The future of the cease-fire in Gaza, along with the territory’s long-term fate, hung in the balance on Tuesday as President Trump prepared to meet with King Abdullah II of Jordan amid a public spat between the American leader and Hamas.

Mr. Trump envisions taking over Gaza and expelling Palestinians to nearby countries, including Jordan. He has threatened to end American financial support for Jordan if the king refuses to accept that vision.

The dispute is one of several that imperil the fragile truce between Hamas and Israel. Hamas threatened to derail the agreement on Monday night, warning that it would delay the release of some hostages on Saturday if Israel did not send more aid to Gaza.

In response, Mr. Trump threatened “all hell” if every hostage was not released by the weekend. Hamas later softened its position, before the Israeli prime minister, Benjamin Netanyahu, warned on Tuesday night that Israel would resume fighting if hostages were not released on Saturday. He did not immediately clarify how many hostages had to be released, leaving himself with room to maneuver.

While the immediate crisis is still likely to be resolved soon, analysts said, another hurdle looms in March, when the cease-fire is set to elapse unless Hamas and Israel negotiate an extension.

“It’s likely that they will reach a compromise before Saturday,” said Ibrahim Dalalsha, director of the Horizon Center, a political research group in Ramallah, in the West Bank. “But this crisis is a prelude for a much bigger crisis that is coming in early March.”

All the major players have made negotiations harder.

Mr. Netanyahu has delayed the talks, wary of an extension that would allow Hamas to remain the dominant military force in Gaza.

Hamas, though nominally willing to share control with other Palestinian factions, has given no sign that it will disarm.

And Mr. Trump’s pronouncements — including his threats to expel Gaza’s residents — have angered Hamas and amplified the sense of chaos surrounding the negotiations.

Mr. Trump has repeatedly called for the United States to occupy and rebuild Gaza, threatening on Monday to withdraw financial support for Egypt and Jordan if they do not house Palestinians displaced by that effort.

Such a forced migration would destabilize both countries, and King Abdullah is expected to present alternatives to Mr. Trump.

Analysts are divided about whether Mr. Trump’s idea is serious, but the dispute highlights the growing unpredictability about Gaza’s future.

The current standoff stems partly from Hamas’s accusation that Israel has not upheld its promises for the first phase of the cease-fire. Israel was required to send hundreds of thousands of tents into Gaza, a promise that Hamas says Israel has not kept.

Speaking on the condition of anonymity to discuss a sensitive matter, three Israeli officials and two mediators said that Hamas’s claims were accurate.

But COGAT, the Israeli military unit that oversees aid deliveries, said in a written response that Hamas’s claims were “completely false accusations. Hundreds of thousands of tents have entered Gaza since the beginning of the agreement, as well as fuel, generators and everything Israel pledged.”

Regardless, officials and commentators say this dispute can be resolved relatively easily if Israel allows more aid to Gaza.

The more serious issue is the widespread perception that Mr. Netanyahu is undermining the negotiations over an extended truce.

Those talks were meant to begin early last week. Instead, Mr. Netanyahu delayed sending a team to Qatar, which is mediating talks, until early this week.

That delegation consisted of three officials who have not previously led Israel’s negotiating effort, according to five Israeli officials and an official from one of the mediating countries. And their mandate was only to listen, not to negotiate.

That created the perception that Mr. Netanyahu was playing for time rather than trying to extend the truce.

All of the officials spoke on the condition of anonymity to discuss the private talks more freely.

Asked for comment, Omer Dostri, a spokesman for the prime minister, said that Mr. Netanyahu was “working tirelessly to return all hostages held by the Hamas terrorist organization.” Mr. Dostri added that Israel would send negotiators to discuss the deal’s extension after Israel’s position had been set by the cabinet.

Mr. Netanyahu has often said that Hamas will not remain in power after the war. And key members of Mr. Netanyahu’s governing coalition have expressed a desire to resume the war to oust Hamas, despite calls from much of the Israeli public for an extension of the truce to free more hostages, even if it leaves the militant group in power.

A Hamas official, Mahmoud Mardawi, said that the group’s warning on Monday had been in response mainly to the disagreements over humanitarian aid. But analysts said that it was also an attempt to force Mr. Netanyahu to negotiate earnestly and was probably a reaction to Mr. Trump’s recent statements about depopulating Gaza.

Michael Milshtein, an Israeli analyst of Palestinian affairs, said, “There’s an anger among Hamas about the demands of both Netanyahu and Trump that Hamas will be kicked out of Gaza.”

“The announcement yesterday was a kind of a signal that, if you continue demanding this, there will be several dramatic crises,” Mr. Milshtein added.

Natan Odenheimer, Gabby Sobelman and Adam Rasgon contributed reporting.

Prime Minister Benjamin Netanyahu of Israel warned Hamas on Tuesday that if hostages were not released by noon on Saturday, the cease-fire in the war in the Gaza Strip would end and Israeli troops would resume “intense fighting.”

Mr. Netanyahu said that Hamas’s threat on Monday to postpone the next round of hostage releases amounted to a decision to violate the cease-fire agreement. Mr. Netanyahu did not specify how many hostages would have to be freed to stop a renewed war. The prime minister’s office declined to confirm how many hostages he was referring to.

His statements closely echoed President Trump’s ultimatum on Monday evening to Hamas that said if all remaining Israeli hostages were not released from Gaza by 12 o’clock on Saturday, then the cease-fire agreement with Israel should be canceled and “all hell is going to break out.”

