The heat flow in nonlinear Hodge theory |
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Authors: | Christoph Hamburger |
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Institution: | Hohle Gasse 77, D-53177, Bonn, Germany |
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Abstract: | We study the nonlinear Hodge system dω=0 and δ(ρ(|ω|2)ω)=0 for an exterior form ω on a compact oriented Riemannian manifold M, where ρ(Q) is a given positive function. The solutions are called ρ-harmonic forms. They are the stationary points on cohomology classes of the functional
with e′(Q)=ρ(Q)/2. The ρ-codifferential of a form ω is defined as δρω=ρ−1δ(ρω) with ρ=ρ(|ω|2).We evolve a given closed form ω0 by the nonlinear heat flow system
for a time-dependent exterior form ω(x,t) on M. This system is the differential of the normalized gradient flow
for E(ω) with ω=ω0+du. Under a technical assumption on the function 2ρ′(Q)Q/ρ(Q), we show that the nonlinear heat flow system
, with initial condition ω(·,0)=ω0, has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of ω0. This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative. |
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Keywords: | ρ -Harmonic exterior forms Nonlinear Hodge theory Heat flow method Exterior curvature operator Mean curvature flow Gas dynamics |
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