While their curly-haired rescue dog napped on the floor, an Italian couple logged on to a late-night video call with their American surrogate, sunbathing in her garden in Oregon. The fathers-to-be cooed as she told them she was playing fairy tales close to her belly that they had recorded for their future daughter. “And she is kicking!” she said.

But the men, both civil servants, said they had not dared to share their excitement with almost anyone around them. They did not talk about the pregnancy with many friends, colleagues or neighbors or post about it on social media. They asked to remain anonymous for this article.

They have reason for caution. Surrogacy is already illegal if conducted in Italy. But the government of Prime Minister Giorgia Meloni wants to expand the prohibition. It has promoted a bill that would also punish Italians who make use of surrogacy even in places abroad where it is legal, like in parts of the United States. Those Italians who do could face up to two years in prison and be fined the equivalent of about a million dollars.

Italy’s lower chamber of Parliament approved the bill last summer, and the Senate’s justice committee greenlighted it last month. The Senate is expected to vote on it as soon as the fall.

On the international stage, Ms. Meloni has presented herself as a pragmatic partner for mainstream European leaders and aligned herself with Western democracies on the issues that matter to them, like support of Ukraine.

But at home, Ms. Meloni has asserted her conservative credentials on cultural issues such as abortion, gender, gay rights and surrogacy.

“No one can convince me that it is an act of freedom to rent one’s womb,” she said in the spring at an event in Rome. “No one can convince me that it is an act of love to consider children as an over-the-counter product in a supermarket.”

“Uterus renting is a shameful, inhuman practice,” she said. “It will become a universal crime.”

While Ms. Meloni’s stance may reflect her own deeply held convictions, analysts say such rhetoric also serves to please the right-wing base of her Brothers of Italy party. Polls show that her party’s voters are disproportionately more opposed to surrogacy and adoption by gay couples than the general population is.

“On the economy and foreign policy, she took completely mainstream positions,” said Roberto D’Alimonte, a political scientist at Luiss Guido Carli University in Rome. “And she compensates her mainstream positions with the fact that she still uses an old-right rhetoric on things that don’t matter to define her international profile.”

Issues like surrogacy and same-sex parenthood “still define her as a right-wing person sensitive to traditional values,” Mr. D’Alimonte said. “And she exploits this when she can.”

Many of those positions — for instance, against gay parenthood and favoring abortion prevention rather than access — place Ms. Meloni in much the same ranks as other social conservatives and the Roman Catholic Church.

But her Italian critics say that by pressing for further restrictions on gay families in a country that already ranks near last in Europe when it comes to such civil liberties, Ms. Meloni has taken a particularly hard line.

Her moves on the cultural front have often been subtle, like her tinkering with Italy’s abortion law. When it comes to surrogacy, many feminists also oppose it, and other European countries also outlaw it, though it is allowed in some, like Britain or Greece, under certain conditions.

But analysts and opponents say Italy’s proposed new law is especially perplexing because it is tailored to penalize a relatively small number of Italians and is so far-reaching that some experts are skeptical it could withstand legal challenges.

“It’s pure propaganda,” said Susanna Lollini, a lawyer for L.G.B.T.Q. families, “but it’s spreading absolute panic.”

Most Italian couples who use surrogacy are believed to be heterosexual, and they could also be affected by the proposed new law. But because same-sex couples need a third party to have children, many gay Italians feel that the change in the law would leave them vulnerable to special scrutiny. Also, adoption is allowed only for heterosexual couples, leaving gay Italians with few options.

“I can’t tell I am about to become a father,” one of the future fathers said. The other added, “We can’t tell our story, because my government is persecuting me and my family.”

Ms. Meloni’s lawmakers have not hidden whom the law is targeting. Carolina Varchi, who presented the anti-surrogacy bill, wrote on Facebook in June that with the new law, her party was working against L.G.B.T. “ideology.”