Originally, three Israelis were to be freed this week in the latest hostage-for-prisoner exchange as required by the cease-fire deal to end the war that began with the Hamas-led attack on Israel on Oct. 7, 2023. Most of the exchanges so far have taken place on Saturdays.

So far, 16 of 33 Israeli hostages set to be released in the first phase of this cease-fire have been freed. About 60 other hostages, some of whom are believed to be dead, were to be released later this spring under a second phase of the deal.

In a video posted after a four-hour meeting with his security cabinet on Tuesday, Mr. Netanyahu said that he and his top advisers had been shocked by the emaciated appearances of three Israeli men who were freed last Saturday.

“The decision I passed in the cabinet, unanimously, is this: If Hamas does not return our hostages by Saturday noon, the cease-fire will end, and the I.D.F. will resume intense fighting until Hamas is decisively defeated,” Mr. Netanyahu said in the video, referring to the Israel Defense Forces.

Mr. Netanyahu also reiterated his order on Monday night to reinforce troops in and around Gaza but did not specifically say they were planning to recapture territory from which Israel had recently withdrawn. “This operation is currently underway,” Mr. Netanyahu said. “It will be completed as soon as possible.”

In a statement Tuesday evening, the Israeli military said it was mobilizing reservists among the troops that would be part of the operation.

In its threat on Monday to delay the next round of hostages to be released, Hamas accused Israel of violating parts of the cease-fire agreement, including by slowing sufficient humanitarian aid in and around Gaza. Israel has denied the claim.

On Tuesday night, Hamas said in a statement that it was committed to the agreement “as long as the occupation adheres to it.” The statement added, “We emphasize that the occupation is the party that has not fulfilled its commitments, and it bears responsibility for any complications or delays.”

But more broadly, Hamas has been infuriated by Mr. Trump’s repeated proposal to relocate about two million Palestinians from Gaza and rebuild the war-torn territory into the “Riviera of the Middle East.”

Mr. Trump also said the enclave’s residents would not be allowed to return once they leave. The forced deportation of a civilian population is a war crime under international law.

After Hamas’s 2023 attack, in which about 1,200 people were killed and 250 others abducted to Gaza, Israel had bombarded the territory, aiming to decimate the militants. About 48,000 Palestinians have been killed during the fighting, according to Gazan health officials, who do not distinguish between combatants and civilians. Millions of Palestinians have also been displaced in the enclave, and many are returning to neighborhoods in ruins.

Mr. Trump has said he would “make a deal” with Jordan and Egypt to take in the Palestinians — a move that analysts say would destabilize both countries, and that has been flatly rejected by their leaders.

On Tuesday, Egypt said in a statement released by a spokesperson for the foreign ministry that it intended to present to the United States a “comprehensive vision for the reconstruction of the Gaza Strip” that “ensures the Palestinian people remain in their homeland.” Egypt’s president, Abdel Fattah el-Sisi, earlier scrapped tentative plans to visit Mr. Trump in Washington until further notice.

King Abdullah II of Jordan, meeting with Mr. Trump at the White House on Tuesday, said: “I think the point is, how do we make this work in a way that is good for everybody. Obviously we have to look at the best interests of the United States, of the people in the region, especially to my people of Jordan.”

He said that Jordan was willing to take in 2,000 Palestinian children with cancer or very ill “right away.” Mr. Trump called it “a beautiful gesture.”

Mr. Trump’s insistence that the United States had the authority to “take” Gaza and that other countries in the region would absorb the Palestinians who live there has ignited widespread anger among Arab states in the Middle East and even some American allies in Europe. It also raised concerns in Israel that Mr. Netanyahu now had a political escape hatch to end the cease-fire negotiations and instead return to war.

Hamas, in a statement on Tuesday, called Mr. Trump’s broader proposal of removing Gazans “ethnic cleansing.”

Responding to Mr. Netanyahu on Tuesday, a group representing the hostages’ families urged him to continue the diplomatic talks.

“You made the decision to bring all our hostages home through an agreement,” the Hostages and Missing Families Forum said in a statement. “We must not go backwards. We cannot allow the hostages to waste away in captivity.” It concluded: “Complete the full negotiations immediately, and bring back every last hostage with utmost urgency.”

The first phase of the cease-fire deal was struck last month between Israel and Hamas and was set to expire on March 2. Some Israeli officials have opposed a second stage of the deal that would include talks on how to fully end the war, urging the government instead to have the military continue fighting Hamas.

“Either all of the hostages are released by Saturday — no more phases, no more games — or we will open the gates of hell upon them, and this means no electricity, no water, no fuel, no humanitarian aid,” Bezalel Smotrich, the ultranationalist finance minister, said on social media after the security cabinet meeting on Tuesday.

Others want the negotiations, which have been held in Doha with the United States, Egypt and Qatar serving as intermediaries, to continue in order to usher in a lasting peace.

“Netanyahu, go to Doha,” the opposition leader Yair Lapid wrote on social media earlier on Tuesday. “Bring everyone home. Time is running out.”

The cease-fire agreement was already on shaky ground before Mr. Netanyahu’s warning on Tuesday, along with the dueling threats from Hamas and Mr. Trump leading up to it. With Mr. Netanyahu’s right-wing government in jeopardy if the war ends with Hamas still in control in Gaza, it has been widely expected in Israel that he would try to delay moving toward a permanent cease-fire.

