| Among the class of 4-dimensional Riemannian manifolds, the product spaceS~2×S~2, which is K(a|¨)hler-Einstein with constant scalar curvature, may be the mostimportant space aside from the 2-dimensional complex space forms. The methods andresults on the surfaces in S~2×S~2are very rich, especially on the Lagrangian surfaces ofdi?erent families: minimal, with parallel mean curvature tensor, Hamiltonian-stable,Hamiltonian-minimal, etc.In this paper, we mainly study the Lagrangian surfaces in S~2×S~2achieving theminimum of a geometric inequality. We derive some local properties of such La-grangian surfaces via studying the associated Jacobian function in depth, and show thatunder some typical geometric conditions, there exist no surfaces satisfying |σ|~2 = 3|H|~2except the totally geodesic sphere M0 or the totally geodesic torus T.In addition, we also study the global problems of the Lagrangian surfaces with|σ|~2 = 3|H|~2. We derive a restriction on the genus of such surfaces, and constructan example of a Lagrangian sphere. The sphere is not totally geodesic, which showsM0 is not the only case of Lagrangian surfaces satisfying the geometric equality, incontrast with the torus. The sample implies the obstruction of the classification of suchLagrangian surfaces in S~2×S~2. Comparing the results of this paper with the ones ofminimal Lagrangian surfaces by I. Castro and F. Urbano, we see these two families ofLagrangian surfaces in S~2×S~2have many similarities. |
