If the systole is defined as the length of the shortest essential simple closed curve are there any known upper bounds for hyperbolic surfaces with punctures?



Upper bounds for systoles on punctured surfaces
Yes, since the injectivity radius (defined as the max of injectivity radii over all points of the surface) is bounded by roughly the log of the area (think "embedded disk"). For interesting papers on this subject, check out:


@article {MR1269424,
 AUTHOR = {Buser, P. and Sarnak, P.},
 TITLE = {On the period matrix of a {R}iemann surface of large genus},
 NOTE = {With an appendix by J. H. Conway and N. J. A. Sloane},
 JOURNAL = {Invent. Math.},
 FJOURNAL = {Inventiones Mathematicae},
 VOLUME = {117},
 YEAR = {1994},
 NUMBER = {1},
 PAGES = {27--56},
 ISSN = {0020-9910},
 CODEN = {INVMBH},
 MRCLASS = {22E40 (14H15 14H42 32G20)},
 MRNUMBER = {1269424 (95i:22018)},
MRREVIEWER = {Jos{\'e} M. Mu{\~n}oz Porras},
 DOI = {10.1007/BF01232233},
 URL = {<http://dx.doi.org/10.1007/BF01232233>},
}


@article {MR3065183,
 AUTHOR = {Basmajian, Ara},
 TITLE = {Universal length bounds for non-simple closed geodesics on
 hyperbolic surfaces},
 JOURNAL = {J. Topol.},
 FJOURNAL = {Journal of Topology},
 VOLUME = {6},
 YEAR = {2013},
 NUMBER = {2},
 PAGES = {513--524},
 ISSN = {1753-8416},
 MRCLASS = {30F40 (30F45 58E10)},
 MRNUMBER = {3065183},
MRREVIEWER = {Makoto Masumoto},
 DOI = {10.1112/jtopol/jtt005},
 URL = {<http://dx.doi.org/10.1112/jtopol/jtt005>},
}


