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In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_{\alpha}, g_{\alpha})$ be a sequence of compact gradient Ricci solitons of dimension $n\geq 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$ norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded volume and with uniformly upper bounded diameter, then there must exists a subsequence $(M_{\alpha}, g_{\alpha})$ converging to a compact orbifold $(M_{\infty}, g_{\infty})$ with finitly many isolated singularities, where $g_{\infty}$ is a gradient Ricci soliton metric in an orbifold sense., Comment: 21pages

In this paper, we study free boundary-value problems for $p$-harmonic maps on half simple spaces of Euclidean space, and obtain some Liouville-type theorems.

The purpose of this paper is to establish an intersection formula in equivariant complex geometry, in the presence of an excess normal bundle. The contribution of the excess normal bundle to the formula appears through an additive genus K R. In a forthcoming paper, an infinite dimensional analogue of this formula will be shown to be the result of Bismut-Lebeau on the behaviour of Quillen metrics under complex immersions.

In this article a geometric process to compare spinors for different metrics is constructed. It makes possible the extension to spinor fields of a variant of the Lie derivative (called the metric Lie derivative), giving a geometric approach to a construction originally due to Yvette Kosmann. The comparison of spinor fields for two different Riemannian metrics makes the study of the variation of Dirac operators feasible. For this it is crucial to take into account the fact that the bundle in which the sections acted upon by the Dirac operators take their values is changing. We also give the formulas for the change in the eigenvalues of the Dirac operator. We conclude by giving a few cases in which an eigenvalue is stationary.

Based on a self-contained, coordinate-free exposition of the necessary concepts and tools of spray and Finsler geometry (with detailed proofs), we derive new results among others on the consequences of the direction-independence of the Landsberg tensor and the stretch tensor of a Finsler manifold. We show that an at least 3-dimensional Finsler manifold with vanishing projected Berwald tensor is a Berwald manifold. To obtain consequences in two dimensions, we transcript Berwald's classical theory on 2-dimensional Finsler manifolds in our setup. We prove criteria and necessary conditions for Finsler-metrizability and projective Finsler-metrizability of a spray. We present new proofs for such classical results as the Berwald - del Castillo - Szab\'o theorem on isotropic Finsler manifolds, the Finslerian Schur lemma, and the uniqueness of the Berwald connection of a Finsler manifold., Comment: Final version of the author's PhD Thesis

pp.139-163 We give an explicit lower bound for almost psh functions on some Fano manifolds. These manifolds generalize those introduced by Calabi in [5], and also provide a generalization of the concept of the blowing-up of $\mathbb P_m\mathbb C$ at one point. To this end, we use a method introduced in [4], which consists of studying the behavior of psh functions along some well-chosen holomorphic curves.

Using a link between Einstein-Sasakian structures and Killing spinors we prove a general construction principle of odd-dimensional Riemannian manifolds with real Killing spinors. In dimensionn=7 we classify all compact Riemannian manifolds with two or three Killing spinors. Finally we classify nonflat 7-dimensional Riemannian manifolds with parallel spinor fields.

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 $C^2$-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., 34 pages

In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constant and a finite dimensional map associated to it has non-zero degree., 27 pages

We show that the unit tangent bundle of S^4 and a real cohomology CP^3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature., Comment: 37 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper14.abs.html

We study the existence of positive solutions for the equation $\Delta u+ K(x) u^{\frac{n+2}{n-2}}=0$ in $\mathbb{R}^n$ ($n \geq 3$) which decay to $0$ at infinity like $|x|^{2-n}$; $K(x)$ is a function which is bounded from above and below by positive constants, and no symmetry assumption on $K$ is made. We find conditions that guarantee existence for a large class of $K$'s. As a consequence one can explicitly show that the set of coefficients for which a solution exists is dense, in $C^1$ norm, in the space of positive bounded $C^1$ functions. These conditions also allow us to display a radial $K$ such that the previous problem has nonradial solutions but no radial solution. Some new results for the corresponding problem on $\Sn$ are also proved.