Already, some world leaders were bracing on Tuesday amid the signs that it was falling apart.

“We must avoid at all costs resumption of hostilities in Gaza that would lead to immense tragedy,” urged António Guterres, the United Nations secretary general. “I appeal to Hamas to proceed with the planned liberation of hostages. Both sides must fully abide by their commitments in the cease-fire agreement & resume serious negotiations.”

Reporting was contributed by Patrick Kingsley, Gabby Sobelman, Natan Odenheimer, Ephrat Livni and Aaron Boxerman.

President Trump has vowed to end the fighting in Ukraine. Just how he could do that remains unclear, given that President Vladimir V. Putin of Russia believes he is winning. But in his own blunt way, Mr. Trump has opened up the possibility of some kind of negotiations for a cease-fire.

If a deal is to be reached, analysts say, Mr. Trump is likely to ask Europe to put it in place and to take responsibility for Ukraine, wanting to reduce the American commitment.

But a key question remains: How to secure what is left of Ukraine and prevent Mr. Putin from restarting the war, even several years from now?

The prospect of a deal has accelerated debate over so-called European boots on the ground to keep the peace, monitor a cease-fire and help deter Russia from future aggression. The question is whose boots, and how many, and whether Mr. Putin would ever agree.

It is a topic sure to be a central focus for discussion this week at the annual Munich Security Conference, which Vice President JD Vance and Secretary of State Marco Rubio are scheduled to attend.

Some European countries, among them the nations of the Baltics, as well as France and Britain, have raised the possibility of including some of their own troops in a force in Ukraine. Senior German officials have called the idea premature.

Short of NATO membership for Ukraine, which seems unlikely for many years, the idea of having large numbers of European troops from NATO nations seems reckless to many officials and analysts.

Without clear American involvement in such an operation — with American air cover, air defenses and intelligence, both human and technical — European troops would be at serious risk from Russian probing and even attacks.

President Volodymyr Zelensky of Ukraine has indicated that he is ready for serious talks about a deal to end the war, so long as his allies provide security guarantees, not just assurances.

In the absence of NATO membership, which he prefers, Mr. Zelensky has spoken of as many as 200,000 foreign troops on the ground in Ukraine. But that is nearly three times the size of the entire British Army and is regarded by analysts as impossible.

A senior European official said that the continent doesn’t even have 200,000 troops to offer, and that any boots on the ground must have American support, especially faced with the world’s second-largest nuclear power, Russia. If not, they would be permanently vulnerable to Russian efforts to undermine the alliance’s political and military credibility.

Even a more modest number of European soldiers like 40,000 would be a difficult goal for a continent with slow economic growth, troop shortages and the need to increase military spending for its own protection. And it would likely not be enough to provide realistic deterrence against Russia.

A real deterrent force would typically require “well over 100,000 troops assigned to the mission” for regular rotations and emergencies, said Lawrence Freedman, emeritus professor of war studies at King’s College London.

The danger would be a policy of what Claudia Major, a defense expert with the German Institute for International and Security Affairs, called “bluff and pray.”

“Providing too few troops, or tripwire forces without reinforcements, would amount to a bluff that could invite Russia to test the waters, and the NATO states would hardly be able to counter this,” she wrote in a recent paper with Aldo Kleemann, a German lieutenant colonel, about how to secure a Ukrainian cease-fire.

That is why Poland, which neighbors Ukraine and is deeply involved in its security, has so far dismissed taking part in such a force.

“Poland understands it needs the United States to be involved in any such proposal, so wants to see what Trump wants to do,” said Alexandra de Hoop Scheffer, acting director of the German Marshall Fund. “It wants guarantees from Trump that there will be U.S. security help to support Europeans in the front line.”

But that is not at all clear, she said. “Trump will do the deal and look for a Nobel Prize and then expect the Europeans to pay for it and implement it,” she said.

Still, European “willingness to be ready to do something useful” for Ukraine without the Americans will be important to ensure that Europe has a seat at the table when negotiations finally happen, said Anthony Brenton, a former British ambassador to Russia.

Mr. Putin’s stated aims have not changed: the subordination of Ukraine into Russia, a halt to NATO enlargement and a reduction in its forces, to force the creation of a new buffer zone between the Western alliance and the supposed Russian zone of influence.

Nor is it likely that Russia would agree in any deal to the deployment of NATO or NATO-country forces in Ukraine in any case, even if they were ostensibly there to train Ukrainian soldiers. The Russian Foreign Ministry has already stated that NATO troops in Ukraine would be “categorically unacceptable” and escalatory.

Mr. Freedman described three possible models — peacekeeping, tripwire and deterrence — all of which have significant flaws.

Peacekeepers, intended to reinforce agreed-upon cease-fires and keep belligerents apart, are lightly armed for self-defense and often contain troops from many countries, usually under the United Nations. But given that the line of contact in Ukraine is some 1,300 kilometers, or more than 800 miles, he said, “a huge number of troops” would be required.

Before the 2022 invasion, there was an international monitoring mission of the Organization for Security and Cooperation in Europe, with Russian agreement, to supervise observance of a much shorter cease-fire line in eastern Ukraine. It was a failure, said Michael Bociurkiw, who was its spokesman from 2014 to 16.

“The Russians did everything to block the mission,” he said. “They pretended to cooperate, limited access and hid various nefarious activities. When things don’t work the way they want, they shut it down.”

