symplectic Geometry, in the case of Complex Multivariate Gaussian Law (Complex Circular Process): ˆ −1 −n + p( X/R ,m ) = (2π ) R e −Tr [R .R ] with Rˆ = ( X − m )( X − m ) such that E Rˆ = R n

n

n

n

−1 n

n

n

[ ]

n

n

n

and only consider random process with zero mean m n = E [ X ] = 0 , the Fisher matrix

[(

)(

is reduced to g ij (θ ) = Tr Rn .∂ i Rn−1 . Rn .∂ j Rn−1

)] and metric

[(

ds 2 = Tr Rn .dRn−1

)] 2

In case of Complex Autoregressive models (CAR model), we can exploit the following specific blocks structure of covariance matrices : R

(1 ) n

⎤ ⎡ 1 − An(1−)1+ . Rn(1−)11 / 2 ⎥ − An(1−)1+ . Rn(1−)1 ⎤ ⎢ ( 1 )1 / 2 ( 1 )1 / 2 + (1)1/2 (1) . Rn with R n = ⎢ α n−1 ⎥ = Rn ⎥ Rn(1−)1 ⎦ ⎥⎦ ⎢⎣ 0 Rn(1−)11 / 2 ⎤ ⎡ An( −−1) ⎤ α n( 2−)1 . An( 2−)1+ ⎡ An−1 ⎤ 2 −1 −1 where 1 . and . α μ α μ A = − = + ⎥ ⎢ ⎥ n n n − 1 n n ⎢ 0 ⎥ + α n( 2−)1 . An( 2−)1 An( 2−)1+ ⎦ ⎣ ⎦ ⎣ 1 ⎦

⎡α (1)−1 + An(1−)1+ . Rn(1−)1 . An(1−)1 = ⎢ n −1 − Rn(1−)1 . An(1−)1 ⎣

⎡ α ( 2) R (n2 )−1 = ⎢ ( 2 ) n −1( 2 )+ ⎣α n−1 . An−1

Rn( 2−)1

[

]

We can deduce the expression of the Siegel metric previously defined :

[(

ds = Tr Rn .dR 2 n

)]

−1 2 n

[(

2

⎛ dα ⎞ = ⎜⎜ n −1 ⎟⎟ + α n −1 .dAn+−1 . Rn −1 .dAn −1 + tr Rn −1 .dRn−−11 ⎝ α n −1 ⎠

)] 2

2

⎛ dα ⎞ that leads to : ds = ds + ⎜⎜ n −1 ⎟⎟ + α n −1 .dAn+−1 . Rn −1 .dAn −1 ⎝ α n −1 ⎠ We can then define a new hyperbolic distance between CAR models as Inferior Bound 2 n

2 n −1

⎛ dα of this metric : ds > ∑ ⎜⎜ k k =0 ⎝ α k n −1

2 n

n − 1 dμ ⎞ k ⎟⎟ + ∑ k =1 1 − μ k ⎠ 2

2 2

INFORMATION METRIC FROM ERICH KÄHLER GEOMETRY Natural extension of Riemannian Geometry to Complex Manifold has been introduced by a seminal paper of Erich Kähler during 30th ‘s of last century. We can easily apply this geometric framework for information metric definition. Let a complex Manifold M n of dimension n, we can associate a Kählerian metric, which n

can be locally defined by its definite positive Riemannian form : ds 2 = 2 ∑ g ij .dz i dz j

[ ]

with g ij

i, j

i , j =1

an Hermitian definite positive matrix. Kähler assumption assumes that

we can define a Kähler potential Φ , such that g ij = given

∂ 2Φ

∂z i ∂z j tensor can

. Fundamental relation,

by Erich Kähler, is that Ricci be expressed by : 2 n ∂ log(det g kl ) with the associated scalar curvature R = ∑ g kl Rkl . One Rij = − ∂z i ∂z j k , l =1 important geometric flow, in physic & mathematic, is the Kähler-Ricci flow which ∂g ij 1 = − Rij + Rg ij . This flow converges to a drive the evolution of the metric by : ∂t n ∂ 2 log (det g kl ) ∂ 2Φ Kähler-Einstein metric Rij = k 0 g kl , or equivalently to : − = k0 i j , ∂z i ∂z j ∂z ∂ z known as Monge-Ampère equation : det( g kl ) = ψ e − k 0 Φ where Φ is a 2