Even before Ms. Meloni took office at the end of 2022, Italy, home to the Vatican, was one of the few European Union countries not to recognize same-sex marriage, and the couples interviewed for this article were joined by civil union or unmarried.

Still, L.G.B.T.Q. activists say, while they had to challenge previous governments to advance their rights, now they are playing defense.

In another step aimed at same-sex couples, Ms. Meloni’s government this spring appealed a court decision that allowed parents to be identified as “parent” on their children’s IDs, instead of as “mother” and “father.”

“I think it would be wrong to prevent by law a child from having a father and a mother,” Ms. Meloni said on Italian television in July.

Members of Italy’s L.G.B.T.Q. community are especially concerned by what they consider to be a tone by the government that singles them out.

“No member of the L.G.B.T.Q. community could say that this government has a moderate attitude,” said Emanuela Bruno, a lesbian mother, who is trying to prevent Italian authorities from removing her name from her twin children’s birth certificates.

Ms. Meloni’s government has sought to vigorously enforce a court decision that had barred a mayor from registering children born through surrogacy abroad as having two fathers. Cities that used to issue such certificates, like Milan, stopped doing so.

The government’s directive has had the ripple effect of encouraging Italian prosecutors in several cities to also revise the birth certificates of children born to lesbian couples.

In the northern city of Padua, prosecutors are now trying to remove at least 38 mothers from the birth certificates of their children.

Among the mothers is Brona Kelly, an Irish teacher who has a child with her Italian wife, Alice Bruni.

Ms. Bruni, who is the child’s biological mother, is undergoing treatment for breast cancer, and she worries that if Ms. Kelly were to lose her recognition in court, their 1-year-old son would become an orphan in the event of her death.

Like other gay parents, Ms. Bruni is considering leaving the country, but for now she has to stay put because of her cancer treatment.

When her case went to court, Ms. Bruni said, she listed to the judges what she considered the serious problems facing the world: wars, violence, abuse.

“And we are here for what?” she asked. “Because I have a child with the woman I love?”

Their windows broken and roofs smashed, the abandoned homes in an otherwise bucolic valley carpeted with cornfields and orchards near Serbia’s border with Bosnia look like the wreckage of the Balkan wars of the 1990s.

But the houses are actually the casualties of a current struggle freighted with geopolitics: where and how Europe can get the materials it needs to make electric car batteries and break its dependence on sources like China.

The houses, in the Jadar Valley in the west of Serbia, were bought up years ago by the minerals behemoth Rio Tinto, which planned to tear them down and start mining and processing lithium, a crucial element for electric car batteries. Its plans stalled by vociferous opposition, the company left the properties to crumble.

The project has been supported by the United States and the European Union, which is in desperate need of lithium to meet its climate goals. But it has generated a wave of public fury in Serbia, where fears that the mine will poison the air and water have set off huge street protests against President Aleksandar Vucic.

Europe has plenty of lithium and more than 20 mining projects for the mineral at various stages of development. But none have started producing battery-grade lithium. The giant project in Serbia was aimed at filling that hole.

“There is no green transition in Europe without this lithium,” said Chad Blewitt, the head of Rio Tinto’s Serbian operations, adding that the company planned to invest more than $2.55 billion in the project.

The Serbian government gave preliminary approval in 2019, but, worried about losing votes during protests against Rio Tinto before a 2022 election, canceled it.

Under pressure from the European Union, which Serbia aspires to join, the government changed its mind in July, allowing Rio Tinto to revive the project. The British-Australian multinational says it has already invested nearly $600 million to buy land, dig 500 exploratory holes, commission studies and make donations to the local soccer club and other entities.

Serbia’s mining minister, Dubravka Djedovic Handanovic, said mining probably would not start for another two years, but once it did, lithium from the Jadar Valley would allow Serbia to manufacture batteries and electric cars, providing about 20,000 jobs.

