Critical points of a function subject to a constraint can be either detected
    by restricting the function to the constraint or by looking for critical points
    of the Lagrange multiplier functional. Although the critical points of the two
    functionals, namely the restriction and the Lagrange multiplier functional are
    in natural one-to-one correspondence this does not need to be true for their
    gradient flow lines. We consider a singular deformation of the metric and show
    by an adiabatic limit argument that close to the singularity we have a
    one-to-one correspondence between gradient flow lines connecting critical
    points of Morse index difference one. We present a general overview of the
    adiabatic limit technique in the article [FW22b].

    The proof of the correspondence is carried out in two parts. The current part
    I deals with linear methods leading to a singular version of the implicit
    function theorem. We also discuss possible infinite dimensional generalizations
    in Rabinowitz-Floer homology. In part II [FW22a] we apply non-linear methods
    and prove, in particular, a compactness result and uniform exponential decay
    independent of the deformation parameter.

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