Frank Pacard

Refined asymptotics for constant scalar curvature metrics with isolated singularities

Abstract. We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, , in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohožaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.

Constant scalar curvature metrics with isolated singularities

We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen [12], but the proof we give here, based on the techniques of [6], is more direct, and provides more information about their geometry. When Λ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in [7] and [8]

Extremal metrics on blowups

In this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of Kahler manifolds which already carry an extremal metric. As a particular case, we construct extremal metrics on

Partial Regularity for Weak Solutions of a nonlinear elliptic equation.

mathbb {P}^2

A Variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

blown-up k points in general position, with

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

k lt m+2

Existence and convergence of positive weak solutions of −Δu=\(u^{\frac{n}{{n - 2}}} \) in bounded domains of ℝ n ,n≥3

.

Constant curvature foliations in asymptotically hyperbolic spaces.

For scalar non-linear elliptic equations, stationary solutions are defined to be critical points of a functional with respect to the variations of the domain. We consideru a weak positive solution of −Δu=uα in -Δu=uα in Ω ⊂ ℝn, which is stationary. We prove that the Hausdorff dimension of the singular set ofu is less thann−2α+1/α−1, if α≥n+2/n−2.

Bubbling along boundary geodesic near the second critical exponent

We establish new existence and non-existence results for positive solutions of the Einstein–scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein–scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.

ATTACHING HANDLES TO CONSTANT-MEAN-CURVATURE-1 SURFACES IN HYPERBOLIC 3-SPACE

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrodinger type equations ?u-u+f(u)=0 in R N , u?H 1 (R N ) , where N=2 . Under natural conditions on the nonlinearity f , we prove the existence of infinitely many nonradial solutions in any dimension N=2 . Our result complements earlier works of Bartsch and Willem (N=4 or N=6 ) and Lorca-Ubilla (N=5 ) where solutions invariant under the action of O(2)×O(N-2) are constructed. In contrast, the solutions we construct are invariant under the action of D k ×O(N-2) where D k ?O(2) denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with k sides invariant, for some integer k=7 , but they are not invariant under the action of O(2)×O(N-2) .