Samy Skander Bahoura

HARNACK INEQUALITIES FOR SOLUTIONS OF PRESCRIBED SCALAR CURVATURE TYPE EQUATIONS

We give Harnack inequalities for solutions of equations of type prescribed scalar curvature in dimensions n ≥ 4.

INTRODUCTION AND MAIN RESULTS

We consider on a Riemannan manifold (M, g) of dimension n ≥ 4, the equation:

∆u + hu = V u (n+2)/(n-2) , u > 0, (E)
with, h a smooth function and

0 < a ≤ V (x) ≤ b < +∞, ||∇V || ∞ ≤ A.
Equation of this type were considered by many authors, see . This equation arise in physics and astronomy. Here we look to a priori estimates of type sup, inf which are characteristic of this equation.

Let (u k ) a sequence of regular solutions of (E).

We fix a compact set K of M . We want to prove that: for each compact, for all terms of the sequence (u k ): sup K u k and inf M u k are linked. Here we prove a weaker inequality for blow-up solutions of the previous equation. Equations of previous type are called, Yamabe equation, prescribed scalar curvature equation, of type prescribed scalar curvature and Schrodinger equation. Here we prove that for blow-up solutions, precisely for each sequence (u k ), there is a positive function c > 0, such that for all compact set, there is a sequence of positive numbers, 0 < ǫ k (K) ≤ 1, which link sup K u k and inf M u k .

Here we have two possibilites up to a subsequence we have a compactness result or an inequality bertween sup K u k and inf M u k . Note that for Li-Zhang result in dimension 3 and 4, they consider the problem aronund a point, thus the compactness result. Also, for the Harnack inequality, also, see the introduction of [START_REF] Bagchi | Exotic origins of tensionless superstrings[END_REF].

We obtain: Theorem 1.1. We have:

1) There is a compact K 0 of M and a subsequence i j and a positive constant C > 0, such that:

sup K0 u ij ≤ C, ∀j.

Or,

2) For all compact K, sup K u k → +∞ and:

n = 4, (sup K u k ) 1-ǫ ≤ c(a, b, A, inf M u k (sup K u k ) ǫ , K, M, g),
with ǫ > 0, and, n ≥ 5, (sup

K u k ) 1-ǫ ≤ ǫ 1-ǫ k c(a, b, A, ǫ ǫ k inf M u k (sup K u k ) ǫ , K, M, g), with, ǫ > (n-4) n-2 1
We can see that sup K u k and inf M u k are linked. There is a relation which link these two quantities. For all compact K and all k ∈ N, sup K u k and inf K u k are linked by the previous relations. There is a positive function

c(•, •, •) > 0, such that for all k, sup K u k = c(K, ǫ k , inf M u k ). If we denote F = {u k }, G = {ǫ k = ǫ k (K)}, F × G = {(u k , ǫ k )}, 0 < ǫ k = ǫ k (K) ≤ 1, we have: ∃ c(•, •, •) > 0, ∀K ⊂ M, ∀(u, ǫ ′ ) ∈ F × G, sup K u = c(K, ǫ ′ (K), inf M u) or, ∃ c(•, •, •) > 0, ∀K ⊂ M, ∀(u, ǫ ′ ) ∈ F × G, (sup K u) 1-ǫ ≤ c(K, ǫ ′ (K), inf M u (sup K u) ǫ ).
We write this to highlight the rolling-up phenomenon and the distortion. Remarks: a) In the previous theorem, the point 1) assert that, up to a subsequence we have a compactness result.

The point 2) assert that we have a relation between the local supremum and the global infimum. Also, see the introduction of [START_REF] Bagchi | Exotic origins of tensionless superstrings[END_REF].