A tripwire force is essentially what NATO has deployed in eight member countries closest to Russia. There are not enough troops to stop an invasion or to be seen by Moscow as provocative, but the concept only works if there is a clear, unbreakable link between the troops on the ground and larger reinforcements committed to fight once the wire is tripped.

But there are always doubts about the absolute nature of that guarantee. And an attacking force would gain significant territory before any reinforcements arrive, which is why NATO itself is increasing the size of its tripwire forces from battalion to brigade level, to enhance deterrence against a newly aggressive Russia.

The third type, a deterrent force, is by far the most credible, but needs to be very large and well-equipped, and would require up to 150,000 well-equipped troops, plus significant commitments of air defense, intelligence and weaponry — and American help with the strategic enablers Europe continues to lack, from air transport to satellites to missile defense.

But it would be hard to imagine that Russia would agree to any such force for precisely the same reasons that Mr. Zelensky wants one, Mr. Freedman said.

So the best answer for the near future after a potential cease-fire may be some version of the “porcupine” model: giving the Ukrainian military enough weaponry, resources and training — including by Western forces — to convince Russia not to try again. Such a commitment, however, would have to be for the long term.

But first Ukraine must stop Russia’s slow advance in the east and Mr. Putin must be convinced to end the war, with further battlefield losses and economic pressure. How to do that will be one of the main tests for Mr. Trump if he is to have success in ending the killing, as he promises to do.

Luis Alberto Castillo, a father of one from Venezuela, entered the United States on Jan. 19, one day before Donald Trump became president for a second term — swept into office on a promise to treat undocumented migrants with a heavy hand.

By Feb. 4, Mr. Castillo was on a plane to a U.S. Navy base at Guantánamo Bay in Cuba, best known for a detention center that has long held terrorism suspects accused of launching the deadliest attack on American soil.

That day, the Department of Homeland Security declared that those who had been transferred to the island represented “the worst of the worst” and were all members of a Venezuelan criminal group, the Tren de Aragua.

But in an interview from her home in Colombia, Mr. Castillo’s sister Yajaira Castillo said her brother was not a gang member to be feared, but rather an everyday Venezuelan who had fled his country because of its economic crisis.

She broke down repeatedly during the conversation, crying as she described her pain and confusion around her brother’s situation.

“My brother is not a criminal,” she said. “This is all discrimination and xenophobia, just because he’s Venezuelan.”

Given that Mr. Castillo had spent such little time in the United States, she questioned how the U.S. government could have determined that he was a member of the Tren de Aragua, or was worthy of such harsh treatment.

After he entered the United States, officials suspected Mr. Castillo of being a member of the gang because of his tattoos, according to two people familiar with his case who were not authorized to discuss it publicly.

He did not admit to being a part of the criminal group. Later, officials assessed his tattoos and interviewed him, and they found that he did not appear to have ties with the Tren de Aragua, the people said.

In an initial email, a spokeswoman for the Department of Homeland Security said that Mr. Castillo was in the United States illegally and had final deportation orders issued by a federal judge.

“This administration abides by the rule of law,” said the spokeswoman, Tricia McLaughlin. “During further assessment, intelligence officers could not definitively determine whether the individual is or is not a confirmed member of TDA,” or Tren de Aragua. “He may very well be a member of this vicious gang. He may not be.”

In a follow-up message, Ms. McLaughlin said that the department had received new information that Mr. Castillo was a member of the gang. She did not provide evidence.

“TDA is a pathetic gang for human trafficking, drug trafficking and kidnap for ransom among other heinous crimes,” Ms. McLaughlin said. “The New York Times is more interested in writing sob stories about its disgusting members than justice for its victims.”

In the interview, Mr. Castillo’s sister shared a screenshot that indicated that he had not tried to evade authorities when he entered the country. The image included details of an appointment her brother had secured to present himself at the border to claim asylum on Jan. 19 at 7 a.m.

The Biden administration had set up a system for migrants to make these claims and enter the United States legally through an app called CBP One. Mr. Trump ended the program on Jan. 21.

Over the last week, the United States government has sent more than 80 men to Guantánamo Bay as part of a larger plan by the Trump administration to hold as many as 30,000 migrants at the Naval base. So far, all the detainees are believed to be Venezuelans. Some are being held in a prison building on the base, while others are detained at the Migrant Operations Center, a dormitory-style facility. (Mr. Castillo is being held at the migrant center, according to D.H.S.)

U.S. troops are building a tent camp at the site in an effort to vastly expand its capacity to hold detained migrants.

The Pentagon has described the first few Guantánamo arrivals as “high-threat illegal aliens” and their detention at the base as “a temporary measure.” But the administration has not released any details proving the men have criminal records or described how officials determined they were a threat.

Mr. Castillo is believed to be among the first 10 men who were sent there from El Paso, Texas, on Feb. 4, because his sister recognized him in a photograph of migrants being sent to Guantánamo that was published on social media by Kristi Noem, the new head of the Department of Homeland Security.

The image, in which Mr. Castillo’s head is bowed low and an officer in camouflage and gloves holds his back, was later shared widely on social media. Ms. Castillo happened upon it on TikTok.

The Tren de Aragua, a multinational group born in Venezuela’s Aragua state, has expanded to other parts of Latin America and as far as the United States. The Trump administration recently began the process of designating the group as a foreign terrorist organization.

Mr. Castillo has a Michael Jordan tattoo on his neck, which his sister believes border authorities took as a sign that he was a member of the gang. In one of the last messages he sent to her before detention, Mr. Castillo said that he had made it to the border and that officials “treated him badly because of the tattoos,” she said.

