In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold M n is bounded from above by (n + 2)!d/4, where d is the diameter of a manifold M n . The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold M n is bounded from above by (n + 2)! FillRad(M n ) ≤ (n + 2)!(n + 1)n n (n + 1)!(vol(M n )) 1/n , where FillRad(M n ) is the filling radius of a manifold, and vol(M n ) is its volume.

Computations in Dynamical Triangulation Models of Four-Dimensional Quantum Gravity involve weighted averaging over sets of all distinct triangulations of compact four-dimensional manifolds. In order to be able to perform such computations one needs an algorithm which for any given N and a given compact four-dimensional manifold M constructs all possible triangulations of M with ≤ N simplices. Our first result is that such algorithm does not exist. Then we discuss recursion-theoretic limitations of any algorithm designed to perform approximate calculations of sums over all possible triangulations of a compact four-dimensional manifold.

In this paper we will show that for any pair of points p, q ∈ M , where M is a Riemannian manifold diffeomorphic to the 2-dimensional sphere, there always exist at least k geodesics connecting them of length at most 22kd, where d is the diameter of M .Introduction.

Let M n be a closed Riemannian manifold of diameter d. Our first main result is that for every two (not necessarily distinct) points p, q ∈ M n and every positive integer k there are at least k distinct geodesics connecting p and q of length ≤ 4nk 2 d.We demonstrate that all homotopy classes of M n can be represented by spheres swept-out by "short" loops unless the length functional has "many" "deep" local minima of a "small" length on the space Ω pq M n of paths connecting p and q. For example, one of our results implies that for every positive integer k there are two possibilities: Either the length functional on Ω pq M n has k distinct non-trivial local minima with length ≤ 2kd and "depth" ≥ 2d; or for every m every map of S m into Ω pq M n is homotopic to a map of S m into the subspace Ω 4(k+2)(m+1)d pq M n of Ω pq M n that consists of all paths of length ≤ 4(k+ 2)(m + 1)d.

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