b) For the point 1) we have one parameter, the local supremum is controled by its self. For the point 2) we have two parameters, the local supremum and the global infimum, there is a relation which link those two quantities. In the paper "Estimations du type sup × inf sur une variété compacte", we have 3 paramaters, the local supremum, the local infimum and the global supremum. Here, we have, 1 parameter or 2 paramaters each time. At most 2 parameters and at least 1 parameter. Also, see the introduction of [START_REF] Bagchi | Exotic origins of tensionless superstrings[END_REF]. c) In the transformation u → v = λu(λ 2/(n-2) x), for the rescaling or blow-up, we have approximatively, the constant c(m/λ) ≡ c(m)/λ. Because, v ≥ m ⇒ v(0) ≤ c(m), but this it is equivalent to, u ≥ m/λ ⇒ u(0) ≤ c(m)/λ, which imply that c(m/λ) ≡ c(m)/λ. Thus, the rescaling by λ imply a relation of the type (approximatively): c(m/λ) ≡ c(m)/λ. d) In general as in the paper of Li-Zhang, for the dimensions 3 and 4, of the Yamabe equation, we look to the estimate around a point. The compactness result is important, but also, we look to the solutions which blow-up as mentionned by the example x → [ǫ/(ǫ 2 + |x| 2 )] (n-2)/2 , ǫ → 0, thus the point 2). Also, see the introduction of [START_REF] Bagchi | Exotic origins of tensionless superstrings[END_REF].

PROOF OF THE RESULT

For the proof, we use the computations of previous papers with modifications, see [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Bahoura | Estimations uniformes pour l'équation de Yamabe en dimensions 5 et 6[END_REF][START_REF] Bahoura | A uniform estimate for scalar curvature equation on manifolds of dimension 4[END_REF][START_REF] Bahoura | An estimate on Riemannian manifolds of dimension 4[END_REF]. I) blow-up analysis: Let (u k ) a sequence of solutions of (E). We fix a compact set K of M . We want to prove that: for each compact, for all terms of the sequence (u k ): sup K u k and inf M u k are linked.

1) If there is a "big" compact K 0 for which there is a subsequence sup K0 u ij is bounded, then we have a compactness result for a "big" K 0 and for K ⊂ K 0 the sequence (u ij ) is bounded.

2) If for all compact K, sup K u k → +∞. We do a blow-up. We consider

sup K u k = u k (y k ). Consider R k → 0, R (n-2)/2 k = [u k (y k )] -ǫ with 0 < ǫ < 1. Then: R (n-2)/2 k sup BR k (y k ) u k ≥ c k = [u k (y k )] 1-ǫ → +∞.
We use the blow-up technique to have,

∃t k , tk , u k (t k ) ≥ u k ( tk ) ≥ u k (y k ) → +∞. tk , sup BR k (y k ) u k = u k ( tk ) ≥ u k (y k ) > 0, We consider s k (y) = (R k -d(y, tk )) (n-2)/2 u k (y)
, and,

t k , sup BR k ( tk ) s k = s k (t k ) ≥ s k ( tk ) = R (n-2)/2 k u k ( tk ) ≥ R (n-2)/2 k u k (y k ) > 0,
We do a blow-up, then we consider:

n = 4, v k (y) = r k u k (t k + r k y) = r k u(exp t k (r k y)), r k = [u k (y k )] -ǫ ,
with, ǫ > 0, and,

n ≥ 5, v k (y) = r k u k (t k + (r k ) 2/(n-2) y) = r k u k (exp t k (r 2/(n-2) k y)), r k = [u k (t k )] -ǫ ,
with, ǫ > (n-4) n-2 . Note that, here we have considered all terms of the sequence (u k ). Let's consider the blow-up functions (v k ) defined previousely with the exponential maps for n ≥ 4, exp t k (y), like in the previous papers for the dimensions, 4, 5, 6. Because we consider the compact sets K, 2K, and t k ∈ 2K, the injectivity radius is uniformly bounded below by a positive number. Thus, we can consider all the terms of the sequence (u k ) without extraction.(After supposing the assertion inf v k ≥ m > 0 infinitly many times, we can use extraction, for the points (t k )).

We consider, δ 0 = δ 0 (K) = inf{δ P /4, P ∈ K}, with δ P continuous in P and smaller than the injectivity radius in P for each P . We have a finite cover of K by small balls of radius δ 0 /2, we have a finite set of points

z j ∈ K: K ⊂ ∪ {j=0,...,l} B(z j , δ 0 /2) ⊂ ∪ {j=0,...,l} B(z j , 3δ 0 ) = K δ0 is compact. We take R (n-2)/2 k = inf{u k (y k ) -ǫ , (δ 0 /2) (n-2)/2 }.
The small balls are all compact, thus, tk exist and t k exist. We take for n = 4, r k = inf{δ P /4, P ∈ K δ0 , u k (y k ) -ǫ } and, for n ≥ 5, r k = inf{δ P /4, P ∈ K δ0 , u k (t k ) -ǫ }. Thus tk , t k and exp t k (•) and v k are defined for all k ≥ 0.