Ronna Rísquez, a Venezuelan investigative journalist who has written a book about the criminal group, said the authorities would be wrong to assume that someone with a Michael Jordan tattoo is a member.

Some individuals in the Tren de Aragua might wear the symbol, she said, but this has more to do with the fact that basketball — which can be played with limited resources — has become enormously popular in poorer parts of Venezuela over the last few decades.

“The passion for Michael Jordan, because he is the ultimate symbol of basketball, has been around for generations,” she said.

Ms. Castillo said the tattoos were simply part of her brother’s look, which often included shorts and sneakers and Jordan gear. She shared a past photo of him wearing a sweatshirt with the “jumpman” logo associated with Mr. Jordan.

She said she believed her brother’s affinity for Mr. Jordan had turned him into a “guinea pig” for the Trump administration’s expanding deportation program.

Like many Venezuelans, Mr. Castillo had left his country years ago and was living in Colombia, washing cars. His sister said he was barely scraping by and had left for the United States in the hopes of making more money “to give everything to his son, to work and work for his son.”

His journey to the U.S. southern border began in late 2023, she said. With limited funds, it took him until January of this year to reach Texas. There, she said, he was apprehended by the authorities.

A review of public police records indicates that Mr. Castillo was not convicted of any crimes while in Colombia.

Mr. Castillo is the eighth of nine siblings; four live in Colombia and four others in Venezuela. He will turn 30 on Feb. 23.

The Guantánamo base is best known for its post-9/11 detention facility, operated by the Pentagon. It today holds 15 foreigners as wartime prisoners, separately from the Venezuelans being held there in migrant detention.

Ms. Castillo said she had known little about Guantánamo, only that it was “a high-danger cell for major terrorists” and that once sent there, prisoners “have no rights, that it’s for the worst criminals.”

All she wanted now for her brother, she said, “is for him to be returned.”

“I don’t want him in the hands of any government,” she continued. “What I want is for him to be returned to us.”

Carol Rosenberg contributed reporting from Florida and Simón Posada from Bogotá, Colombia.

He was just off the autobahn, beaming at a rush-hour crowd, and Friedrich Merz’s mind had steered to speedometers.

“If you’ve recently bought a new car, have you noticed what kind of automatic systems it’s equipped with now?” the man in the driver’s seat to be Germany’s next chancellor asked on Friday afternoon. “If you drive two kilometers per hour too fast, the thing starts beeping.”

Those beeps are the product of a European Union regulation. For Mr. Merz, they were a timely and tidy example of the government intrusions that he blames for stymying the German economy and frustrating its citizens.

They were also a handy segue into the issues Mr. Merz hopes to lounge in, like a nice leather captain’s chair, over the final stretch before Germany’s parliamentary elections on Feb. 23.

Mr. Merz and his party, the conservative Christian Democrats, endured two nervous weeks after he took a political gamble and broke a decades-old taboo by voting with rivals on the far right in a failed bid to toughen migration laws.

Outcry followed. Rival candidates sensed an opening. But polls taken since the hubbub indicate that Mr. Merz has emerged relatively unscathed. Even if he is now seen as a more polarizing figure, the former businessman and longtime conservative stalwart seems once again to be cruising toward the chancellorship.

Mr. Merz is refocusing his stump speech on E.U. regulation, federal red tape, work ethics, energy costs and other ingredients of what business leaders call the components of a German competitive crisis. He is telling voters that an increasingly volatile world needs a stronger and steadier chancellor at the wheel than Germany’s current leader, Olaf Scholz, of the center-left Social Democrats.

The price of Mr. Merz’s fraught detour into immigration politics, and the benefits of turning the campaign back to more friendly and familiar turf, were on display during the stop he made last week in the small western town of Stromberg, where the only restaurant open for lunch downtown was an ice-cream parlor.

About 1 in 6 workers are employed in manufacturing in the heavily wooded, wine-producing state, Rhineland-Palatinate. The state’s economy shrank by nearly 5 percent in 2023, government statistics show. A party official said the venue had been chosen in part because of its highway proximity, easy for attendees and Mr. Merz to reach by car. Many of the attendees said they had driven in from out of town.

They were greeted, as is increasingly the case for Mr. Merz these days, by protesters. Since Mr. Merz broke the taboo of working with the Alternative for Germany, or AfD, parts of which are classified by German intelligence as extremist, outraged voters have taken to following him from campaign stop to campaign stop. Some accuse Mr. Merz of working with Nazis. Others call him one.

“He is not suitable as a chancellor,” said Walter Witzke, one of about 150 protesters who gathered in near-freezing temperatures outside the gymnasium in Stromberg where Mr. Merz spoke. “He has made the biggest mistake by voting with the AfD now.”

Mr. Witzke carried a sign that read “five minutes until 1933,” a reference to the dawn of Germany’s Nazi era. His wife, Heike Witzke, who joined him at the protest, said she feared for the nation’s democracy — and was saddened by a the backlash against immigrants. “You should never give up hope, but at the moment it is very, very bad,” she said.

Ms. Witzke said most of her friends came from abroad, and that she bakes cookies to celebrate holidays with Muslim neighbors. “It works, we have no problems at all,” she said.

Inside the gymnasium, where an 11-piece jazz band warmed up the crowd with lounge-act hits, Mr. Merz’s supporters were far more concerned with immigrants who receive social assistance.

