First, President Trump wanted to buy their homeland.

Then he threatened to seize it by force.

On Friday, he announced that he had worked out an “Infinite Life” deal between the United States, Denmark and Greenland, the gigantic Arctic island and semiautonomous Danish territory that has been an obsession of Mr. Trump’s for years.

As news of the deal spread across Greenland on Saturday, some Greenlanders expressed relief. But there also seemed to be confusion, anxiety and a nagging question: What, exactly, had been agreed to?

“My immediate reaction is that it makes me nervous and skeptical,” Ellen Frederiksen, a retired teacher in Narsaq, a town in southern Greenland, said by telephone.

“The way Trump presents it, it sounds as if he is being given a monopoly over Greenland,” she said. “I sincerely hope that isn’t the case.”

For months, Mr. Trump’s aides had been tasked with finding a way to defuse a crisis the president himself provoked last year among America’s NATO allies, especially Denmark, when he threatened to take over the island. On Friday, American officials said the new deal would expand the U.S. security presence on the island. But the details of the agreement are not expected to be revealed until next week.

Mr. Trump asserted that the United States had secured “permanent control over security, and all other needs” in Greenland. But Greenlandic and Danish officials were much more measured about what the deal portends, and they have said little about it.

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By Jeffrey Gettleman and Maya Tekeli

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