A report by The Hague Center for Strategic Studies estimates that if it is to reach its goal of carbon neutrality by 2050, Europe will need 60 times more lithium by that year than what it imported in 2020 from China and elsewhere.

Michael Schmidt, a lithium expert at Germany’s Federal Institute for Geosciences and Natural Resources, said Europe might be able to reach its targets without supplies from Serbia. But, he said, “the Serbian project is one of the largest, and that is why it is so significant.” He added, “We need each and every project to reach targets.”

The success of the projects ultimately depends on the price for lithium on the global market and whether companies like Rio Tinto can recoup their investments. The price has collapsed over the past 18 months as Chinese demand has slackened and its output soared.

The proposed mine in Serbia has not only provoked fury among farmers, environmental activists and ordinary citizens, it also has become a proxy battleground in the West’s efforts to extract the country from the orbits of Russia, its traditional ally, and China.

Geoffrey R. Pyatt, the U.S. assistant secretary of state for energy resources, this past week cheered the Serbian lithium project on social media as “an opportunity to contribute to the green transition at home & abroad.”

For those who view Serbia as a partner for the United States and Europe rather than a Moscow-aligned and authoritarian regional bully, Mr. Vucic’s support for Rio Tinto, along with his assent to Serbian-made weapons being sold covertly to Ukraine, is evidence he was serious about disengaging from Russia.

Russia has strong support among hard-line Serb nationalists, and some diplomats and analysts say Moscow has been stirring the unrest over the mine. Mr. Vucic, however, has said Moscow told him that the West is orchestrating the protests because it wants to topple him.

“Unfortunately, it has become a political fight, a big political battle,” said the mining minister, Ms. Djedovic Handanovic.

Among those taking part in recent nationwide demonstrations against Rio Tinto have been leaders of People’s Patrol, an ultranationalist group aligned with Moscow. Social media accounts known for spreading Russian disinformation have been active in promoting horror stories about the planned lithium mine.

But leftists and middle-of-the-road pro-Europeans have also joined the protests, chanting opposition to a project that has become a lightning rod for diverse grievances against the government.

“He sold out Kosovo but is not going to take away our clean water,” read a sign denouncing Mr. Vucic that was held by Angela Rojovic, 25, at a recent protest in Belgrade, the capital. She said the president had not done enough to defend the interests of Serbs living in mainly ethnic Albanian Kosovo.

And she said Mr. Vucic was sacrificing Serbia’s environment to serve Europe’s climate goals. “I don’t need green cars,” she said. “I need green apples and green grass.”

In Gornje Nedeljice, a Jadar Valley village that sits atop Europe’s biggest known deposit of high-grade lithium, the project has alienated Mr. Vucic’s previously stalwart rural base.

Dragan Karajcic, the district head for a cluster of small settlements around the proposed mine, said he was a member of Mr. Vucic’s governing party but still joined a local protest group hostile to Rio Tinto and the government.

“We are not trying to bring down the government,” he said. “The government is doing that itself.”

Goran Tomic, a native of Gornje Nedeljice who now lives mostly in Germany, said he understood the need to combat climate change by moving away from gasoline-powered cars, but he was still appalled that his older brother had agreed to sell his house and land to Rio Tinto.

“He allowed himself to betray himself for money, and in doing that he betrayed us all,” Mr. Tomic said, sitting on his front stoop with his mother, who was also angry but proud that two of her three sons refused to sell to Rio Tinto.

Its assurances over safety undermined by past misbehavior, Rio Tinto has tried to counter what it dismisses as lies and disinformation spread on social media by recently disclosing preliminary findings of an environmental impact assessment. It was carried out by Serbian and foreign scientists who debunked much of what protesters believe about lithium mining.

Wild claims on social media included one last week that an exploratory hole bored by Rio Tinto was belching radioactive fluid.

Mr. Vucic, rattled by the scale and intensity of public anger, has also veered into fear-mongering, claiming protests were led by “anarchists, Marxists and hidden fascists.”