Kähler

potential and ψ a non specified holomorphic function, but that could be reduced to

unity : if k 0 ≠ 0 ) by choice of a new Φ potential, or if k0 = 0 by local holomorphic coordinates selection so that volume det( g kl ) is reduced to 1 (cancellation of Ricci tensor is existence condition of this coordinates system). In case of Complex Auto-Regressive (CAR) models, if we choose as Kähler ∂ 2Φ potential Φ with g ij = i j , the Entropy of the process expressed according to ∂z ∂z reflexion coeffient in the unit Polydisk {z / z k < 1 ∀k = 1,...n}, the Kähler potential is

given

:Φ =

by n −1

(

K D ( z, z) = ∏ 1 − z k k =1

)

n −1

∑ρ k =1

2 ρk

k

[

ln 1 − z k

2

] = ln K

D

( z, z )

and

Bergman

kernel

. Very surprisingly, this case was the first example of

potential studied by Erich Kähler in his seminal paper, named by Erich Kähler HyperAbelian case. This choice of Kähler potential as Entropy of CAR model, can be justified by remarking that Entropy Hessian along one direction in the tangent plane of parametric manifold is a definite positive form that can be considered as a Kählerian n ∂ 2 H ( Pθ ) differential metric: g ij( H ) (θ ) = − ⇒ ds H2 = ∑ g ijH (θ ).dθ i .dθ j . ∂θ i .∂θ j i , j =1 In case of Complex Autoregressive models, with as previously Whishart density, Entropy is given by : T H n = − ∫ P(X n /m n ,Rn ) ln[P(X n /m n ,Rn )]dX n = ln Rn + n ln(π .e ) with X n = [ x1 L x n ] If we use the blocks structure of covariance matrix in case of CAR models, we obtain the Entropy expressed according to reflection coefficients : n −1 1 n 2 2 H n = ∑ (k − n). ln 1 − μ k + n. ln π .e.α 0−1 with α 0−1 = P0 = ∑ x k n k =1 k =1 The Kähler metric is given by Hessien of Entropy, where Entropy is considered as ∂ 2 Φ( p ) with Φ( p) = ∫ p( x / θ ). ln p( x / θ ).dx Kähler potential g ij (θ ) = − *

[

]

[

]

∂θ i ∂θ j

Let θ ( n ) = [P0

μ1

μ n −1 ]T , we have then g11 = nP0−2 , g ij =

( n − i ).δ ij

(1 − μ )

2 2

and

i

n −1 dμ i ⎛ dP ⎞ metric of this Hyper-Abelian Case : ds n2 = n.⎜⎜ 0 ⎟⎟ + ∑ ( n − i ) i =1 ⎝ P0 ⎠ 1 − μi 2

(

2

)

2 2

INFORMATION RICCI & CALABI FLOWS FOR COMPLEX AR Historical Root of Ricci flow can be found in Hilbert work on General Relativity. The “Hilbert Action” S is defined as the integral of scalar Curvature R on the Manifold Mn with volume V ( g ) = ∫ dη and : S ( g ) = ∫ R. det( g ) .d n x = ∫ R.dη Mn

