which sentence is correct?

  1. The theory of floating island, supported by minds competent to draw a conclusion, was abandoned.

  2. The theory of floating island, supported by minds that are competent to draw a conclusion, was abandoned.

They are both possible. The use of “are” in #2 seems to imply a recent abandonment, else it would be “were”.

“The theory of floating island” reads oddly in isolation. It might make more sense in the full context.

Appreciate your sincere answer.Thks, Dozy!

That’s a misquote:
“The theory of the floating island, and the unapproachable sandbank, supported by minds little competent to form a judgment, was abandoned.”
Chapter 2, ‘20,000 Leagues Under The Sea’ by Jules Verne.

verne.thefreelibrary.com/20000-L … ing+island

In 1866 a mysterious phenomenon was taking place regularly at sea around both Europe and America. Vessels had been met by “an enormous thing,” a long object, spindle-shaped, occasionally phosphorescent, and infinitely larger and more rapid in its movements than a whale.
One of many theories regarding what this might be was some sort of a floating island.