“This poverty migration, we are simply overburdened by it now,” said Elke Müller, an executive at a cosmetics company.

She said she was a fan of Mr. Merz and that it was acceptable for him to push the tougher immigration measures that the AfD voted for. “I think he has economic expertise,” she said. “He can assert himself. And you can rely on him. And he is trustworthy. And I think he is the right man for the time we have now.”

Polls suggest a plurality of voters agree. They show Mr. Merz and his party hovering around 30 percent support in the German electorate, a relatively low number for a would-be chancellor, but well ahead of his closest rival.

Some surveys suggested that the migration gambit cost Mr. Merz slightly with voters. Others found a slight gain. None suggest it fundamentally altered the race. According to the latest Politbarometer survey, 30 percent of Germans say they will vote for Mr. Merz’s party, 1 percentage point more than in the end of January. The AfD sits second, with the Social Democrats and Greens lagging behind.

Mr. Merz addressed the migration-vote controversy near the end of his speech, which stretched more than an hour. He defended his decision but vowed to never form a government with the AfD — a distinction that supporters like Ms. Müller said was important to them.

Mr. Merz called migration one of the critical issues facing the country, but he leaned more into his economic pitch, vowing to reduce taxes and regulations for businesses and to build new nuclear power reactors to reduce energy costs.

The local candidate who introduced Mr. Merz apologized that fire safety regulations had capped the number of attendees. She acknowledged the protesters, calling them a sign of democracy.

Mr. Merz told his audience that this month’s election would be a “directional election,” for Germany and the world.

“Perhaps we should take a quick look across national borders and take a brief moment to consider the situation around us,” he said at one point. He then listed global challenges, including “the war in Ukraine, an increasingly aggressive China, major problems in the cohesion of the European Union” and the new administration of President Trump.

Amid those challenges, he asked, “Where is Germany, actually?”

As Mr. Merz wrapped up, the last daylight faded over the autobahn. Cheers grew in the gymnasium. Outside, a few residents walked their dogs, gazing with befuddled expressions into the glow of the rally. Police officers huddled in twos and threes.

The protesters had all cleared out.

Nations targeted by President Trump’s tariffs on steel and aluminum decried the measures as unfair Tuesday and threatened to retaliate in a growing trade dispute that threatens to further roil economic markets and strain the United States’ relations with major allies.

Prime Minister Justin Trudeau of Canada, the United States’ biggest supplier of both metals, described Mr. Trump’s 25 percent tariffs on all steel and aluminum imports as “unjustified” and “unacceptable.”

Mr. Trudeau said he hoped to avert the imposition of the tariffs — which will not go into effect until March 12 — by highlighting their “negative impacts on Americans and Canadians.”

He added that Canada would “also be working with our international partners and friends” to pressure the Trump administration. Asked whether he was ready to respond with countertariffs, Mr. Trudeau said Canada “will stand up strongly and firmly if we need to.”

European countries also warned of retaliation in response to Mr. Trump’s tariffs. Ursula von der Leyen, the president of the European Commission, called the levies “unjustified.’’

Leaders in Mexico said the tariffs were “unjustifiable” and risked undermining the economic integration of the North American market that was forged over the past two generations.

Noting that the United States has a trade surplus with Mexico in steel and aluminum, Marcelo Ebrard, the country’s secretary of the economy, said he planned to appeal to Mr. Trump’s common sense to try to fend off the tariffs.

“We take his word for it — common sense — not shooting ourselves in the foot, not destroying what we have built in 40 years,” Mr. Ebrard told reporters.

Mr. Trump’s announcement of the tariffs on Monday took leaders in Canada and Mexico, the United States’ biggest trading partners, especially by surprise. Last month, the president threatened 25 percent tariffs on all imports from both nations, prompting them to warn of retaliation.

But a trade war was averted at the last minute when Mr. Trump and the leaders of both countries reached separate deals that yielded a 30-day postponement of the tariffs last week. Both agreements involve beefing up security at the borders the United States shares with Mexico and Canada.

The sudden imposition of tariffs on Canada’s steel and aluminum weeks before the end of the reprieve left political and business leaders in the country feeling whipsawed. Canada’s economy is much smaller than the United States’ and would suffer significantly from the tariffs. It now faces a repeat of the down-to-the-wire negotiations that have dominated news coverage in recent weeks.

The tariffs are likely to deepen Canada’s mistrust of the Trump administration, heighten feelings of betrayal among Canadians and further fuel discussions about the need to strengthen ties to countries other than the United States.

Mr. Trump revealed that he was prepared to impose 25 percent tariffs on all steel and aluminum imports into the United States on Sunday while speaking to reporters aboard Air Force One en route to the Super Bowl.

Canada was the biggest supplier of steel to the United States last year, followed by Brazil and Mexico. It is also a major supplier of aluminum to the United States.

Canada’s steel industry, which employs 23,000 people, exports almost all of its products to the United States. There is little demand for Canadian steel in other markets, which are increasingly dominated by China.

The aluminum sector, which employs 9,000 people in Canada, is more competitive globally, though the United States is its major buyer.

During his first term, Mr. Trump levied tariffs on steel and aluminum imports from all over the world, angering Canada, Mexico and other allies. He eventually exempted major steel-producing countries like Brazil, South Korea and Australia in exchange for quotas limiting their exports to the United States. He removed the barriers on steel and aluminum for Canada and Mexico with the signing of a revised trade agreement among the three countries.