We fix m > 0, we prove the result by assuming inf v k ≥ m > 0, like for the dimensions 4 and 6. After we take m = r k inf u k > 0. Suppose by contradiction, that there are infinitly many

(v k ) with inf v k ≥ m > 0, the proof imply that [u k (•)] 1-ǫ = v k (0) ≤ c(m) < +∞ which is impossible. Thus, there is a finite number of terms such that inf v k ≥ m > 0, k 1 , . . . , k i(m) . Thus we have also, v k (0) ≤ c(m) when inf v k ≥ m > 0.
In all cases, we have the following assertion:

inf v k ≥ m > 0 ⇒ [u k (•) 1-ǫ ] = v k (0) ≤ c(m) < +∞,
we obtain: There is a non-increasing positive function m → c(m) > 0, such that inf v k ≥ m > 0 ⇒ (u k (•)) 1-ǫ ≤ c(m). then we apply this with m = r k inf M u k , we obtain for all terms of the sequence (u k ):

n = 4, [u k (y k )] 1-ǫ ≤ c(a, b, A, inf M u k [u k (y k )] ǫ , K, M, g), and, n ≥ 5, [u k (t k )] 1-ǫ ≤ c(a, b, A, inf M u k [u k (t k )] ǫ , K, M, g).
For n ≥ 5, we set,

0 < ǫ k = u k (y k ) u k (t k ) ≤ 1, ǫ k = ǫ k (K), we obtain: n ≥ 5, [u k (y k )] 1-ǫ ≤ ǫ 1-ǫ k c(a, b, A, ǫ ǫ k inf M u k [u k (y k )] ǫ , K, M, g). We have u k (y k ) = sup K u k .
Here in the blow-up analysis, we supposed that there are infinitly many v k with inf v k ≥ m > 0, without loss of generality we assume that the subsequence is the sequence, also for the points t k there is a subsequence which converge to a point t ∈ K δ0 , without loss of generality we assume that the subsequence is the sequence. In the blow-up function (for k large), the blow-up of v k is the blow-up of u k , then we use the diagonal process to extract a subsequence which converge on compact sets to v and we use Caffarelli-Gidas-Spruck result to have v = ( 1 1+V ( t)|x| 2 ) (n-2)/2 . Without loss of generality we assume that this subsequence is the sequence and V ( t) = 1. Also we have:

inf v k ≥ m > 0, t k → t ∈ K δ0 .

II) Auxiliary function and moving-plane method:

We use the computations of previous papers with modifications, see [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Bahoura | Estimations uniformes pour l'équation de Yamabe en dimensions 5 et 6[END_REF][START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF][START_REF] Bahoura | A uniform estimate for scalar curvature equation on manifolds of dimension 4[END_REF][START_REF] Bahoura | An estimate on Riemannian manifolds of dimension 4[END_REF]. We consider:

z k (t, θ) = e (n-2)t/2 u k (exp t k (e t θ)),
and the blow-up function,

w k (t, θ) = e (n-2)t/2 v k (e t θ) = z k (t + 2 n -2 log r k , θ),
We have

λ k = -2 n-2 log v k (0), N = 2n n-2 . We have: Let, b 1 = J(t k , e t , θ) = det(g ij,t k )(e t θ), a(t k , t, θ) = log J(t k , e t , θ).
Lemma 2.1. The function z k is solution of:

(1) -∂ tt z k -∂ t a∂ t z k + ∆ θ z k + cz k = V k z (n+2)/(n-2) k , with, c = c(t k , t, θ) = (n -2) 2 4 + ∂ t a + he 2t .
Proof of the lemma, see [START_REF] Bahoura | Estimations uniformes pour l'équation de Yamabe en dimensions 5 et 6[END_REF][START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF][START_REF] Bahoura | A uniform estimate for scalar curvature equation on manifolds of dimension 4[END_REF][START_REF] Bahoura | An estimate on Riemannian manifolds of dimension 4[END_REF].

