21:9 2560x1080 on Mini late 2012
According to MacTracker.app:


Display Connection    1 - HDMI port or Thunderbolt digital video output
Display Support   Supports an HDMI-compatible device while using one Thunderbolt display or support for two Thunderbolt displays
Display Modes Dual display extended, video mirroring, and AirPlay Mirroring
External Resolution   Up to 2560 by 1600 (Thunderbolt) or 1920 by 1200 (HDMI)


Concluding: You need a Thunderbolt display adapter for your new display