Mn

Mn

R=

g μν . R μν . Fundamental theorem of Hilbert said that for ∑μ ∑ ν

n ≥ 3 , S ( g ) is

minimal with V ( g ) = cste if R( g ) is constant and g is an Einstein metric : 1 R ij = R . g ij . So the more natural geometric flow that converges to Einstein metric is n ∂g ij 1 ⎡ ⎤ = 2⎢− Rij + Rg ij ⎥ , but unfortunately this flow exhibits some given by : ∂t n ⎣ ⎦ convergence problems in finite time. That the main raison why R. Hamilton has introduced the following normalized Ricci flow : ⎛ ⎞ ⎛ ⎞ ∂g ij 1 ⎡ ⎤ = 2⎢− Rij + r . g ij ⎥ with r = ⎜⎜ ∫ Rdη ⎟⎟ / ⎜⎜ ∫ dη ⎟⎟ . This flow can be interpreted n ∂t ⎣ ⎦ ⎝ Mn ⎠ ⎝ Mn ⎠ ∂g ij = Δ g g ij , by using isothermal as Fourrier Heat Operator acting on metric g , ∂t coordinates introduced by G. Darmois. More specifically A. Lichnerowicz has proved ∂ 2 g ij kl that in isothermal coordinates, we have : − 2. R ij = ∑ ∑ g . k l + Q ij g −1 , ∂g . ∂x ∂x k l Recently, Ionnis Bakas has established connection between 2 distinct classes of geometric deformations, Ricci and Calabi Flows, respectively of 2nd and 4th order. Calabi flow preserves the Kähler class and minimize the quadratic curvature for extremal metrics. All geometric flows share some common qualitative features with the linear heat flow equation. The Ricci and Calabi flows correspond to intrinsic deformations. Let M denote a complex n-dimensional manifold, that admits a Kähler metric g with : ds 2 = 2∑ g ij dz i dz *j . The Calabi flow, defined for Kähler Manifolds,

(

i, j

is given by :

∂ g ij ∂t

=

(

)

)

∂ 2φ z , z * ∂ R = g . If we use that , then it is equivalent to ij ∂z i ∂z *j ∂z i ∂z *j 2

∂φ = R − R . It preserves the volume and the Kähler class of the metric. Critical ∂t points of the flow are called extremal metrics. The following functional S ( g ) = ∫ R 2 ( g )dV ( g ) decreases monotonically along the Calabi Flow and the M

minimum of the functional S ( g ) is given by Schwarz’s inequality : 2

⎞ ⎛ S ( g ) = ⎜⎜ ∫ R( g )dV ( g ) ⎟⎟ / ∫ dV ( g ) . Bakas [1] has recently proved that there is a ⎠ M ⎝M relation between the Ricci and Calabi Flows on Kähler manifolds of arbitrary dimension that manifests by squaring the time evolution operator. The proof is given by taking time derivative of the Ricci Flow :

⎧ ∂ 2 g ij ∂ R ij = − ⎪ ∂g ij ∂t ⎪ ∂t 2 = − Rij ⇒ ⎨ ⇒ 2 ∂t ⎪ R = − ∂ log (det( g ) ) ⎪ ij ∂z i ∂z *j ⎩

⎧ ∂ 2 g ij ∂ 2 ⎛ ∂ log (det( g ) ) ⎞ ⎪ 2 = ⎜ ⎟ ∂ 2 g ij ∂t ∂z i ∂z *j ⎝ ∂2R ⎪ ∂t ⎠ ⇒ = − ⎨ ∂t 2 ∂z i ∂z *j ⎪ ∂ log (det( g )) ij = − = − g R R ∑ ij ⎪ ∂t i, j ⎩

If the second derivative of the metric with respect to the Ricci time is identified with ∂2 ∂t minus its first derivative to the Calabi time, the two flows are the same : 2 = − ∂t C ∂t R Considering the two dimensions case, and a system of conformally flat (Kähler) * coordinates : ds 2 = 2e Φ ( z , z ,t ) dz .dz * . The only non-vanishing components of the Ricci ∂ 2 log (det( g )) ∂ 2Φ because R = − and g zz * = e Φ curvature tensor is : R zz * = − * * zz * ∂z ∂ z ∂ z∂ z For the Ricci flow : 2 ∂ g ij ∂e Φ ∂ 2Φ ∂ 2Φ ∂Φ Φ ∂Φ −Φ ∂ . = − R ij ⇒ = ⇒ = ⇒ = ΔΦ with Δ . = e e ∂t ∂t ∂t ∂t ∂z∂ z * ∂z∂ z * ∂ z∂ z * For the Calabi flow : 2 ∂ g ij ∂2R ∂e Φ ∂ 2 ΔΦ ∂Φ −Φ ∂ Φ ij = and R = ∑ g Rij = − e = − ΔΦ ⇒ =− ⇒ − ΔΔΦ * * * ∂t ∂t ∂t ∂z ∂ z ∂ z∂ z ∂z ∂ z i, j If we use definition of metric g as previously for CAR models, as derived from Entropic Kähler potential in case of a Complex Autoregressive models, then we can