The real leaders, however, were people like Nebojsa Petkovic, a villager from Gornje Nedeljice and an activist who traveled to Belgrade to help organize a demonstration on Saturday, Aug. 10, that attracted tens of thousands of people.

“Let the Germans save the planet,” Mr. Petkovic said. “We need to save ourselves.”

Eager to get mining started, Chancellor Olaf Scholz of Germany and executives of Mercedes Benz, which has big electric vehicle plans, visited Belgrade last month to applaud the Rio Tinto project.

Germany’s role, however, has only amplified opposition.

Mr. Karajcic, the district head, said he was infuriated by German assurances that the mine would be safe, recalling Nazi atrocities in a nearby town in 1941 that the Germans had promised would be left unhurt.

He said his great-grandfather fought nearby against Austrian troops during World War I. “He fought to keep our land, and now I’m supposed to give it away to Rio Tinto. No way,” he said. “There is a lot of bad blood in these hills.”

They came into the village just past sundown, dozens of Israeli settlers wearing masks, dressed in dark clothes and armed with rocks, Kalashnikovs and M-16s, witnesses said.

A local resident, Muawiya al-Sidee, said his 13-year-old daughter was one of the first to spot them as she and her younger siblings were playing on Thursday in their front garden in the village of Jit, in the Israeli-occupied West Bank.

“‘Baba, settlers are coming!’” the girl screamed.

Mr. al-Sidee and his wife, who was breastfeeding their 2-year-old daughter at the time, piled their five children into their car and drove off just as the settlers reached their front door.

Seconds later, he said, the Israelis from the nearby Eli settlement smashed the windows of his family’s home and threw in three Molotov cocktails, burning rooms where, moments earlier, the family had gathered.

As the family fled, a call went out over mosque loudspeakers in the village of some 3,000 people, imploring young men to come out and defend against the rampaging settlers.

When Mr. al-Sidee returned hours later, after the settlers had withdrawn, he found the sofas in his house were charred husks and the overhead lamps had melted.

Elsewhere in the village, Rasheed al-Seda, 23, awoke when the call for defenders sounded from the mosques. He joined a group determined to defend the village, armed with nothing but stones.

It would cost him his life, his mother and the Palestinian Ministry of Health said.

The Israeli military confirmed the attack on the village.

“Dozens of Israeli civilians, some of them masked, entered the town of Jit and set fire to vehicles and structures in the area, hurled rocks and Molotov cocktails,” a military statement said, adding that the military had opened an investigation and was looking into reports of a fatality.

More than 2.7 million Palestinians reside in ancestral cities, towns and farming villages in the West Bank, where, for generations, many have lived off the land. But that existence is increasingly under threat as more Israelis move to the territory — they now number nearly 500,000 — to live in settlements considered illegal under international law.

Since the Oct. 7 Hamas-led attack on Israel and the start of the war in Gaza, attacks by Jewish settlers on Palestinians across the West Bank have become common. There have been about 1,250 such attacks in this time, according to the United Nations — 25 in the past week alone.

More than 589 Palestinians have been killed by Israeli forces or Jewish settlers in the West Bank since Oct. 7, according to Palestinian health officials. Eighteen Israelis have been killed in the territory in the same time period, according to the United Nations.

On Thursday night, Mr. al-Seda became one of the latest Palestinian casualties.

He had just joined the other men who were running in the direction of the rampaging settlers when, according to residents and Palestinian health officials, he was shot in the chest by a settler.

Multiple residents said the Israeli army was preventing ambulances and fire trucks from entering the village. The Israeli military denied the accusations.

Other men carried Mr. al-Seda to a car — his blood staining the pavement. Residents said he was driven to the entrance of the village, where he was transferred to an ambulance that had been blocked from entering. From there he was taken to a hospital, where he was later pronounced dead.

In total, four houses and six vehicles were burned, according to Yesh Din, an Israeli rights group.