Now we have, ∂ t a = ∂ t b 1 b 1 , b 1 (t k , t, θ) = J(t k , e t , θ) > 0,
We can write,

- 1 √ b 1 ∂ tt ( b 1 z k ) + ∆ θ z k + [c(t) + b -1/2 1 b 2 (t, θ)]z k = V k z (n+2)/(n-2) k , where, b 2 (t, θ) = ∂ tt ( √ b 1 ) = 1 2 √ b 1 ∂ tt b 1 - 1 4(b 1 ) 3/2 (∂ t b 1 ) 2 . Let, zk = b 1 z k ,
and the blow-up function (rescaled function) and the function with the auxiliary function:

wk = ( b 1 )(t + 2 n -2 log r k , θ) • w k , wk (t, θ) = wk (e t θ) - m 2 e (n-2)t/2 ,
we have:

Lemma 2.2. The function zk is solution of:

-∂ tt zk + ∆ θ (z k ) + 2∇ θ (z k ).∇ θ log( b 1 ) + (c + b -1/2 1 b 2 -c 2 )z k = (2) = V k 1 b 1 N -2 z(n+2)/(n-2) k ,
where, c 2 = [ Proof of the lemma, see [START_REF] Bahoura | Estimations uniformes pour l'équation de Yamabe en dimensions 5 et 6[END_REF][START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF][START_REF] Bahoura | A uniform estimate for scalar curvature equation on manifolds of dimension 4[END_REF][START_REF] Bahoura | An estimate on Riemannian manifolds of dimension 4[END_REF]. We have,

c(t k , t, θ) = (n -2) 2 4 + ∂ t a + he 2t , (α 1 ) b 2 (t, θ) = ∂ tt ( b 1 ) = 1 2 √ b 1 ∂ tt b 1 - 1 4(b 1 ) 3/2 (∂ t b 1 ) 2 , (α 2 ) c 2 = [ 1 √ b 1 ∆ θ ( b 1 ) + |∇ θ log( b 1 )| 2 ], ( α 3 )
We have if we denote the previous operator:

L(t, θ) = -∂ tt (•)+∆ θ (•)+2∇ θ (•).∇ θ log( √ b 1 )+ (c + b -1/2 1 b 2 -c 2 )(•) and, b1 = b 1 (t + 2 n -2 log r k , θ),
We have:

L(t, θ)(z k ) = V k 1 b 1 N -2 z(n+2)/(n-2) k ,
and for the blow-up function (the rescaled function), we replace t by t + 2 n-2 log r k :

L(t + 2 n -2 log r k , θ)[ wk (t, θ)] = V k (t + 2 n -2 log r k , θ) 1 b1 N -2 w(n+2)/(n-2) k We set, L(t, θ) = L(t + 2 n -2 log r k , θ), Ṽk = V k (t + 2 n -2 log r k , θ)
Thus,

L(t, θ)[ wk (t, θ)] = Ṽk 1 b1 N -2 w(n+2)/(n-2) k and, wk (t, θ) = wk (e t θ) - m 2 e (n-2)t/2 , Proposition 2.3. We have for λ k = -2 n-2 log v k (0); 1) wk (λ k , θ) -wk (λ k + 4, θ) ≥ k > 0, ∀ θ ∈ S n-1 .
For all β > 0, there exist c β > 0 such that:

2) 1 c β e (n-2)t/2 ≤ wk (λ k + t, θ) ≤ c β e (n-2)t/2 , ∀ t ≤ β, ∀ θ ∈ S n-1 .
We want to apply the Hopf maximum principle.

wk (t, θ) = wk (e t θ) - m 2 e (n-2)t/2 ,
Like in [START_REF] Bahoura | A uniform estimate for scalar curvature equation on manifolds of dimension 4[END_REF] we have the some properties for wk , we have:

Lemma 2.4. There is ν < 0 such that for λ ≤ ν :

wλ k (t, θ) -wk (t, θ) ≤ 0, ∀ (t, θ) ∈ [λ, t 0 ] × S n-1 .
Let ξ k be the following real number,

ξ k = sup{λ ≤ λ k + 2, wλ k (t, θ) -wk (t, θ) ≤ 0, ∀ (t, θ) ∈ [λ, t 0 ] × S n-1 }.
We have the same computations as for the previous papers, see [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Bahoura | Estimations uniformes pour l'équation de Yamabe en dimensions 5 et 6[END_REF][START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF][START_REF] Bahoura | A uniform estimate for scalar curvature equation on manifolds of dimension 4[END_REF][START_REF] Bahoura | An estimate on Riemannian manifolds of dimension 4[END_REF]. We have the increment of functions and operators.