(

express Ricci tensor : Rkl = −2δ kl 1 − μ k

)

2 −2

for k = 2,..., n − 1 and R11 = −2 P0− 2

Identically, we can deduce scalar curvature R = ∑ g kl . Rkl of CAR models : k ,l

n −1 ⎡ ⎤ ⎡ n −1 ⎤ R = −2.⎢ n −1 + ∑ ( n − j ) −1 ⎥ = −2.⎢ ∑ ( n − j ) −1 ⎥ (this curvature diverges when n j =1 ⎣ ⎦ ⎣ j=0 ⎦ tends to infinity).We can observe that we have an Einstein metric but more generally −1 defined as : Rij = B ( n ) g ij with R = Tr B ( n ) where B ( n ) = −2diag .., (n − i ) ,..

[ ]

[ ]

[

{

]

}

If we study Kähler-Ricci flow acting on reflection coefficients, we have :

(

∂ ln 1 − μ i

2

)= −

∂ ln P0 1 1 and = . From which, we obtain the behaviour of ∂t ∂t (n − i ) n refelction coefficient in asymptotic case :

(

)

μ i ( t ) = 1 − 1 − μ i ( 0) .e

−

t ( n− i )

⎯t⎯ ⎯→ μ i ( t ) = 1 that converges to unit circle. →∞ If We introduce Calabi flow for CAR model, solution is defined as steady state of the ∂ψ = Rψ − R with ψ Kähler potential associated to the Kähler PDE equation : ∂t ∂ 2ψ metric g ij = i j , Rψ scalar curvature and R its mean value on the Manifold. ∂z ∂ z This can be used to defined shortest path between two parametric models. If we 2

2

2

consider the path ψ ( t ) (0 ≤ t < 1) and if we assume the existence of Kähler potential {φ ( s, t ) : 0 ≤ t < 1} driven by Calabi Flow : ∂φ = R(φ ) − R and φ (0, t ) = ψ ( t ) . ∂t 1 2

⎛ ⎛ ∂φ ( s , t ) ⎞ ⎞ Length of path L(s ) is then given by : L( s ) = ∫ ⎜ ∫ ⎜ dηφ ( s ,t ) ⎟ dt . ⎟ ⎜ ⎟ ∂t ⎠ 0 ⎝V ⎝ ⎠ If we apply same approach as previously for CAR models, Calabi flow will act on Entropy − H ( p) defined as Kähler potential : 2

1

[

∂ ln 1 − μ k

2

] − n. ∂ ln[π .e.P ] = −2.⎡∑

1 1⎤ + ⎥ ⎢ ∂t ∂t k =1 ⎣ k =1 n − k n ⎦ We then deduce the asymptotic behaviour of PARCOR coefficients submitted to n −1

− ∑ ( k − n).

(

∂ ln 1 − μ k

n

0

2

)= −

∂ ln P0 2 2 with = 2 . We can observe, as 2 ∂t ∂t n (n − k ) previsouly, that Calabi flow will drive PARCOR coefficients evolutions to unit circle. Calabi Flow :

