Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUniversal systole bounds for arithmetic locally symmetric spaces
64
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of a semisimple element x in SL_n(R) in terms of its associated Mahler measure. We use these geometric methods to prove the existence of extensions of number fields in which fixed sets of primes have certain prescribed splitting behavior.
rate research
Read More
We explain how the Transference Principles from Diophantine approximation can be interpreted in terms of geometry of the locally symmetric spaces $T_n=SO(n) backslash SL(n,R) /SL(n,Z)$ with $n>1$, and how, via this dictionary, they become transparent geometric remarks and can be easily proved. Indeed, a finite family of linear forms is naturally identified to a locally geodesic ray in a space $T_n$ and the way this family is approximated is reflected by the heights at which the ray rises in the cuspidal end. The only difference between the two types of approximation appearing in a Transference Theorem is that the height is measured with respect to different rays in $W$, a Weyl chamber in $T_n$. Thus the Transference Theorem is equivalent to a relation between the Busemann functions of two rays in $W$. This relation is easy to establish on $W$, because restricted to it the two Busemann functions become two linear forms. Since $T_n$ is at finite Hausdorff distance from $W$, the same relation is satisfied up to a bounded perturbation on the whole of $T_n$.
In this paper, we classify three-locally-symmetric spaces for a connected, compact and simple Lie group. Furthermore, we give the classification of invariant Einstein metrics on these spaces.
We find many examples of compact Riemannian manifolds $(M,g)$ whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric $g$ is replaced with $g$ in a neighbourhood of $g$. Our examples $(M,g)$ consist of certain minimal isoparametric hypersurfaces of spheres; their focal manifolds; the Lie groups $SU(n)$ for $nleq 17$, and $Sp(n)$ for all $n$; and all quaternionic Grassmannians.
In this article we show that for any given Riemann surface $Sigma$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $Sigma$ in terms of the genus and the combined extremal lengths on $Sigma$ of $(g-1)$ disjoint, non-homotopic, simple closed compressible curves. This result is used to partially answer a question posed by Maldacena about comparing renormalized volumes of Schottky and Fuchsian manifolds with the same conformal boundary.
We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)le 1. Let g be another metric with scalar curvature k, such that gge g on 2-vectors. We show that kge k everywhere on M implies k=k. Under an additional condition on the Ricci curvature of g, kge k even implies g=g. We also study area-non-increasing spin maps onto such Riemannian manifolds.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
