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Some Nonlinear Problems in Riemannian Geometry
1 Riemannian Geometry.- 2 Sobolev Spaces.- 3 Background Material.- 4 Complementary Material.- 5 The Yamabe Problem.- 6 Prescribed Scalar Curvature.- 7 Einstein-Kahler Metrics.- 8 Monge-AmpereExpand
Nonlinear analysis on manifolds, Monge-Ampère equations
1 Riemannian Geometry.- 1. Introduction to Differential Geometry.- 1.1 Tangent Space.- 1.2 Connection.- 1.3 Curvature.- 2. Riemannian Manifold.- 2.1 Metric Space.- 2.2 Riemannian Connection.- 2.3Expand
The Ricci Curvature
In this chapter we deal with problems concerning Ricci Curvature mainly: Prescribing the Ricci curvature Ricci curvature with a given sign Existence of Einstein metrics.
The Scalar Curvature
We shall deal with some problems concerning the scalar curvature of compact riemannian manifolds. In particular we shall deal with the problem of Yamabe: Does there exist a conformal metric for whichExpand
On the best Sobolev inequality
Abstract We prove that the best constant in the Sobolev inequality ( W 1 , P ⊂ L p* with 1/ p * = 1/ p − 1/ n and 1 p n ) is achieved on compact Riemannian manifolds, or only complete under someExpand
Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature
Abstract Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 2, let p(1, n) real, and let H1p (M) be the standard Sobolev space of order p. By the Sobolev embedding theorem, H1p(M) ⊂ Lp* (M)Expand
A course in differential geometry
Background material Differentiable manifolds Tangent space Integration of vector fields and differential forms Linear connections Riemannian manifolds The Yamabe problem-An introduction to researchExpand
The Mass according to Arnowitt, Deser and Misner
Abstract For asymptotically Euclidean manifolds of order τ > ( n − 2 ) / 2 , under the hypothesis that the mass m (according to Arnowitt, Deser and Misner) exists (in particular if the scalarExpand