L(t, θ)[ wξ k k (t, θ)-wk (t, θ)] = [ L(t, θ)-L(t ξ k , θ)][ wξ k k (t, θ)]+ L(t ξ k , θ)[ wξ k k (t, θ)]-L(t, θ)[ wk (t, θ)] = = [ L(t, θ)-L(t ξ k , θ)][ wξ k k (t, θ)]+ Ṽ ξ k k 1 bξ k 1 N -2 ( wξ k k ) (n+2)/(n-2) -Ṽk 1 b1 N -2 w(n+2)/(n-2) k Thus, L(t, θ)[ wξ k k (t, θ) -wk (t, θ)] = = [ L(t, θ)-L(t ξ k , θ)][ wξ k k (t, θ)]+ Ṽ ξ k k 1 bξ k 1 N -2 ( wξ k k ) (n+2)/(n-2) -Ṽk 1 b1 N -2 w(n+2)/(n-2) k , Thus, L(t, θ)[ wξ k k (t, θ) -wk (t, θ)] = = [ L(t, θ) -L(t ξ k , θ)][ wξ k k (t, θ)]+ + Ṽ ξ k k 1 bξ k 1 N -2 ( wξ k k ) (n+2)/(n-2) -Ṽk 1 b1 N -2 w(n+2)/(n-2) k + +O(1)r 4/(n-2) k e 2t (e (n-2)t/2 -e (n-2)t ξ k /2 ),
We have:

[ L(t, θ) -L(t ξ k , θ)][ wξ k k (t, θ)] = O(1) wξ k k r 4/(n-2) k (e 2t -e 2t ξ k ),
and,

Ṽ ξ k k 1 bξ k 1 N -2 ( wξ k k ) (n+2)/(n-2) -Ṽk 1 b1 N -2 w(n+2)/(n-2) k = (3) = O(1)r 4/(n-2) k wξ k k (e 2t -e 2t ξ k )+O(1) wξ k k r 2/(n-2) k (e t -e t ξ k )+ Ṽk 1 b1 N -2 [( wξ k k ) (n+2)/(n-2) - w(n+2)/(n-2) k ] Thus, L(t, θ)[ wξ k k (t, θ) -wk (t, θ)] = = Ṽk 1 b1 N -2 [( wξ k k ) (n+2)/(n-2) - w(n+2)/(n-2) k ] + O(1)r 4/(n-2) k wξ k k (e 2t -e 2t ξ k )+ (4) +O(1) wξ k k r 2/(n-2) k (e t -e t ξ k ) + O(1)r 4/(n-2) k e 2t (e (n-2)t/2 -e (n-2)t ξ k /2 ),
We want to prove that by using the Hopf maximum principle, (like in [START_REF] Bahoura | Différentes estimations du sup u×inf u pour l'équation de la courbure scalaire prescrite en dimension n ≥ 3[END_REF][START_REF] Bahoura | Estimations du type sup × inf sur une variété compacte[END_REF][START_REF] Bahoura | Lower bounds for sup + inf and sup * inf and an extension of Chen-Lin result in dimension 3[END_REF][START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF]):

min θ∈Sn-1 wk (t 0 , θ) ≤ max θ∈Sn-1 wk (2ξ k -t 0 , θ),
For this, we argue by contradiction and we assume that:

min θ∈Sn-1 wk (t 0 , θ) > max θ∈Sn-1 wk (2ξ k -t 0 , θ),
Thus, our assumption is:

wk (2ξ k -t 0 , θ) -w k (t 0 , θ) < 0, ∀θ ∈ S n-1 .
Now, we want to prove that:

[ wξ k k (t, θ) -wk (t, θ)] ≤ 0 ⇒ L(t, θ)[ wξ k k (t, θ) -wk (t, θ)] ≤ 0, For this:
1) The biggest term is the term of V (for n ≥ 6):

wξ k k r 2/(n-2) k (e t -e t ξ k ), t 0 ≥ t ≥ ξ k .
Because we must compare: 2) , and, we have used the mean value theorem for f (t) = t (n+2)/(n-2) , and wξ k k ≤ wk to have:

