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Steve Karam

Publications and source records attributed to Steve Karam.

5 recordsLinked to original sources
arXiv2018

Macroscopic Schoen conjecture for manifolds with non-zero simplicial volume

We prove that given a hyperbolic manifold endowed with an auxiliary Riemannian metric whose sectional curvature is negative and whose volume is sufficiently small in comparison to the hyperbolic one, we can always find for any radius at least $1$ a ball in its universal cover whose volume is bigger than the hyperbolic one. This result is deduced from a non-sharp macroscopic version of a conjecture by R. Schoen about scalar curvature, whose proof is a variation of an argument due to M. Gromov and based on a smoothing technique. We take the opportunity of this work to present a full account of this technique which involves simplicial volume and deserves to be better known.

math.DG↗
arXiv2018

Minimal length product over homology bases of manifolds

Minkowski's second theorem can be stated as an inequality for $n$-dimensional flat Finsler tori relating the volume and the minimal product of the lengths of closed geodesics which form a homology basis. In this paper we show how this fundamental result can be promoted to a principle holding for a larger class of Finsler manifolds. This includes manifolds for which first Betti number and dimension do no necessarily coincide, a prime example being the case of surfaces. This class of manifolds is described by a non-vanishing condition for the hyperdeterminant reduced modulo $2$ of the multilinear map induced by the fundamental class of the manifold on its first ${\mathbb Z}_2$-cohomology group using the cup product.

math.GT↗
arXiv2013

Short Homotopically independent loops on surfaces

In this paper, we are interested in short homologically and homotopically independent loops based at the same point on Riemannian surfaces and metric graphs. First, we show that for every closed Riemannian surface of genus $g \geq 2$ and area normalized to $g$, there are at least $\ceil{\log(2g)+1}$ homotopically independent loops based at the same point of length at most $C\log(g)$, where $C$ is a universal constant. On the one hand, this result substantially improves Theorem $5.4.A$ of M. Gromov in \cite{G1}. On the other hand, it recaptures the result of S. Sabourau on the separating systole in \cite{SS} and refines his proof. Second, we show that for any two integers $b\geq 2$ with $1\leq n\leq b$, every connected metric graph $Γ$ of first Betti number $b$ and of length $b$ contains at least $n$ homologically independent loops based at the same point and of length at most $24(\log(b)+n)$. In particular, this result extends Bollobàs-Szemerédi-Thomason's $\log(b)$ bound on the homological systole to at least $\log(b)$ homologically independent loops based at the same point. Moreover, we give examples of graphs where this result is optimal.

math.DG↗
arXiv2013

Growth of balls in the universal cover of surfaces and graphs

In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant $δ>0$ such that if $(M,hyp)$ is a closed hyperbolic surface and $h$ another metric on $M$ with $\area(M,h)\leq δ\area(M,hyp)$ then for every radius $R\geq 1$ the universal cover of $(M,h)$ contains an $R$-ball with area at least the area of an $R$-ball in the hyperbolic plane. This positively answers a question of L. Guth for surfaces. We also prove an analog theorem for graphs.

math.DG↗