Einstein structures: Existence versus uniqueness

Abstract

Two well-known questions in differential geometry are "Does every compact manifold of dimension greater than four admit an Einstein metric?" and "Does an Einstein metric of a negative scalar curvature exist on a sphere?" We demonstrate that these questions are related: For every n _> 5 the existence of metrics for which the deviation from being Einstein is arbitrarily small on every compact manifold of dimension n (or even on every smooth homology sphere of dimension n) implies the existence of metrics of negative Ricci curvature on the sphere S n for which the deviation from being Einstein is arbitrarily small. Furthermore, assuming either a version of the PalaisSmale condition or the plausible looking existence of an algorithm deciding when a given metric on a compact manifold is close to an Einstein metric, we show for any n > 5 that: 1) If every n-dimensional smooth homology sphere admits an Einstein metric then S" admits infinitely many Einstein structures of volume one and of negative scalar curvature; 2) If every compact n-dimensional manifold admits an Einstein metric then every compact n-dimensional manifold admits infinitely many distinct Einstein structures of volume one and of negative scalar curvature. [S]. A study of Einstein structures is motivated by possible applications to General Relativity and also, according to [Be], by the following question of R. Thom: "Are there any best (or nicest) Riemannian structures on a given compact manifold M?" (Einstein structures are natural candidates to be considered as nice structures.) One of our results (Theorem 3) is that there is no quite satisfactory positive answer to Thom's question. More precisely, assume that one defines nice structures in such a manner that: 1) (Existence) A nice structure exists on every compact manifold of a fixed dimension n _> 5; 2) (Scale-invariance) If a metric is nice, then metrics obtained by the multiplication of this metric by a positive constant are nice; and 3) (Recognizability) It is possible to recognize when a given Riemannian structure is close to a nice structure.Then the set of nice structures of volume one for every compact manifold

A biased view of symplectic cohomology

“…The original reference is [39]; for more recent expositions, see [54,Chapter 2] (informal) or [37] (more detailed). Let P = g 1 , .…”

Representación finita de variedades compactas

“…Una variación del teorema anterior se debe a A. Nabutovsky [11], quien probó que es imposible reconocer S n , n ≥ 5, en la clase de hipersuperficies algebraicas no-singulares en R n+1 .…”

Some Fundamental Issues Concerning Degrees of Unsolvability

“…There I heard a talk by Shmuel Weinberger, a prominent topologist and geometer. At the time Weinberger was trying to learn something about the recursively enumerable Turing degrees, R T , with an eye to applying them in the study of moduli spaces in differential geometry [48], using recursion-theoretic methods pioneered by Nabutovsky [33,34]. Weinberger was visibly frustrated by the fact that R T does not appear to contain any specific, natural examples of recursively enumerable Turing degrees, beyond the two standard examples due to Turing, namely 0 ′ = the Turing degree of the Halting Problem, and 0 = the Turing degree of solvable problems.…”