How can my iMac support 2560x1080 external monitor (LG 29EA93)

You shouldn't have any problem on using the 2560x1080 resolution, as the graphic card supports it correctly. You will only have to buy a Mini DisplayPort to HDMI adapter and a HDMI cable, as it's the only cable that can handle this resolution. See > http://support.apple.com/kb/HT4241?viewlocale=en_US

A mini-DisplayPort to Dual-Link DVI would also work:

http://store.apple.com/us/product/MB571Z/A/mini-displayport-to-dual-link-dvi-ada pter?fnode=51

though if HDMI will work the adapter would be less expensive.

Regards.

My guess is that the adapter you have doesn't support the higher resolutions. You might try this adapter:

http://www.monoprice.com/products/product.asp?p_id=9475&format=2

Regards.