This time, Mr. Trump has laced tariff threats against Canada with talk of annexing the country and turning it into the 51st state. While the government of Prime Minister Trudeau initially dismissed Mr. Trump’s comment as a joke, he told business leaders on Friday that he regarded the threat as real.

Mr. Trump, in an interview with Fox News on Sunday, said that he was serious about annexing Canada.

On Tuesday, Mr. Trudeau pushed back against Mr. Trump.

“This is a moment of pride,” Mr. Trudeau said, speaking in Paris where he was attending a conference on artificial intelligence. “This is a moment of pulling together. This is a moment for Canada to be solid in our identity.”

Mexico was the third largest supplier of steel to the United States in 2024, according to the American Iron and Steel Institute.

A spokesman for the Mexican steel trade association, Canacero, said it would wait to assess the impact on the industry, which is concentrated in northern Mexico and supplies steel for everything from cars to washing machines to construction materials.

When Mr. Trump imposed tariffs in his first term — 25 percent on steel and 10 percent on aluminum — Mexico retaliated with countertariffs on key American products, including pork, apples, cheese, cranberries, bourbon whiskey and Harley Davidson motorcycles, according to Valeria Moy, the general director of the Mexican Institute for Competitiveness.

The tariffs were lifted about a year later with the signing of the revised free trade agreement among the United States, Canada and Mexico.

While the short-lived tariffs on steel and aluminum did not lead to significant gains for American manufacturers, Ms. Moy said that it was not surprising that Mr. Trump was considering turning to them again.

“It represents a win that’s easy for Trump to communicate,” Ms. Moy said. “It benefits — in the short term — an industry that’s become emblematic in the United States.”

Brazil exported $4.5 billion worth of steel to the United States last year. The United States was by far the largest international market for Brazilian steel, but that represented only about 11 percent of Brazil’s steel sales, most of which are domestic.

The last time Mr. Trump implemented tariffs against steel imports, he largely exempted Brazil. Experts believed at the time that because the United States had a trade surplus with Brazil, Mr. Trump did not see it necessary to target the country with tariffs, said Carla Beni, an economist who tracks international trade at the Fundação Getulio Vargas, a university in Brazil.

Tariffs on Brazilian steel could backfire on the United States, Ms. Beni said. Brazil imports more than $1 billion a year in coal from the United States — much of it to make steel — so harming the Brazilian steel industry could hit the American coal industry, she said.

“This could be very interesting for Brazil at the negotiating table,” she said. “Because if I produce less steel, I’m going to need less coal.”

For years, the Palestinian administration in the occupied West Bank has doled out hundreds of millions of dollars in stipends to the families of Palestinians jailed or killed by Israel — including those involved in violent attacks.

The United States and Israel have long condemned the payments and pressured the Palestinian Authority to end them. And on Monday, the Authority announced that it was backing away from the practice — a shift that analysts saw as an attempt to curry favor with President Trump and bring much-needed foreign aid into Palestinian coffers.

Palestinian officials, speaking on condition of anonymity to discuss a sensitive issue, said the move was aimed to bring the Palestinian administration into compliance with American law and to allow for more foreign aid to flow. A U.S. law banned direct American economic assistance to the Palestinian Authority as long as it carried out the practice.

The ban has only deepened the economic distress of the cash-strapped Palestinian Authority in recent years and it has increasingly struggled to make ends meet and to pay its employees’ monthly salaries.

Mahmoud Abbas, the aging Palestinian Authority president, issued a decree on Monday night that overhauled the payment system. The stipends have been one of the most emotionally charged issues in Palestinian politics.

A body set up to manage social welfare payments to needy Palestinians, known as the Palestinian National Economic Empowerment Institution, said in a statement that the families of prisoners would receive funds based only on financial needs and social welfare criteria, “without regard to political affiliations or past actions.”

The law means that the families of prisoners would still be eligible for social welfare payments as long as they demonstrate a financial need, as opposed to being compensated for fighting against Israel’s rule.

The new system would abide by 43 internationally recognized criteria for assessing social welfare needs, the statement said.

Both U.S. and Israeli officials will closely monitor the implementation of the new policy to see whether it leads to a genuine shift.

Palestinians were quick to criticize Mr. Abbas’s decision. Many in the West Bank and Gaza view those imprisoned by Israel as either the victims of fundamentally unjust Israeli military courts or freedom fighters who fought back against their occupiers.

But Mr. Abbas is gambling on a new beginning with Mr. Trump after years of bad blood, and he is hoping for a muted domestic response, said Ibrahim Dalalsha, a Palestinian political analyst.

“This is the Trump effect. The Palestinian Authority wants to start off well with Trump,” Mr. Dalalsha said in a phone interview.

Since the U.S. election in November, Mr. Abbas’s government has sought to rebuild its relationship with the American president after his tumultuous first term. But Mr. Trump’s recent insistence that the roughly two million Palestinians should be transferred out of the Gaza Strip has already added new strains.

During his first term, Mr. Trump outraged the Palestinian leadership by moving the U.S. Embassy in Israel from Tel Aviv to the contested capital of Jerusalem, freezing most aid and peddling a peace plan they saw as deeply slanted in Israel’s favor.

Israel has argued that paying benefits to the families of prisoners who have been involved in deadly stabbings, shootings and suicide bombings against Israelis creates a financial incentive for terrorism. It has labeled the policy a “pay-for-slay” arrangement, in which Palestinians with longer sentences get higher stipends, effectively rewarding people for committing deadlier attacks in Israel’s view.

