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b"We study the initial mass function (IMF) of NGC 3603, one of the most massive galactic star-forming regions, to answer a fundamental question in current astrophysics - is the IMF universal, or does it vary?\\n\\nUsing our very deep high angular resolution images obtained with the NAOS-CONICA adaptive optics system at the VLT/ESO, we have successfully revealed the low-mass stellar population in the cluster core down to about 0.4 Msun (50 % completeness limit). Based on the JHKsL' color-magnitude and color-color diagrams, we first derive an average age 0.7 Myr for the pre-main sequence stars, and an upper limit of ~2.5 Myr for the main sequence stars. We find an average foreground extinction of Av = 4.5 +- 0.5 mag, with a radial increase of Delta_Av ~ 2.0 mag towards larger radii (r < 50''). \\nFrom the infrared excess emission identified in the Ks - L' vs J - H color-color diagram, we measure a disk fraction of ~25 % for stars with M > 0.9 Msun in the cluster center (r < 10'').\\n\\nApplying a field star rejection and correcting for incompleteness, we derive the Ks-band luminosity function (LF) for stars simultaneously detected in the JHKs-bands. The LF follows a power-law with an index of alpha ~ 0.27, and shows no turnover or truncation within the detection limit. The IMF for stars within r < 110'' is reasonably fitted by a single power-law with index Gamma ~ -0.74 in the mass range of $0.4 - 20 Msun. This is substantially flatter than the Salpeter-like IMF (Gamma = -1.35). \\nThe IMF power-law index decreases from Gamma ~ -0.31 at r < 5'' to Gamma ~ -0.86 at 30'' < r < 110''. This radial steepening of the IMF mainly occurs in the inner r < 30'' field, indicating mass segregation at the very center of the starburst cluster. \\n\\nAnalyzing the radial mass density profile, we derive a cluster core radius of ~4''.8 (~0.14 pc), and a lower limit of ~110'' (~3.2 pc) for the cluster size. We also derive an upper limit of r ~ 1260'' (~37 pc) for the cluster size adopting an estimate of the tidal radius of the cluster. \\nBased on the de-projected stellar density distribution, we estimate the total mass and the half-mass radius of NGC 3603 to be about 1.0 - 1.6 x 10^4 Msun and 25'' - 50'' (~0.7 - 1.5 pc), respectively. The derived core radius is > 6 x 10^4 Msun pc^-3.\\n\\nThe estimate of the half-mass relaxation time for stars with a typical mass of 1 Msun is 10 - 40 Myr, suggesting that the intermediate- and low-mass stars have not yet been affected significantly by the dynamical relaxation in the cluster. The relaxation time for the high-mass stars is expected to be much smaller, and is comparable to the age of the cluster. We can thus not conclude if the mass segregation of the high-mass stars is primordial or caused by dynamical evolution. \\nOur observation covers at least ~67 % of intermediate- and low-mass stars in NGC 3603, and the stars residing outside the observed field can merely steepen the IMF by Delta_Gamma < 0.16. Therefore, because of the almost constant IMF beyond a radius r > 30'', we are confident that our IMF adequately describes the whole NGC 3603 starburst cluster.\\n\\nWe also thoroughly analyze the systematic uncertainties in our IMF determination. We conclude that the power-law index of NGC 3603 including the systematic uncertainties is Gamma = -0.74^{+0.62}_{-0.47}. Our result thus supports the hypothesis of a top-heavy IMF in starbursts, especially in combination with other studies of similar clusters such as the Arches cluster and the Galactic Center cluster."