The military said that its forces, along with Israeli Border Police, had been dispatched to the site and dispersed the rampage by firing shots into the air and “removing the Israeli civilians from the town” within about 30 minutes from the time it began.

But rights groups and Palestinians have said in the past that the Israeli military often does nothing to stop such attacks. And Jit residents said that the military had not arrived at the scene until more than an hour after the settler rampage had begun.

The office of Prime Minister Benjamin Netanyahu of Israel, whose coalition government includes West Bank settlers in top positions, said the leader “takes seriously the riots that took place this evening in the village of Jit, which included injury to life and property by Israelis who entered the village.”

But far-right members of Mr. Netanyahu’s government, including Bezalel Smotrich, the finance minister, and Itamar Ben-Gvir, the minister of national security, have made inflammatory statements about Palestinians before and have advanced policies to expand Israel’s hold on the West Bank.

In October, Mr. Ben-Gvir, who oversees the police, promised to provide thousands of guns to Israelis, including to settlers. He posted photographs that showed him handing out assault rifles to civilians.

Jit residents said Mr. Netanyahu’s government bore some responsibility for the attack.

“Ben-Gvir gave them these weapons to attack us,” said Oomyma al-Sidee, a relative of Muawiya al-Sidee. She said she was holed up in her home with her six sons and other relatives as settlers tried to break through the metal front door.

“This is terrorism,” she said.

From the roof of their home, her sons threw rocks at the settlers, trying to push them away from the home and two vehicles parked out front. Some of the settlers smashed the vehicles’ windows and set them on fire with Molotov cocktails, she said.

Despite the danger, she said, Ms. al-Sidee’s husband ran outside with a hose to try to put out the fire, worried that the vehicles would explode and ignite their home.

“We escaped death,” she said.

This was not the first time that the family had been targeted, she said. In October, Israelis from the same settlement kidnapped her husband for an hour, beat him with guns and threatened to shoot him.

Since then, Ms. al-Sidee said, she has kept their IDs, important documents and gold jewelry in a lunchbox that she carries with her every time she leaves her house.

On Thursday night, after the attack, she and her family slept at a relative’s house.

“Tonight, I don’t know where we will sleep. They might come back,” Ms. al-Sidee said on Friday, expressing a widespread fear throughout the village.

She had just returned from the wake for Mr. al-Seda, who had been a student in an Arabic class she teaches.

Throughout the village, mourning posters had gone up for Mr. al-Seda. At his family’s home, a banner hung outside as villagers streamed in to attend the wake. Recitations from the Quran played in the background as women came in, offering their condolences and sipping bitter coffee.

In one corner, his mother, Iman al-Seda, sat reciting prayers and lamenting the loss of her son.

“My love, my life,” she said, weeping and wiping her bloodshot eyes with a crumpled tissue.

Mr. al-Seda, who had worked in computers, was a sociable person who brought life to their home, his mother said. He would always kiss on her cheeks and hands, a common sign of respect for elders in Arab culture.

“What am I going to do?” she said. “I wish he hadn’t gone to help.”

Secretary of State Antony J. Blinken was heading to Israel on Sunday to try to clinch a deal that could end the war in Gaza, even as the Middle East remained on edge amid the looming threat of wider regional conflict.

The visit, part of an intensive diplomatic campaign led by the Biden administration, comes days after Israel’s negotiating team held talks in Qatar with senior American officials, as well as Qatari and Egyptian representatives who are mediating between Israel and Hamas.

Those talks ended without a major breakthrough, but the White House said in a statement on Friday that the United States had put forward a “bridging proposal,” with Egyptian and Qatari support, intended to close remaining gaps between the sides. It said teams would continue to hash out details for implementing the deal and that senior negotiators hoped to reconvene in Cairo before the end of this week to finalize an agreement.