( wξ k k ) (n+2)/(n-2) r 2/(n-2) k (e t -e t ξ k ) and ( wξ k k ) (n+2)/(n-2) -( wk ) (n+2)/(n-
( wξ k k ) (n+2)/(n-2) -( wk ) (n+2)/(n-2) ≤ c( wξ k k ) 4/(n-2) ( wξ k k -wk ), and, ( wξ k k ) 4/(n-2) ( wξ k k -wk ) ≤ c( wξ k k ) 4/(n-2) (e (n-2)t ξ k /2 -e (n-2)t/2
). Now, we write:

e t = e (n-2)t/2 e (4-n)t/2 ≤ e (4-n)ξ k /2 e (n-2)t/2 ,
we integrate between t and t ξ k , we obtain:

(e t -e t ξ k ) ≤ ce (4-n)ξ k /2 (e (n-2)t/2 -e (n-2)t ξ k /2 ), But, wξ k k ≤ ce (n-2)(ξ k -λ k )/2
, Thus the biggest term is:

wξ k k r 2/(n-2) k (e t -e t ξ k ) ≤ cr 2/(n-2) k e (ξ k -(n-2)λ k )/2 (e (n-2)t/2 -e (n-2)t ξ k /2 ), but ξ k ≤ λ k + 2
, we obtain:

wξ k k r 2/(n-2) k (e t -e t ξ k ) ≤ cr 2/(n-2) k
e -(n-4)λ k /2 (e (n-2)t/2 -e (n-2)t ξ k /2 ), Thus,

n ≥ 5, wξ k k r 2/(n-2) k (e t -e t ξ k ) ≤ c [u k (t k )] ǫ-(n-4)/(n-2) (e (n-2)t/2 -e (n-2)t ξ k /2 ), n = 4, wξ k k r 2/(n-2) k (e t -e t ξ k ) ≤ c [u k (y k )] ǫ (e (n-2)t/2 -e (n-2)t ξ k /2 ),
These terms are controled by the term: -m 2 (e (n-2)t/2 -e (n-2)t ξ k /2 ).

2) Also, we have for n ≥ 6:

Because we must compare:

wξ k k r 4/(n-2) k
(e 2t -e 2t ξ k ) and ( wξ k k ) (n+2)/(n-2) -( wk ) (n+2)/(n-2) , We must look to the term:

( wξ k k ) 1-(4/(n-2)) r 4/(n-2) k (e 2t -e 2t ξ k ) = ( wξ k k ) (n-6)/(n-2) r 4/(n-2) k (e 2t -e 2t ξ k ), t 0 ≥ t ≥ ξ k ,
We write: e 2t = e (n-2)t/2 e (6-n)t/2 ≤ e (6-n)ξ k /2 e (n-2)t/2 , But,

wξ k k ≤ ce (n-2)(ξ k -λ k )/2 , Thus, ( wξ k k ) (n-6)/(n-2) r 4/(n-2) k (e 2t -e 2t ξ k ) ≤ cr 4/(n-2) k
e -(n-6)λ k /2 (e (n-2)t/2 -e (n-2)t ξ k /2 ), we obtain:

( wξ k k ) (n-6)/(n-2) r 4/(n-2) k (e t -e t ξ k ) ≤ cr 4/(n-2) k
e -(n-6)λ k /2( (e (n-2)t/2 -e (n-2)t ξ k /2 ), Thus,

n ≥ 6, ( wξ k k ) (n-6)/(n-2) r 4/(n-2) k (e 2t -e 2t ξ k ) ≤ c [u k (t k )] ǫ-(n-6)/(n-2) (e (n-2)t/2 -e (n-2)t ξ k /2 ), But, ǫ > n-4
n-2 , these terms are controled by the term: -m 2 (e (n-2)t/2 -e (n-2)t ξ k /2 ).

