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The mechanisms governing the star formation rate in spiral galaxies are notyet clear. The nearby, almost face-on, and interacting galaxy M51 offers anexcellent opportunity to study at high spatial resolutions the local starformation laws. In this first paper, we investigate the correlation of H2, HI,and total gas surface densities with the star forming activity, derived fromthe radio continuum (RC), along radial averages out to radii of 12kpc.  We have created a complete map of M51 in 12CO 2-1 at a resolution of 450kpcusing HERA at the IRAM-30m telescope. These data are combined with maps of HIand the radio-continuum at 20cm wavelength. The latter is used to estimate thestar formation rate (SFR), thus allowing to study the star formation efficiencyand the local Schmidt law. The velocity dispersion from CO is used to study thecritical surface density and the gravitational stability of the disk.  The critical gas velocity dispersions needed to stabilize the gas againstgravitational collapse in the differentially rotating disk of M51 using theToomre criterion, vary with radius between 1.7 and 6.8 km/s. Observed radiallyaveraged dispersions derived from the CO data vary between 28 km/s in thecenter and 8 km/s at radii of 7 to 9 kpc. They exceed the critical dispersionsby factors Q_gas of 1 to 5. We speculate that the gravitational potential ofstars leads to a critically stable disk.Comment: accepted by A&

A complete $^\mathsf{{12}}$CO 2-1 map of M 51 with HERA