Amid the lush vegetation that covers this Caribbean island, the commodity that White House officials most hope will flourish is Jimmy Carter's image as a forceful world leader.

Almost from the moment the American party left Washington for the four-power summit conference that ended here today, Carter has been discreetly portrayed by his aides as the towering figure at such gatherings of Western leaders.

The aides said, for example, that President Carter tends to dominate private conversations such as those he held here with French President Valery Giscard d' Estaing, British Prime Minister James Callaghan and West German Chancellor Helmut Schmidt, and that the other three invariably look to him for leadership.

At the same time, Carter has been portrayed as a man who is unconcerned with image, supremely self-assured and therefore not at all interested in dominating whatever news the summit happens to produce.

The fact is that while the personal relationships among the four men are a subject of endless fascination to observers of such international meetings, only the sketchiest kind of information is available on how the talks have gone since they began Thursday night.

There is no question that Carter gets along best with Callaghan, who has about him an air of perpetual informality befitting his nickname of "Sunny Jim." Giscard, the haughty Frenchman, appears uniformly aloof to everyone he encounters, but there is no sign of any tension between him and the president.

If there is tension, it is between Carter and Schmidt, the compact, chain-smoking West German chancellor.

There are matters of importance to press secretaries and other image molders as well as to the leaders for whom they work. Thus it was that on Friday White House Press Secretary Jody Powell confronted his West German counterpart, Klaus Boelling, with a complaint about supposed leaks from the Bonn delegation quoting Schmidt as criticizing certain aspects of American foreign policy.

Boelling was described as "personally offended" by the encounter, one of the many byplays that occur at such conferences but seldom become known publicly.

As the summit broke up today, it was also clear that the downplaying of whatever differences do exist among them was of equal importance to the four principals. Each stressed the "friendly" nature of the talks and said differences over emphasis were relatively unimportant.

Carter said the talks were marked by an "almost unprecedented harmony" among the four and Schmidt said he wished reporters had been invited to one of the sessions "so you could see for yourselves how friendly the atmosphere was."

Throughout the talks here, American reporters were told that the president was extremely confident of his position, certain he could explain U.S. positions and persuade the others of the soundness of those positions. Carter was variously described as "stimulated" by the talks and as thriving on the "give and take" of discussions among world leaders, with the clear suggestion that he was also leading those discussions.

Undoubtedly, the British, French and German reporters here were being told much the same thing about the men they cover, for even a supposedly informal and private summit conference is a stage to play on according to each leader's domestic political needs. But for President Carter, now midway through his term, the question of leadership is of special importance.

While he is the leader of the West's most powerful nation, Carter is the least imposing physically and in public manner of the four men. Moreover, at least during the early period of the Carter administration there were widespread doubts in Europe about the direction of U.S. policy under a president with so little experience in foreign policy.

More recently, Carter has received praise abroad and at home for some of his foreign policy initiatives. But the problems confronting the president have not disappeared, with each of them representing a test of his leadership of the Western alliance.

Carter's most notable foreign policy achievement, the Camp David Middle East peace agreements, still has not produced a peace treaty between Israel and Egypt. There also have been doubts about U.S. policy in Iran.

A SALT agreement now seems near, but Senate approval, necessary if it is to mean anything, is far from certain.

When the president leaves the brilliant sunshine of this French-ruled island Tuesday, these and numerous domestic problems will be awaiting him in Washington. It is the outcome of such matters, rather than Carter's private persuasiveness or his cordial relationships, that will determine how his leadership is viewed both in Europe and the United States.