A Curve Flow on an Almost Hermitian Manifold Evolved by a Third Order Dispersive Equation

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We present a time-local existence theorem of solutions to the initial value problem for a third-order dispersive evolution equation for open curves into compact almost Hermitian manifolds. Our equations geometrically generalize a two-sphere valued physical model describing the motion of vortex filament. These equations cause the so-called loss of one-derivative since the target manifold is not supposed to be a Kähler manifold. We overcome this difficulty by using a gauge transformation of a multiplier on the pull-back bundle to eliminate the bad first order terms essentially.

We consider a curve flow for maps from a real line into a compact almost Hermitian manifold, which is governed by a third order nonlinear dispersive equation. This article shows short-time existence of a solution to the initial value problem for the equation. The difficulty comes from the lack of the Kähler condition on the target manifold, since the covariant derivative of the almost complex structure causes a loss of one derivative in our equation and thus the classical energy method breaks down in general. In the present article, we can overcome the difficulty by constructing a gauge transformation on the pull-back bundle for the map to eliminate the derivative loss essentially, which is based on the local smoothing effect of third order dispersive equations on the real line.


  • Funkcialaj Ekvacioj

    Funkcialaj Ekvacioj 55(1), 137-156, 2012

    Mathematical Society of Japan - Kobe University


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