35 research outputs found
Linear bounds for constants in Gromov's systolic inequality and related results
Let be a closed Riemannian manifold. Larry Guth proved that there
exists with the following property: if for some the volume of each
metric ball of radius is less than , then there exists a
continuous map from to a -dimensional simplicial complex such that
the inverse image of each point can be covered by a metric ball of radius
in . It was previously proven by Gromov that this result implies two
famous Gromov's inequalities: and,
if is essential, then also with
the same constant . Here denotes the length of a shortest
non-contractible closed curve in .
We prove that these results hold with . We demonstrate that for essential Riemannian manifolds . All previously known upper bounds for were
exponential in .
Moreover, we present a qualitative improvement: In Guth's theorem the
assumption that the volume of every metric ball of radius is less than
can be replaced by a weaker assumption that for every point
there exists a positive such that the volume of the
metric ball of radius centered at is less than (for ).
Also, if is a boundedly compact metric space such that for some and
an integer the -dimensional Hausdorff content of each metric ball
of radius in is less than , then there exists a
continuous map from to a -dimensional simplicial complex such that
the inverse image of each point can be covered by a metric ball of radius