conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

compactness for conformal scalar flat metrics on umbilic boundary manifolds abstract let m g a compact riemannian n dimensional manifold with umbilic boundary it is well known that under certain hypothesis in the conformal class of g there are scalar flat metrics that have m as a constant mean curvature hypersurface in this paper we prove that these metrics are a compact set provided n 8 and the weyl tensor of the boundary is always different from zero or if n 8 and the weyl tensor of m is always different from zero on the boundary

herglotz nevanlinna functions in several variables in this thesis we investigate different aspects of the class of herglotz nevanlinna functions in several variables these are holomorphic functions on the poly upper half plane having non negative

a pointwise finite dimensional reduction method for einstein lichnerowicz type systems the six dimensional case abstract we construct blowing up solutions for smooth perturbations of a focusing einstein lichnerowicz system on a closed 6 dimensional manifold our construction combines asymptotic analysis techniques with a finite dimensional reduction approach following the scheme of proof developed in premoselli 2016 the blow up examples constructed here highlight in particular the role of the coupling field in dimension 6

the lin ni conjecture in negative geometries abstract we prove see theorem 1 1 that the lin ni conjecture for closed manifolds is false in dimensions n 4 5 when the scalar curvature is negative but that it holds true in the bounded energy setting when n 6 see theorem 1 2 as a corollary to theorem 1 1 we prove see theorem 2 2 that contrary to the 3 dimensional case the sup inf inequality does not hold in general with a sole control on the size of the potential for positive solutions of stationary critical schrodinger equations

blow up phenomena for scalar flat metrics on manifolds with boundary abstract let m n g be a compact riemannian manifold with boundary m this article is concerned with the set of scalar flat metrics on m which are in the conformal class of g and have m as a constant mean curvature hypersurface we construct examples of metrics on the unit ball b n in dimensions n 25 for which this set is non compact these manifolds have umbilic boundary but they are not conformally equivalent to b n

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

nevanlinna representations in several variables abstract we generalize to several variables the classical theorem of nevanlinna that characterizes the cauchy transforms of positive measures on the real line we show that for the loewner class a large class of analytic functions that have non negative imaginary part on the upper polyhalf plane there are representation formulae in terms of densely defined self adjoint operators on a hilbert space we find four different representation formulae and we show that every function in the loewner class has one of the four representations corresponding precisely to four different growth conditions at infinity

operator monotone functions and lowner functions of several variables we prove generalizations of lowner s results on matrix monotone functions to several variables we give a characterization of when a function of d variables is locally monotone on d tuples of commuting self adjoint n by n matrices we prove a generalization to several variables of nevanlinna s theorem describing analytic functions that map the upper half plane to itself and satisfy a growth condition we use this to characterize all rational functions of two variables that are operator monotone

a global theory of algebras of generalized functions abstract we present a geometric approach to defining an algebra g m the colombeau algebra of generalized functions on a smooth manifold m containing the space d m of distributions on m based on differential calculus in convenient vector spaces we achieve an intrinsic construction of g m g m is a differential algebra its elements possessing lie derivatives with respect to arbitrary smooth vector fields moreover we construct a canonical linear embedding of d m into g m that renders c m a faithful subalgebra of g m finally it is shown that this embedding commutes with lie derivatives thus g m retains all the distinguishing properties of the local theory in a global context

positive clusters for smooth perturbations of a critical elliptic equation in dimensions four and five abstract we construct clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension n 4 5 such a construction is already available in the literature in dimensions n 6 see for instance 10 14 30 32 36 and not possible in dimension 3 by 25 this also provides new patterns for the lin ni 26 problem on closed manifolds and completes results by brezis and li 8 about this problem

a variational analysis of einstein scalar field lichnerowicz equations on compact riemannian manifolds 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

system of nonlinear volterra s integral equations with polar kernel and singularities abstract we study an existence uniqueness theorem for a system of nonlinear volterra type integral equations with polar kernel and singularities to do so we use a method of regularization of fractional derivatives with a delta sequence to avoid singularities over diagonal y x y 0 the other singularities we regularize with delta sequences with different growth as a framework we use colombeau special algebra

compactness results in conformal deformations of riemannian metrics on manifolds with boundaries this paper is devoted to the study of a problem arising from a geometric context namely the conformal deformation of a riemannian metric to a scalar flat one having constant mean curvature on the boundary by means of blow up analysis techniques and the positive mass theorem we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the c2 norm and the total leray schauder degree of all solutions is equal to 1 then we deduce from this compactness result the existence of at least one solution to our problem

the effect of linear perturbations on the yamabe problem in conformal geometry the compactness conjecture asserts that the set of yamabe metrics on a smooth compact aspherical riemannian manifold left m g right is compact established in the locally conformally flat case by schoen lecture notes in mathematics vol 1365 pp 120 154 springer berlin 1989 surveys pure application and mathematics 52 longman science technology pp 311 320 harlow 1991 and for n le 24 by khuri marques schoen j differ geom 81 1 143 196 2009 it has revealed to be generally false for n ge 25 as shown by brendle j am math soc 21 4 951 979 2008 and brendle marques j differ geom 81 2 225 250 2009 a stronger version of it the compactness under perturbations of the yamabe equation is addressed here with respect to the linear geometric potential frac n 2 4 n 1 mathrm scal g mathrm scal g being the scalar curvature of left m g right we show that a priori l infty bounds fail for linear perturbations on all manifolds with n ge 4 as well as a priori gradient l 2 bounds fail for non locally conformally flat manifolds with n ge 6 and for locally conformally flat manifolds with n ge 7 in several situations the results are optimal our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g

examples of non isolated blow up for perturbations of the scalar curvature equation on non locally conformally flat manifolds solutions to scalar curvature equations have the property that all possible blow up points are isolated at least in low dimensions this property is commonly used as the first step in the proofs of compactness we show that this result becomes false for some arbitrarily small smooth perturbations of the potential

blow up phenomena for linearly perturbed yamabe problem on manifolds with umbilic boundary abstract we build blowing up solutions for linear perturbation of the yamabe problem on manifolds with umbilic boundary provided the weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n 11

on a conjecture of lin ni for a semilinear neumann problem let q be a bounded domain in rn n 3 and a 0 we consider au au u n 2 n 2 in q2 u 0 in q au av 0 on i2 and show that for a sufficiently small the minimal energy solutions are only constants

bubble tower solutions to asymptotically critical nonlocal elliptic equations on conformal infinities abstract assume that x g is an asymptotically hyperbolic manifold with conformal infinity m h we construct positive and sign changing bubble tower solutions to low order perturbations of yamabe equations 0 1 on m h these solutions exhibit isolated but non simple blow up in the sense of schoen 30