Mr. Moon may see an opportunity in the surprise offer by the North Korean leader, Kim Jong-un, conveyed in person by Mr. Kim’s sister, to hold their first summit meeting in Pyongyang. Mr. Kim seized on Mr. Moon’s peace overtures before the Olympics to send his sister, Kim Yo-jong, to the opening ceremony and a large contingent of cheerleaders and athletes to the Games in Pyeongchang, South Korea.
But Mr. Moon also knows he must convince the Americans to give him a chance. In a sign of how hard that will be, and how deeply the United States and North Korea distrust each other, Mr. Pence, who was Washington’s envoy to the opening ceremony, and Ms. Kim would not even look at each other despite being seated only a few feet apart.
On Friday, Mr. Moon argued for a South Korean-brokered peace and for the United States-North Korea talks when he met with President Trump’s daughter Ivanka Trump, who arrived to attend the Games’ closing ceremony. He told her he wanted to improve ties “in parallel” with efforts to denuclearize the North.
Analysts said that once the Olympics ended, Mr. Moon would be left to sort out how much of the North’s so-called charm offensive, in which it refrained from provocations like missile tests, could last.
“South and North Korea used the Olympics to use each other,” said Yoo Dong-ryul, director of the Korea Institute for Liberal Democracy in Seoul. “The South was desperate to ease tensions. The North wanted to soften its image and weaken international sanctions. Now comes the hard part for Moon, after the Olympics.”
Without a solution to the nuclear issue, relations between the two Koreas “will eventually revert to the same crisis mode before the Olympics,” Mr. Yoo said.
So far, Washington and Pyongyang seem unable to talk. The North Koreans, including Mr. Kim’s sister, canceled a meeting with Mr. Pence in Pyeongchang after he refused to soften his criticisms of the North’s weapons and human rights.
North Korea seeks to be accepted as a nuclear power and win economic concessions in return for not advancing its nuclear programs any further, analysts say. But the United States insists it will never enter any serious negotiations or ease sanctions until the North commits itself to nuclear disarmament.
Over the weekend, Mr. Trump announced harsh new sanctions against North Korea and warned of tougher measures if the North fails to relinquish its nuclear arsenal.
The hard line out of Washington has stood in conspicuous contrast with the more conciliatory gestures from Seoul.
David Straub, a former American diplomat who is now a fellow at the Sejong Institute in South Korea, said there were rising frustrations in Washington that Mr. Moon was “apparently working at cross purposes with the Trump administration’s effort to apply ‘maximum pressure’ on North Korea.”
This “could result in a serious clash of wills between the two allied leaders,” Mr. Straub said. “The odds are increasing that relations between the two will worsen.”
Mr. Moon, a dogged advocate of dialogue with the North, had spent last year helplessly watching the Korean Peninsula edge toward a possible war, as the North test-launched intercontinental ballistic missiles and conducted its most powerful nuclear test, and Mr. Trump threatened to rain down “fire and fury” on North Korea.
Desperate to avoid a possible military conflict, Mr. Moon seized upon the Olympics to craft what some analysts called an “Olympic truce.”
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“It dramatically lowered the pitch of tension on the Korean Peninsula, replacing tests, threats and tweets with face-to-face talks, and it restored Seoul as a key player in the game,” said John Delury, a North Korea expert at Yonsei University in Seoul.
The question, Mr. Delury said, is the extent to which the Trump administration is willing to let South Korea be a mediator with the North, especially as expectations in Washington have dwindled that the Chinese president, Xi Jinping, can persuade the North to disarm.
Some Trump administration officials said the United States had to accept that there was now a viable diplomatic channel between North and South Korea, and figure out how to use it for American goals. The first step, according to these officials, is for the administration to settle on a more consistent message toward North Korea.
The United States has veered in recent weeks from expressing openness to diplomacy to reiterating threats of military action if the North does not curb its nuclear and missile programs. The net effect, according to analysts, has been to confuse both North and South Korea.
White House officials will also have to decide how to react if the inter-Korean diplomacy accelerates. The Trump administration is determined not to make undue concessions to North Korea, a mistake it says both Democratic and Republican presidents made. On the other hand, if the United States takes too hard a position, analysts said it could prompt Mr. Moon to conclude that the Americans will never come around.
That could prompt Mr. Moon to set off on his own, isolating the United States by driving away one of its most important allies in Asia, which could end up drifting closer to China.
“Once the ball gets rolling, it’s very hard to stop, in large part because Seoul and Beijing want talks so badly,” said Evan S. Medeiros, a managing director at Eurasia Group who w as a senior Asia adviser to President Barack Obama.
Analysts said Mr. Moon was trying to get that forward momentum by seizing on this chance to move the peninsula toward peace with South Korea in the driver’s seat.
“Moon Jae-in is not a naïve waif,” said Gordon Flake, the chief executive of the Perth USAsia Center at the University of Western Australia. “The South Koreans are desperately seeking to change the trajectory, which was leading to a conflict, and at the same time to do that while not giving too much space between them and the United States.”
For both the Trump and Moon administrations, the first big test will be what to do about joint United States-South Korean military exercises that were postponed during the Olympics and the Paralympics, which will take place from March 9 to 18.
North Korea has indicated that it will restart weapons tests if the drills resume, scuttling Mr. Moon’s efforts to broker a peace.
Conservatives in both South Korea and the United States fear that anything less than the full resumption of the war games would only advance the North’s ultimate goal of ridding the peninsula of the American military presence, which they say the South needs for protection.
But progressive South Koreans who support Mr. Moon would most likely see a push by Washington to resume the exercises as “throwing cold water over the South’s Olympic party,” and as an effort to derail Mr. Moon’s push for inter-Korean rapprochement, said Lee Byong-chul, senior fellow at the Institute for Peace and Cooperation in Seoul.
South Koreans who attended Olympic events were cautious about how much the Olympic diplomacy could accomplish.
Yeon Ju-lee, 21, who recently watched a joint Korean team in women’s ice hockey, said she felt relieved that the North’s participation in the Olympics was easing fears of possible war. But, she said, actual political unification “will take a long time.”
Another spectator, Lee Hae-man, 62, said recent developments were promising “because we are all one family together, North and South.”
I will feel very betrayed if Kim Jong-un goes back to missile tests,” he added.