In a statement issued on Saturday night, after the sabbath, the office of Israel’s prime minister, Benjamin Netanyahu, said the Israeli negotiating team had expressed “cautious optimism” over the possibility of advancing toward a deal based on the bridging proposal. It did not offer further details, and portrayed Hamas as the obstacle to reaching an agreement.

Mr. Blinken is scheduled to meet on Monday with Mr. Netanyahu, whom some officials have accused of stalling by adding new conditions for a deal.

The potential deal would be carried out in three phases and is based on principles that were laid out by President Biden on May 31 and subsequently endorsed by the United Nations Security Council. It would usher in a cease-fire in Gaza and involve the release of the hostages being held captive in the enclave in exchange for Palestinian prisoners and detainees held in Israel.

The Biden administration has created a degree of linkage between the cease-fire efforts and the threat of Iranian-led retaliation against Israel for the back-to-back assassinations of senior figures from Hezbollah, the Iranian-backed Lebanese militia, and Hamas in Beirut and Tehran in late July.

Amid fears that any reprisals and subsequent Israeli counterstrikes could escalate into a broader regional war, American officials have expressed hope that progress on the diplomatic front could stave off a broader conflagration.

In the meantime, the fighting in Gaza has continued. The Israeli military said on Sunday that its troops were operating in the central and southern Gaza Strip.

Ukraine pressed ahead with its offensive inside Russian territory on Sunday, pushing toward more villages and towns nearly two weeks into the first significant foreign incursion in Russia since World War II.

But even as the Ukrainian army was advancing in Russia’s western Kursk region, its troops were steadily losing ground on their own territory. The Russian military is now about eight miles from the town of Pokrovsk in eastern Ukraine, according to open-source battlefield maps. The capture of Pokrovsk, a Ukrainian stronghold, would bring Moscow one step closer to its long-held goal of capturing the entire Donetsk region.

That underscored the gamble Ukraine’s army took when it crossed into Russia: throwing its forces into a daring offensive that risked weakening its own positions on the eastern front.

Whether that strategy will prove advantageous remains to be seen, analysts say.

On the political front, the offensive has already had some success: Ukraine’s rapid advance has embarrassed the Kremlin and has altered the narrative of a war in which Kyiv’s forces had been on the back foot for months.

Here’s what to know about Ukraine’s cross-border operation, which President Biden said last week was creating a “real dilemma” for the Russian government.

What happened?

Ukrainian troops and armored vehicles stormed into the Kursk region of western Russia on Aug. 6, swiftly pushing through Russian defenses and capturing several villages.

The assault, prepared in the utmost secrecy, opened a new front in the 30-month war and caught not only Russia off guard: Some Ukrainian soldiers and U.S. officials also said they lacked advance notice.

Analysts and Western officials estimate that Ukraine deployed about 1,000 troops at the start of the incursion. But military analysts say that it has since poured more troops into the operation to try to hold and expand its positions.

How far into Russia have Ukrainian troops advanced?

Gen. Oleksandr Syrsky, Ukraine’s top commander, said last week that his army now controlled more than 80 Russian settlements in the Kursk region, including Sudzha, a town of 6,000 residents. His claims could not be independently verified, although analysts say that Sudzha is highly likely to be under full Ukrainian control.

Ukraine’s advance in the Kursk region has slowed in recent days, according to open-source maps of the battlefield based on combat footage and satellite images, as Russia sends in more reinforcements. The Ukrainian army appears to be trying to dig in along the border area rather than pushing deeper into Russia.

President Volodymyr Zelensky of Ukraine acknowledged that on Saturday, saying: “Now we are reinforcing our positions. The foothold of our presence is getting stronger.”

Why is this significant?

Kyiv has regularly bombarded Russian oil refineries and airfields with drones since Moscow’s full-scale invasion began in February 2022. It has also helped stage two other ground attacks in Russia. Those, however, were smaller forays by Russian exile groups backed by the Ukrainian army, and they ended in quick retreats.

