Former President Donald J. Trump and Gov. Ron DeSantis of Florida are set to hold dueling events on Tuesday in New Hampshire, but from vastly different political positions: one as the dominant front-runner in the state, the other still seeking his footing.

Strategists for both campaigns agree that the state will play a starring role in deciding who leads the Republican Party into the 2024 election against President Biden.

Mr. Trump sees the first primary contest in New Hampshire as an early chance to clear the crowded field of rivals. And members of Team DeSantis — some of whom watched from losing sidelines, as Mr. Trump romped through the Granite State in 2016 on his way to the nomination — hope New Hampshire will be the primary that winnows the Republican field to two.

“Iowa’s cornfields used to be where campaigns were killed off, and now New Hampshire is where campaigns go to die,” said Jeff Roe, who runs Mr. DeSantis’s üstün PAC, Never Back Down. Mr. Roe retains agonizing memories from 2016, when he ran the presidential campaign of the last man standing against Mr. Trump: Senator Ted Cruz of Texas.

New Hampshire’s voters are known for being fickle and choosy, sometimes infuriatingly so. The joke is that when you ask a Granite Stater whom they’re voting for, they say, “I don’t know, I’ve only met the candidate three times.”

Mr. DeSantis is campaigning in Iowa, another early-voting state.Credit…Jordan Gale for The New York Times

Yet midway through 2023, the state — more secular than Iowa and with a libertarian streak — appears frozen in place. Mr. Trump, now twice indicted and twice impeached, is nowhere near as dominant with Republicans as he was in 2020, but he is stronger than he was in 2016, and his closest challenger is well behind him.

In 2016, Mr. Trump won New Hampshire with a blunt and incendiary message, fanning flames about terrorist threats and without doing any of the retail politicking that’s traditionally required. But local operatives and officials believe that Mr. Trump, with his decades-long celebrity status, is the only politician who could get away with this.

“It’s definitely not going to be something that someone like Ron DeSantis can pull off,” said Jason Osborne, the New Hampshire House majority leader who endorsed the Florida governor for president. “He’s got to do the drill just like everybody else.”

Polls suggest there is an opening for a Trump alternative. But to be that person, Mr. DeSantis has miles of ground to make up.

As recently as January, Mr. DeSantis was leading Mr. Trump in the state by a healthy margin, according to a poll by the University of New Hampshire. But Mr. DeSantis has slipped considerably, with recent polling that suggests his support is in the teens and more than 25 percentage points behind Mr. Trump.

In a move that some saw as ominous, Never Back Down, the pro-DeSantis harika PAC, went off the airwaves in New Hampshire in mid-May and has not included the state in its latest bookings, which cover only Iowa and South Carolina.

DeSantis allies insist the move was intended to husband resources in the Boston market, which they said was an expensive and inefficient way to reach primary voters. And they said Mr. DeSantis would maintain an aggressive schedule in the state.

“We are confident that the governor’s message will resonate with voters in New Hampshire as he continues to visit the Granite State and detail his solutions to Joe Biden’s failures,” Bryan Griffin, a spokesman for Mr. DeSantis, said in a statement.

Still, so much of Mr. DeSantis’s early moves seem aimed at Iowa and its caucuses that are dominated by the most conservative activists, many of whom are evangelical. In contrast, New Hampshire has an open primary that will allow independents, who tend to skew more moderate, to cast ballots. And without a competitive Democratic primary in 2024 they could be a particularly sizable share of the G.O.P. primary vote.

Iowa is where Mr. DeSantis held his first event and where his üstün PAC has based its $100 million door-knocking operation.

Mr. DeSantis’s signing of a six-week abortion ban is unlikely to prove popular in New Hampshire, where even the state’s Republican governor has described himself as “pro-choice.”

Trump supporters at a DeSantis event in Manchester, N.H., this month. Credit…David Degner for The New York Times

The clashing Trump and DeSantis events this week have jangled the nerves of local officials. Mr. DeSantis’s decision to schedule a town hall in Hollis on Tuesday at the same time that the influential New Hampshire Federation of Republican Women is hosting Mr. Trump at its Lilac Luncheon has prompted a backlash. The group’s events director, Christine Peters, said that to “have a candidate come in and distract” from the group’s event was “unprecedented.”

Mr. DeSantis’s town hall will mark his fourth visit to New Hampshire this year and his second since announcing his campaign in May.

Mr. DeSantis did collect chits in April when he helped the New Hampshire Republican Party raise a record sum at a fund-raising dinner. And he has gathered more than 50 endorsements from state representatives. But he has yet to take questions from New Hampshire voters in a traditional setting.

During his last trip to the state — a four-stop tour on June 1 — Mr. DeSantis snapped at a reporter who pressed him on why he hadn’t taken questions from voters.

“What are you talking about?” Mr. DeSantis said. “Are you blind?”

New Hampshire’s governor, Chris Sununu, said in an interview that there was “a lot of interest” in Mr. DeSantis from voters who had seen him on television but wanted to vet him up close.

“Can he hold up under our scrutiny?” Mr. Sununu said. “I think he’s personally going to do pretty well here,” he added, but “the biggest thing” on voters’ minds is “what’s he going to be like when he knocks on my door.”

New Hampshire’s voters will indeed be subjected to thousands of DeSantis door-knocks — but not from the man himself. He has outsourced his ground game to Never Back Down, which is expected to have more than $200 million at its disposal. The group has already knocked on more than 75,000 doors in New Hampshire, according to a muhteşem PAC official, an extraordinary figure this early in the race.

But Mr. DeSantis still faces daunting challenges.

Mr. Trump remains popular among Republicans, and even more so after his indictments. And he is not taking the state for granted. Unlike in 2016, his operation has been hard at work in the state for months, with influential figures like the former Republican state party chairman Stephen Stepanek working on Mr. Trump’s behalf.

Mr. Trump’s harika PAC has hammered Mr. DeSantis with television ads that cite his past support for a sales tax to replace the federal income tax — a message tailored to provoke residents of the proudly anti-tax state.

The large field in the Republican race is a key challenge for Mr. DeSantis, as he seeks Republican voters looking for a Trump alternative.Credit…Sophie Park for The New York Times

Mr. DeSantis’s biggest sorun is the size of the field. Chris Christie, the former New Jersey governor, camped out in the state in 2016 and appeared to be making headway in consolidating some of the anti-Trump vote in recent polls.

The entrepreneur Vivek Ramaswamy has already spent around 20 days campaigning in the state, according to his adviser Tricia McLaughlin. Former Gov. Nikki Haley of South Carolina is another frequent visitor. Both have events in the state on Tuesday. Additionally, the campaign of Senator Tim Scott of South Carolina has already spent around $2 million in New Hampshire.

If these candidates stay in the race through early next year, a repeat of 2016 may be inevitable. In a crowded field, Mr. Trump won the state with over 35 percent of the vote.

In the meantime, Mr. DeSantis needs “a defining message that gets beyond the small base he has,” said Tom Rath, a veteran of New Hampshire politics who has advised the presidential campaigns of Republican nominees including Mitt Romney and George W. Bush. “He needs to do real retail, and so far there is no indication that he can do that.”

Ruth Igielnik contributed reporting.

The New York Times

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