Abstract or Keywords
Let S be a compact hyperbolic Riemann surface of genus
$${g \geq 2}$$
g
≥
2
. We call a systole a shortest simple closed geodesic in S and denote by
$${{\rm sys}(S)}$$
sys
(
S
)
its length. Let
$${{\rm msys}(g)}$$
msys
(
g
)
be the maximal value that
$${{\rm sys}(\cdot)}$$
sys
(
·
)
can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface S
max
a compact Riemann surface of genus g whose systole has length
$${{\rm msys}(g)}$$
msys
(
g
)
. In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating
$${{\rm msys}(\cdot)}$$
msys
(
·
)
of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.