Farid Madani

Equivariant Yamabe problem and Hebey–Vaugon conjecture

Abstract In their study of the Yamabe problem in the presence of isometry groups, E. Hebey and M. Vaugon announced a conjecture. This conjecture generalizes T. Aubins conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubins theorem and we prove the Hebey–Vaugon conjecture in dimensions less or equal to 37.

On the volume of complex amoebas

The paper deals with amoebas of k-dimensional algebraic varieties in the algebraic complex torus of dimension n 2k. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of k-dimensional algebraic variety in (C)n, with n 2k, is finite.

Le problème de Yamabe avec singularités

n−2 where h ∈ Lp(M), p > n/2 and h̃ ∈ R. We give the regularity of φ with respe t to the value of p. Finally, we onsider the results in geometry when g is a singular Riemannian metri and h = n−2 4(n−1) Rg, where Rg is the s alar urvature of g. 1. Introdu tion Let (M,g) be a smooth ompa t Riemannian manifold of dimension n ≥ 3. Denote by Rg the s alar urvature of g. The Yamabe problem is the following: Problem 1. Does there exists a onstant s alar urvature metri onformal to g? If g̃ = φ4/(n−2)g is a onformal metri to g with φ a smooth positive fun tion, then the s alar urvatures Rg and Rg̃ are related by the following equation: (1) 4(n− 1) n− 2 ∆gφ+Rgφ = Rg̃φ N−1 where N = 2n n−2 and ∆g is the geometri Lapla ian of the metri g with nonnegative eigenvalues. To solve the Yamabe problem, it is equivalent to nd a fun tion φ solution of equation above where Rg̃ is onstant. Equation (1) is alled Yamabe equation. Yamabe [11℄ stated the following fun tional, de ned for any ψ ∈ H1(M) − {0} by (2) Ig(ψ) = E(ψ) ‖ψ‖N = ∫M |∇ψ| + n− 2 4(n − 1)Rgψ2dv ‖ψ‖N and he onsidered the in mum of Ig de ned as follow

On toric locally conformally Kähler manifolds

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is

Hebey–Vaugon conjecture II

-\infty

A Detailed Proof of a Theorem of Aubin

-∞, and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.

Conformally related K\"ahler metrics and the holonomy of lcK manifolds

In this paper, we study the amoeba volume of a given

The holonomy of conformally K\"ahler manifolds

k-