Uniqueness and non-existence of metrics with prescribed Ricci curvature

Abstract

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The Cauchy-Kowalevski theorem applied for counting connections with a prescribed Ricci tensor

“…We also consider the case where the trace of the torsion vanishes. A remarkable paper concerning the topic under consideration is [2]. In [2] more advanced techniques (than the Cauchy-Kowalevski theorem) are used and the analyticity assumption can be dropped.…”

Obstruction to prescribed positive Ricci curvature

“…Our results and methods of proof extend [5] from compact to non-compact manifolds. Related, though weaker, results, obtained by different techniques, are those of [0] (a reference kindly pointed out to us by the referee).…”

“…We are interested in finding conditions on h which prevent it from being the Ricci tensor of any Riemannian metric on X. Following [5] we consider the largest eigenvalue λ(h) of the curvature operator acting on covariant symmetric 2-tensors (see [1]). Given any C 2 metric g on X, we let e(g) denote the energy density of the identity map from (X,g) to (X,h).…”

“…Theorem 1 may be viewed as the "true" extension of [5,. Interestingly, Theorem 2 looks somewhat stronger than 12 PH.…”

“…Interestingly, Theorem 2 looks somewhat stronger than 12 PH. DELANOE [5, due to the non-compactness of X; an example here for (X, h) is the Poincare disk, since constant curvature -1 implies at once λ(h) = 1 by [1,Proposition 4.3]. Theorem 3 typically applies when (X, g) is asymptotically flat; as such, it generalizes [8].…”