DEFORMATION BY GAUSS MAP & WEIERSTRASS FORMULA Kenmotsu & Konopelchenko have generalized Weierstrass representation for surfaces in multidimensional Riemann spaces. Theory of immersion and deformations of surfaces has been important part of the classical differential geometry. Various methods to describe immersions and different types of deformations have been considered. The classical Weierstrass formulae for minimal surfaces immersed in the three-dimensional Euclidean space R3 is the best known example of such an approach. Only recently, the Weierstrass formulae have been generalized to the case of generic surfaces in R3. using the two last years the generalized Weierstrass formulae have been used intensively to study both global properties of surfaces in R3 and their integrable deformations (e.g. : modified Veselov-Novikov equation). An analog of the Weierstrass formulae for surfaces of prescribed (non zero) mean curvature have been proposed by Kenmotsu in 1979. The Kenmotsu representation is given by : ⎡z ⎤ r T * i X ( z , z ) = Re ⎢ ∫ ψ .φ i dz'⎥ with φ = 1 − f 2 i 1 + f 2 2 f and the functions ⎢⎣ ⎥⎦ ∂f 2f* * ∂ logψ ∂z =− f and ψ obey the following compatibility condition : and the 2 * ∂z 1+ f r r ∂f * ⎛ 2 2⎞ scalar mean curvature (if we note H = hN ) : h = − / ⎜ψ 1 + f ⎟ with the ⎠ ∂z ⎝ * 2 * * ∂ log h ⎛ ∂f ⎞ ∂ f ∂f 2 2 ⎜ ⎟ = constraint that : . It was proved by − + f 1 / 2 / 1 f ⎜ ∂z ⎟⎠ ∂z∂z * ∂z * ∂z * ⎝ Kenmotsu that any surface in R3 can be represented in such a form :

[

(

)

(

]

( )

)

(

) (

)

r ⎡ ⎡ ∂f * ⎛ T⎤ 2 2 ⎞⎤ 2 2 X ( z , z * ) = Re ⎢ ⎢ − / ⎜ h. 1 + f ⎟ ⎥ ∫ 1 − f dz ∫ i 1 + f dz ∫ 2 f dz ⎥ ⎠⎦ ⎣ ⎣ ∂z ⎝ ⎦ with f the stereographic projection of the Gauss map r T 1 2 N= 2 Re( f ) 2 Im( f ) f − 1 and with the mean curvature h . 2 1+ f The most surprising result, as explained by Hoffman [9], is that a general surface in Rn is essentially determined by its Gauss map. Such a result was unexpected because this result is false for minimal surface. Minimal surfaces in Rn come with a one-parameter family of associated surfaces, all having the same Gauss map, but generally distinct. Even more is true in R3, where essentially any two minimal surfaces have the same Gauss map locally, after a rotation. This representation of surfaces deals basically with the Gauss map for generic surface in R3, as developed by Hoffman[9]. One of the advantages of the generalized Weierstrass formulae is that they allow to construct a new class of deformations of surfaces by use of Geometric Flow on Gauss Map.

(

(

)

)

MEANING OF FOURIER HEAT EQUATION BY INFORMATION GEOMETRY As we have seen previously, heat equation is the main equation of evolution, ∂g ij Kähler flow being interpreted as Heat flow action on differential metric = Δ g g ij . ∂t If we focus on geometric interpretation of heat equation, this analogy could be used to make an amother link with Information geometry by mean of Cramer-Rao inequality. If we consider the classical definition of Laplace operator in Euclidean space :

[θ ( x + ∇x ) − θ ( x )] − [θ ( x ) − θ ( x − ∇x )] , we can observe that is exactly ∂ 2θ = Lim 2 ∇x a 0 ∂x (∇x )2 the difference between the value and its spatial mean:

θˆ ( x ) =

θ ( x + ∇x ) + θ ( x − ∇x ) 2

[

]

∂ 2θ 2 ≈ θˆ ( x ) − θ ( x ) with 2 2 ∂x (∇x )

. This remark could be extended in any dimension :

[

]

∂ 2θ ∂ 2θ 2n ˆ + ... + ≈ θ ( x) − θ (x ) ∂x 12 ∂x n2 (∇x )2 θ ( x + ∇x1 ) + θ ( x − ∇x 1 ) + ... + θ ( x + ∇x n ) + θ ( x − ∇x n ) with θˆ ( x ) = 2n But also for non euclidean space, by extending classical notion of spatial mean in isothermal coordinate : ∂θ ⎞⎟ 1 ∂ ⎛⎜ is equal in isothermal coordinates : Δ gθ = ∑ g g ij ∂x j ⎟⎠ i, j g ∂x i ⎜⎝ Δθ = div (∇θ ) =