Until two weeks ago, Ukrainian forces had not counterattacked in Russia. The gains in Kursk are the quickest for Ukrainian forces since they reclaimed the Kherson region of their own country in November 2022.

How has the Kremlin responded?

As Ukrainian forces pushed deeper into Russia, Moscow scrambled to shore up its defenses, and President Vladimir V. Putin convened his security services to coordinate a response. The Russian military said it was sending more troops and armored vehicles to try to repel the attack, with Russian television broadcasting images of columns of military trucks.

Military analysts and U.S. officials have said the Russian command had so far brought in reinforcements mainly from within Russia so as to not deplete its units on the Ukrainian battlefield, in what they described as a disorganized effort.

“Russia is still pulling together its reaction,” Gen. Christopher G. Cavoli, NATO’s top military commander, said last week during a talk at the Council on Foreign Relations in New York. He described the Russian response as having been “fairly slow and scattered” as the authorities sorted out which military and security forces should take the lead.

And what about Putin?

The incursion has embarrassed Mr. Putin and his military establishment, prompting questions about Russia’s level of preparedness.

Underscoring how the attack rattled the Kremlin, Mr. Putin lashed out last week at the West in a tense televised meeting with his top officials. “The West is fighting us with the hands of the Ukrainians,” he said, repeating his frequent depiction of the war, which he started, as a proxy campaign against Russia by the West.

Ukraine’s incursion has brought the war into Russia like it never has before, and tens of thousands of civilians have evacuated the border area.

What is the goal of Ukraine’s incursion?

Analysts say that Ukraine’s offensive has two main aims: to draw Russian forces from the front lines in eastern Ukraine and to seize territory that could serve as a bargaining chip in future peace talks.

Mykhailo Podolyak, a top Ukrainian presidential adviser, said last week that Russia would be forced to the negotiating table only through suffering “significant tactical defeats.”

“In the Kursk region, we can clearly see how the military tool is being used objectively to persuade” Russia to enter “a fair negotiation process,” he wrote on social media.

The operation has offered a much-needed morale boost for Ukrainians, whose forces have been losing ground to Russian troops for months.

But military analysts have questioned whether Kyiv’s cross-border assault is worth the risk, given that Ukrainian forces are already stretched on the front lines of their own country.

How is it affecting the fight inside Ukraine?

Russian forces have been pummeling Ukrainian troops in the east even as Moscow races to respond to the incursion into Kursk, according to analysts, Western officials and Ukrainian soldiers.

Russia has begun to withdraw small numbers of troops from Ukraine, they said, to try to help repel the Ukrainians, but not enough to significantly affect the overall battlefield for now.

Senior American officials have said privately that they understood Kyiv’s need to change the narrative of the war, but that they were skeptical that Ukraine could hold the territory long enough to force Russia to divert significant resources from the front lines in eastern and southern Ukraine.

While Kyiv’s allies have in the past been wary that Ukrainian incursions in Russia could escalate the war, the European Union’s top diplomat, Josep Borrell Fontelles, said last week that Ukraine had the bloc’s “full support.”

Ukraine has used some Western-supplied weapons in the Kursk operation. But so far, the United States and Britain, two of Kyiv’s closes allies, have said the incursion did not violate their policies.”

What happens next?

As the Ukrainian offensive approaches its two-week mark, analysts say that Kyiv has several options, each with its own challenges.

Ukrainian forces could try to keep pushing farther into Russia, but that will become harder as Russian reinforcements arrive and Ukraine’s supply lines are stretched.

They could keep digging into the territory they now hold and try to defend it, but that could expose fixed Ukrainian positions to potentially devastating Russian airstrikes.

Or, battered by continual losses in eastern Ukraine, they could decide that they have made their point and pull back.

Thibault Fouillet, the deputy director of the Institute for Strategic and Defense Studies, a French research center, said Ukraine’s next move would depend on how Russia responds. “The coming week will be decisive,” he said.

Eric Schmitt contributed reporting.