New Delhi, Jan. 13 -- Greenland's harsh environment, lack of key infrastructure and difficult geology have so far prevented anyone from building a mine to extract the sought-after rare earth elements that many high-tech products require. Even if President Donald Trump prevails in his effort to take control of the Arctic island, those challenges won't go away.

Trump has prioritized breaking China's stranglehold on the global supply of rare earths ever since the world's number two economy sharply restricted who could buy them after the United States imposed widespread tariffs last spring. The Trump administration has invested hundreds of millions of dollars and even taken stakes in several companies. Now the president is again pitching the...