List of Mihawk's Anti-Feats

-Chased a starving, already defeated Krieg Pirate crew to East Blue, the weakest sea, to finish them off

- Claims he doesn't use his all to hunt rabbits, but then proceeds to do just that

- Defeated East Blue Zoro, but not before Zoro's San Zen Sekai made him sweat (look it up)

- Retreated from Don Krieg after realizing against Zoro that maybe these fodders aren't fodders after all

- Refuses to fight Shanks because he "only has one arm"

- Claims to want to measure the "true distance between himself and that man (Newgate) only to backdown when YC3 Jozu steps onto the stage

- Defeats a couple of nameless Newkama swordsmen

- Goes all out against an exhausted pre-skip Luffy at Marineford and fails to land a hit

- Fails multiples times to cut down Buggy

- Postpones his fight against Vista after a few even clashes.

- One shots Daz Bones and then later says "I only signed up to fight the Whitebeard Pirates" when Shanks pulls up. We know this to be a blatant lie because he attacked not only Daz, but Newkama fodders and Luffy as well. Whenever one of Newgates commanders asked him "what's up?" he found an excuse to leave the fight.

- Even clash against Crocodile before presumably "postponing" this encounter off panel as well.

- Acted as though he was excited to be hunted by the Marines again, only to then go on to completely avoid confrontation with them and seek refuge with Crocodile and Buggy

- Subjugated by Buggy and is now carrying out another man's ambitions

- He now lives on an island literally called "Empty Bluffs Island"

- His Seraphim model, S-Hawk, acquired his genetic benefactors greatest technique. The art of postponement

Have I missed anything here? I think this is a pretty comprehensive list of everything Mihawk's accomplished in the story to date. At this point, I'm 100% convinced that Mihawk's fame is just him subsisting off duels he had against a young Shanks who wasn't even in his prime yet.