Asymptotic Analysis for Sacks-Uhlenbeck 𝛼-Harmonic Maps From Degenerating Riemann Surfaces

In this paper, we investigate the asymptotic analysis and qualitative behaviour for a general sequence of Sacks-Uhlenbeck <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-harmonic maps from degenerating Riemann surfaces. This answers an open problem proposed by J. D. Moore, aiming at developing a partial Morse theory for closed parametrized minimal surfaces with arbitrary codimensions in compact Riemannian manifolds. In technical terms, we shall deal with three types of necks which give rise to new difficulties. We establish a generalized energy identity involving some new quantities, such as Pohozaev type constants which measure the extent to which the Pohozaev type identity fails, and some locating constants which characterize the limit position of the blow-up points in long hyperbolic cylinders. Furthermore, we show that the limit of necks are all geodesics in the target manifold where some length formulas are given. Finally, we exploit some geometric and topological conditions to ensure that the limiting neck geodesics appeared in a blow-up sequence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-harmonic maps with finite Morse index are all of finite length, which implies that energy identity hold.