TIMBUKTU, Mali -- France's president, François Hollande, paid a triumphant visit to this ancient city on Saturday, receiving a rapturous welcome from thousands of people who gathered next to a 14th-century mosque to dance, play drums and chant "Vive la France!" The muezzin, whose singing calls residents to pray five times a day, wore a scarf in the colors of the French flag as he shouted, "Vive Hollande!"
It had the trappings of a "mission accomplished" moment.
But even as people outside the mud-and-wood mosque hailed the French leader as the city's, and their country's, savior, questions remain about what France has accomplished aside from chasing Islamic extremists from the cities and into their desert and mountain redoubts.
"These Islamists, they have not been defeated," said Moustapha Ben Essayouti, a member of one of the city's most prominent families who lined up to greet Mr. Hollande here. "Hardly any of them have been killed. They have run into the desert and the mountains to hide."
Even Mr. Hollande, who praised French and Malian troops gathered here for accomplishing "an exceptional mission," acknowledged that "the fight is not over."
Indeed, little is known about the fate of fighters who fled the cities that have been retaken in the lightning northward advance by French and Malian troops to clear Islamists who had taken over the north of the country in recent months. In interviews, residents of cities now abandoned by the Islamist rebels have said that the bulk of the fighters fled in the night long before the French arrived.
Given the fighters' deep familiarity of the vast, forbidding territory between this city and the borders of Algeria and Mauritania, many worry that the Islamist groups will simply regroup and come back to try again.
"If France leaves, they will come back," Mr. Essayouti said.
The spidery network of Islamist militants in Mali numbered about 2,000 hard-core fighters before the French airstrikes and march north, according to American intelligence officials, and there are no clear figures yet on how many died in the fighting. The most dangerous component of that mix is Al Qaeda's affiliate in North Africa, known as Al Qaeda in the Islamic Maghreb, the officials said.
The group has been attracting heavily armed Islamists from about 10 countries across North and West Africa, making Mali the biggest magnet for jihadi fighters other than Syria, one of the senior American intelligence officials said.
The Islamists who advanced toward a pivotal frontier town on Jan. 10 -- leading to worries of a possible march south to the capital and drawing France into the battle in the process -- were well armed, with AK-47s, rocket-propelled grenades and heavy machine guns mounted on vehicles. They also had some armored personnel carriers seized from the Malian military last year.
American military and counterterrorism officials applauded the speed and efficiency of the French-led operation, but they voiced concerns that the militants had ceded the northern cities with little or no resistance in order to prepare for a longer, bloodier insurgency.
"Longer term, and the French know this, it's going to take a while to root out all these cells and operatives," Michael Sheehan, the Pentagon's top special operations policy official, told a defense industry symposium on Wednesday.
The senior United States intelligence official said that the real measure of success would be whether follow-up operations in the north would be able to diminish the strength of Al Qaeda in the Islamic Maghreb and other Islamist groups. Like other American officials, he spoke on the condition of anonymity because operations were continuing.
Whether the Islamists are routed in the end may depend in part on how involved France remains. The Malian troops are considered to be poorly trained, and even the most capable of the other African troops pouring into the country as part of a regional force do not have the same level of weaponry to back them up as the French.
Mr. Hollande refused to give a timetable on Saturday for the withdrawal of the 3,500 French troops currently in Mali. In a speech here he said that it is "not our role to stay," but later in the capital of Bamako he said, "We will be with you to the end, all the way to northern Mali," according to the French news media.
In the past, French officials have talked about handing off the fight in the north to the Malian Army and other African troops.
North Africa specialists and American intelligence officials say the militants might lay low until French forces leave.
"Are they going to dig in and be guerrillas or go to ground and wait?" said Michael R. Shurkin, a former Central Intelligence Agency analyst who is now at the RAND Corporation.
For now, the people of Timbuktu are grateful for France's help. They waved French and Malian flags and sang to the thumping rhythms of djembe drums, which were banned under the harsh version of Shariah imposed by the Islamist group that took control of the city. Men and women danced side by side.
As Mr. Hollande, ringed by security guards, plunged into the crowd to shake hands, some waved banners that said "Papa François, the mysterious city welcomes you."
"Hollande is our savior," said Arkia Baby, a 24-year-old college student, who wore a purple batik dress of a style banned by the Islamists. "He gave us back our freedom."
That sentiment represents a strange twist in France's often troubled history in Africa. France had a vast belt of colonies here that spanned the Sahara, from the Atlantic coast to just short of the Red Sea. After many of its colonies won independence in 1960, many remained bound to France, using a currency pegged to the franc and then the euro, and maintaining close trade, military and diplomatic ties.
France's role has been fraught with moral peril. It pioneered brutal techniques to put down insurgencies in the Algerian war for independence, carpet-bombing villages suspected of harboring nationalist guerrillas. In the early 1990s France staunchly supported the Hutu-dominated government of Rwanda, despite growing signs that a blood bath was in the making.
More recently, French military intervention in Ivory Coast may have heightened ethnic tensions in that country. Even though the French intervened to install the country's democratically elected leader, because the vote was cast along ethnic lines they were seen as favoring northerners and Muslims over southern Christians.
The French lost only one helicopter pilot in Mali, but Mr. Hollande's aides are conscious of the risks of overstaying and becoming targets themselves.
The French hope that with the Islamists in far northern deserts and hills, they can be watched by drones and attacked from the air without harming civilians. The French also expect that the Islamists will have a harder time getting gasoline and food, especially if Algeria, as promised, seals its border with Mali. The French also say the Islamists will find it harder to plan further raids and kidnappings of Westerners that have helped finance their insurgency.
Still, staying and fighting carries risks for France, beyond the safety of its troops. French officials have voiced concerns about charges that the regular Malian Army has been guilty of human rights abuses, including murders of Tuareg and Arab civilians they accuse of ties to the militants.
Mr. Hollande warned the French and African troops here that they must avoid abuses, lest they "tarnish the mission."
Writing in the newspaper Libération, the French columnist Vincent Giret argued that the French face an unhappy choice in Mali. If they remain on the front line they will look, "sooner or later, like white neocolonialists," and any bad episode can turn public opinion quickly sour. But if the French Army "settles for a role supporting the Malian and African troops left on the front line, it then risks being accused of covering up abuses and score settlings."
Steven Erlanger contributed reporting from Munich, Eric Schmitt from Washington, and Scott Sayare from Paris.
Correction: February 2, 2013, Saturday
This article has been revised to reflect the following correction: An earlier version of this article misspelled the surname of one of the Timbuktu residents who lined up to greet the French president. He is Moustapha Ben Essayouti, not Essagouté.
This article originally appeared in The New York Times.