Onthe Lagrangian Willmore Problem The Willmore functional of an immersion is one over
four terms thesquare length of the mean curvature vector field. Over the last two decades,
many variational problems toward the Willmore functional were proposed andsolved. Specially,
in 1995 Minicozzi proved that there exist minimizers amongLagrangian tori in R^4.
In this talk, we will introduce two kinds of geometricallyconstrained variation of the
Willmore functional.The ciritical points arecalled LW surfaces and HW surfaces, respectively.
For noncompact proper HWsurfaces, we get a small energy curvature estimate and gap theorem.
A flowmethod will also be discussed.
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