2 Δ gθ = ∑ g λμ ∂ λμ θ=

2n

g λμ ∑ λ μ ,

(∇x )

λ ,μ

where α λμ =

2

g λμ ∑ g λμ

2

[θˆ( x ) − θ ( x )]

(

⎛ θ ( x + ∇x λ ) + θ x − ∇x μ with θˆ ( x ) = ∑ α λμ ⎜⎜ 2 λ ,μ ⎝

⎟ ⎠

and ∇x = ∇x 1 = ∇x 2 = ... = ∇x n

λ ,μ

Generally, we can then write heat equation

(

) ⎞⎟

)(

)

[

]

∂θ = Δθ = ρ θˆ ( x ) − θ ( x ) ∂t

and

+ d θ dθ + −1 = ρ 2 θ − θˆ θ − θˆ . If we use Cramer-Rao Inequalitty Rθ ≥ I (θ ) , then dt dt −1 E dθ .dθ + = ρ 2 Rθ dt 2 ≥ ρ 2 I (θ ) dt 2 . That could be developed by taking trace of the

[

]

[ (

)]

E Tr I (θ )dθ .dθ + ≥ ρ 2 dt 2Tr ( I n ) . By rewriting last expression

expression :

[

]

as E dθ + I (θ )dθ ≥ nρ 2 dt 2 , we can observe that appears the information metric :

(

[ ]

E ds I2 ≥ nρ 2 dt 2

)(

)

(

)

+ But as we have classically E ⎡d θ − θˆ d θ − θˆ ⎤ = E dθ .dθ + − dθˆ .dθˆ + , we can ⎢⎣ ⎥⎦ extend previous relation to : + E ⎡d θ − θˆ . I (θ ).d θ − θˆ ⎤ + dθˆ + . I (θ ).dθˆ ≥ nρ 2 dt 2 ˆ→ dθ + . I (θ ).dθ = ds I2 = nρ 2 dt 2 ⎢⎣ ⎥⎦ θ aθ

(

)

(

)

+

dθ dθ . I (θ ). = nρ 2 dt dt dθ For a geodesic curve θ = θ (t ) , its tangent vector θ& = (t ) is of constant length with dt dθ i dθ j =nρ 2 . The constant may be chosen to respect to the metric ds I , thus : ∑ g ij dt dt i, j be of value 1 when the curve parameter t is chosen to be the arc-length parameter “s”.

REFERENCES 1. Ionnis Bakas, “The Algebraic Structure of Geometric Flows in Two Dimensions”, Institute of Physics, SISSA, October 2005 2. F. Barbaresco, « Etude et extension des flots de Ricci, Kähler-Ricci et Calabi dans le cadre du traitement de l’image et de la géométrie de l’information », Conf. Gretsi, Louvain la Neuve, Sept. 2005 3. F. Barbaresco, « Calculus of Variations & Regularized Spectral Estimation », Coll. MAXENT’2000, Gif-surYvettes, France, Jul. 2000, published by American Institut of Physics. 4. M. Berger, « Panoramic View of Riemannian Geometry », Springer, 2004 5. G. Besson, “Une nouvelle approche de l’étude de la topologie des varieties de dimension 3 d’après R. Hamilton et G. Perelman », Séminaire Bourbaki, 57ème année, 2004-2005, n°947, Juin 2005 6. J.P. Bourguignon, « The Unabated Vitality of Kählerian Geometry », edited in « Kähler Erich, Mathematical Works », Edited by R. Berndt and O. Riemenschneider, Berlin, Walter de Gruyter, ix, 2003 7. P. Gauduchon, « Calabi’s extremal Kähler metrics : an elementary introduction», Ecole Polytechnique 8. K. Kenmotsu, “Weierstrass formula for surfaces of prescribed mean curvature”, Math. Ann.,n°245, 1979 9. D.A. Hoffman and R. Osserman, „The Gauss map of surfaces in R3 and R4“, Proc. London Math. Soc., vol.3 , n°50, pp.27-56, 1985 10. International Conference “Geometric Flows : Theory & Computation”, IPAM, UCLA, USA, February 3-4 , 2004