In response, Israel has withheld funds from the Palestinian Authority, often more than $100 million each year. The money is drawn from tax revenues Israel collects on behalf of the Palestinian administration.

In 2018, Mr. Trump signed the Taylor Force Act, which ended economic assistance that directly benefited the Palestinian Authority as long as it continued to disburse the stipends. In its statement, the new Palestinian welfare institution said the reform “aligns directly with the goals of the Taylor Force Act.”

The Israeli government quickly dismissed Mr. Abbas’s announcement as a sham, saying it would not end the practice of paying the families of prisoners.

“This is a new deception scheme by the Palestinian Authority, which intends to continue paying terrorists and their families through alternative payment channels,” said Oren Marmorstein, a spokesman for the Israeli foreign ministry.

Mr. Abbas’s decree was unlikely to immediately lead American aid to begin flowing again to the Palestinian Authority. Mr. Trump has yet to say publicly whether he is willing to support the Palestinian Authority.

And other legal hurdles would remain, including the extended process of certifying that Mr. Abbas’s government is in compliance with the Taylor Force Act.

If the Palestinian Authority enforces the changes, it would be a remarkable about-face for Mr. Abbas, who had previously insisted that he would never give up on the payments. In the past, he has gone so far as to say that even if the Palestinian Authority was running out of money, he would spend whatever remained on the stipends.

In late January, Hussein al-Sheikh, a top adviser to Mr. Abbas, informed Steve Witkoff, Mr. Trump’s Middle East envoy, that the Palestinian Authority was prepared to move ahead with revising its prisoner payments system, according to the Palestinian official and another diplomat.

The shift immediately prompted criticism in the West Bank, where the Palestinian Authority administers some areas, including major Palestinian cities. Hamas and Islamic Jihad, Mr. Abbas’s rivals, also condemned the decision.

Qadura Fares, the Palestinian Authority’s commissioner for prisoners’ affairs, called on Mr. Abbas to “immediately retract” the decree during a news conference on Tuesday.

“This move is deeply wrong,” said Esmat Mansour, a former prisoner who said he had served 20 years in prison for involvement in a stabbing attack against an Israeli. “The prisoners are icon. They are the ones who have sacrificed for our freedom.”

Natan Odenheimer and Fatima AbdulKarim contributed reporting to this article.

Italian police officers on Tuesday arrested 181 people believed to be affiliated with the Cosa Nostra, as the Sicilian mafia is known, dealing what officials said was an “important blow” to a criminal organization that has held the region in its grip for generations.

“Cosa Nostra is far from dead,” even after years of being targeted by prosecutors and police raids, Domenico La Padula, a lieutenant colonel with the Carabinieri police who oversaw the investigation, said in a telephone interview. The investigation, he said, showed that the group had reorganized and “had found new energy and new strength,” by enlisting new recruits and setting aside differences to instead focus on profiting from new criminal ventures, like online gambling.

The organization had been able to survive because it remained “strongly tied to the rules of its founding fathers and its ancient rituals,” even as it modernized, the Carabinieri said in a statement — for example, by using encrypted smartphones that “limited the need for traditional meetings and gatherings to the bare minimum.”

Tuesday’s arrests were carried out in Palermo, the Sicilian capital, and neighboring towns, and involved about 1,200 Carabinieri officers. Video released by the police showed hundreds of officers waiting for orders to enter homes as helicopters patrolled the skies during overnight raids. The arrests, which came after two years of investigations, covered a range of charges, including mafia affiliation, drug trafficking, extortion and attempted murder.

Even though police and prosecutors were able to restrict the Cosa Nostra’s activities for years, the Carabinieri said in a statement, the group had not lost its grip and was still “well anchored in its territory over which it exercises constant control, significantly affecting the economic fabric through extortion and the imposition of products.” It “used force, when it saw fit,” and had an ample supply of weapons.

For decades the Cosa Nostra was dominated by the so-called Corleonesi crime family — associated with the town of Corleone, a name first made famous by Mario Puzo’s book, “The Godfather” — whose affiliates included Matteo Messina Denaro, a top boss who died in 2023. But investigators said that Palermo had regained primacy within the organization, and its “clans have become central to the dynamics of Cosa Nostra,” Lt. Col. La Padula said.

The investigation also found that different clans in Palermo had opted, for now, to set aside differences to achieve a common goal: making money. The Cosa Nostra also found synergy with the ’Ndrangheta, the dominant organized crime group in Calabria in southern Italy, when it came to drug trafficking.

Drug dealing also allowed the Cosa Nostra clans to have direct contacts within Sicily, “a form of control, and so, a signal of strength and power,” Mr. La Padula said. Online gambling, which replaced the raffles and lotteries once controlled by the mob, was another profitable enterprise that demonstrated the transition from the traditional to the modern, he added.

The investigations that led to Tuesday’s arrests involved many youths, a sign that “Cosa Nostra continues to exert its appeal in certain environments like suburbs where young people have limited life alternatives and identify with representations of power that the mafia still enjoys,” Palermo’s chief prosecutor, Maurizio De Lucia, told reporters at a news conference on Tuesday. “We have to be particularly careful” about the new recruits, because “that is the future of the mafia,” he said, according to the ANSA news agency.

The investigation also showed that Cosa Nostra still used old-time tricks — like informers inside the prosecutor’s office and lawyers — to stay ahead of investigators, officials said.