3) For n = 5: we have the terms: we use the binomial formula: we write:

( wξ k k ) 7/3 - w7/3 k = (( wξ k k ) 1/3 ) 7 -( w1/3 k ) 7 ,
x 7 -y 7 ≡ (x -y)(x 6 + x 5 y + x 4 y 2 + x 3 y 3 + x 2 y 4 + xy 5 + y 6 ),

x = ( wξ k k ) 1/3 , y = w1/3 k , but, wξi k -wk = (x 3 -y 3 ) ≡ (x -y)(x 2 + xy + y 2 ), Thus, ( wξ k k ) 7/3 - w7/3 k = (x 3 -y 3 ) × (x 6 + x 5 y + x 4 y 2 + x 3 y 3 + x 2 y 4 + xy 5 + y 6 ) (x 2 + xy + y 2 ) ,
Here, we have x ≤ y, thus:

(x 2 + xy + y 2 ) ≤ cy 2 , (x 6 + x 5 y + x 4 y 2 + x 3 y 3 + x 2 y 4 + xy 5 + y 6 ) ≥ c ′ x 2 y 4 , c, c ′ > 0 Thus, because y ≥ m 2 e 3t/2 we obtain:

( wξ k k ) 7/3 - w7/3 k ≤ c( wξ k k -wk )( wξ k k ) 2/3 w2/3 k ≤ -ce t ( wξ k k ) 2/3 (e 3t/2 -e 3t ξ k /2 ), c > 0 For the case: A = ( wξ k k ) 1/3 r 4/3 k (e 2t -e 2t ξ k ), t 0 ≥ t ≥ ξ k We have: |A| ≤ e t ( wξ k k ) 1/3 r 4/3
k (e t -e t ξ k ) The dominant term is:

B = ( wξ k k ) 1/3 r 4/ 3
k (e t -e t ξ k ) We have:

e t = e -t/2 e 3t/2 ≤ ce -ξ k /2 e 3t/2 , w ξ k k ≤ ce 3(ξ k -λ k )/2 Thus, |B| ≤ cr 4/3 k e -λ k /2 (e 3t/2 -e 3t ξ k /2 ) λ k = -(2/3)(1 -ǫ) log u k (t k ), r k = u k (t k ) -ǫ , e -λ k /2 = u k (t k ) (1/3)(1-ǫ) , r 4/3 k = u k (t k ) -4ǫ/3 , r 4/3 k e -λ k /2 = u k (t k ) (1/3)(1-5ǫ) The condition is 1 -5ǫ < 0, ǫ > 1 5 , but ǫ > n-4 n-2 = 5-4 5-2 = 1/3 > 1/5.

4)

We have the same thing for the dimension 4. 5) When we use the auxiliary function m 2 e (n-2)t/2 , there is a term:

r 4/(n-2) k e 2t (e (n-2)t/2 -e (n-2)t ξ k /2 ),
To correct this term, we consider a part of the term :

( wξ k k ) (n+2)/(n-2) -( wk ) (n+2)/(n-2)
We use the binomial formula as for the previous case of dimension 5. We have:

x = ( wξ k k ) 1/(n-2) , y = ( wk ) 1/(n-2) ,
x n+2 -y n+2 = (x -y)(y n+1 + . . .), x n-2 -y n-2 = (x -y)(y n-3 + . . . + x n-3 ) Thus,

x n+2 -y n+2 = (x n-2 -y n-2 ) (y n+1 + . . .) (y n-3 + . . .) Because x ≤ y (y n+1 + . . .) ≥ y n+1 , (x n-3 + . . . + y n-3 ) ≤ cy n-3 , c > 0,

We obtain:

x n+2 -y n+2 ≤ c(x n-2 -y n-2 )y 4 , c > 0, Thus, ( wξ k k ) (n+2)/(n-2) -( wk ) (n+2)/(n-2) ≤ c( wξ k k -wk ) w4/(n-2) k , c > 0 Because, wk ≥ m 4 e (n-2)t/2 , we obtain:

( wξ k k ) (n+2)/(n-2) -( wk ) (n+2)/(n-2) ≤ -ce 2t (e (n-2)t/2 -e (n-2)t ξ k /2 ), c > 0, Thus the term: r 4/(n-2) k e 2t (e (n-2)t/2 -e (n-2)t ξ k /2 ) is controled by the term, -ce 2t (e (n-2)t/2e (n-2)t ξ k /2 ), c > 0.

We obtain the same proof in the previous papers, the dimensions 4, 6, see [START_REF] Bahoura | Estimations uniformes pour l'équation de Yamabe en dimensions 5 et 6[END_REF][START_REF] Bahoura | A uniform estimate for scalar curvature equation on manifolds of dimension 4[END_REF][START_REF] Bahoura | An estimate on Riemannian manifolds of dimension 4[END_REF]. If we use the Hopf maximum principle, we obtain (like in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Bahoura | Estimations uniformes pour l'équation de Yamabe en dimensions 5 et 6[END_REF][START_REF] Bahoura | Harnack inequalities for Yamabe type equations[END_REF][START_REF] Bahoura | A uniform estimate for scalar curvature equation on manifolds of dimension 4[END_REF]): min θ∈Sn-1 wk (t 0 , θ) ≤ max θ∈Sn-1 wk (2ξ k -t 0 , θ), thus for k large:

(u k (•)) 1-ǫ = v k (0) ≤ c,
It is a contradiction. Finaly, for each m > 0 there is a finite v k such that inf v k ≥ m > 0, k 1 , . . . , k m ∈ N.

Here also, we have the existence of c(m) > 0 such that inf v k ≥ m > 0 ⇒ (u k (•)) 1-ǫ = v k (0) ≤ c. We prove this by contradiction, suppose that for fixed m > 0, for all c > 0 there is i c ∈ N with inf v ic ≥ m > 0 and v ic (0) ≥ c, if we take c → +∞, because the number of indices is bounded and we have a sequence of integers, this sequence converge and in fact is constant because we consider integers. Thus there is an index k such that v k (0) ≥ c → +∞ and inf v k ≥ m > 0 , and thus v k is singular at 0, but this is impossible because v k is regular.

We obtain:

There is a non-increasing positive function m → c(m) > 0, such that inf v k ≥ m > 0 ⇒ (u k (•)) 1-ǫ ≤ c(m). then we apply this with m = r k inf M u k , we obtain the inequality for all terms of the sequence (u k ).


Impact of yeast cell cycle specificity on demography inference

Louis OLLIVIER, Flora JAY, Fanny POUYET ~ Université Paris-Saclay, CNRS, INRIA, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France.

[1] Fischer, G, Liti, G, Llorente, B. The budding yeast life cycle: More complex than anticipated? Yeast. 2021; 38: 5-11.

Workflow and methods

The canonical life cycle of Saccharomyces cerevisiae consists of an alternation between mitosis (clonal reproduction) and meiosis events (outcrossing and intratetrad mating). In this life cycle, meiosis events and then recombination events are known to be rare. However, recent studies 1 start to show that heterozygosity in yeast would be greater than previously thought, which implies that maybe recombination events are more frequent. Indeed, these events are estimated at frequencies ranging from 1 per 50000 generations to 1 per 1000 generations.

Using SLiM 2 and dadi 3 (which is based on the SFS), we wanted to see if we can correctly infer demographic parameters given this specific feature (frequency of recombination events). For this, we tested 2 conditions : a population with recombination events every generation (EGR) and a population with recombination events every 1000 generations (1000GR), the recombination rate (ρ) is scaled so that the number of individual events remains the same on average. We simulated 4 simple demographic models to evaluate if the frequency of recombination events has an effect on demography inference. Does recombination frequency (= sexual reproduction events) impact demography inference ?

Context

Results

Conclusions

  [2] Haller, B.C., & Messer, P.W. (2019). SLiM 3: Forward genetic simulations beyond the Wright-Fisher model. Molecular Biologyand Evolution 36(3), 632-637 [3] RN Gutenkunst, RD Hernandez, SH Williamson, CD Bustamante "Inferring the joint demographic history of multiple populations from multidimensional SNP data" PLoS Genetics 5:e1000695 (2009) [4] Duan, SF., Han, PJ., Wang, QM. et al. The origin and adaptive evolution of domesticated populations of yeast from Far East Asia. Nat Commun 9, 2690 (2018).

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  of generations since the demographic event NuB : ration of botteleneck population size to initial (N bot / N initial ) NuF : ration of final population size to initial (N final / N initial ) Evaluate other demography inference softwares (SMC++, fastsimcoal, …) -WIP  Evaluate other simple demograpic models (instantaneous growth, …) -