Asymptotic stability near spheres for the exponential surface diffusion flow

Establish asymptotic stability results for the exponential surface diffusion flow V = Δ_Γ f(−κ) with f(r) = e^r on closed hypersurfaces initially close to a round sphere; specifically, prove global-in-time existence and convergence of the evolving surface to a (possibly shifted) stationary sphere, in analogy with the known stability theory for the conventional surface diffusion flow V = −f'(0) Δ_Γ κ.

Background

The paper recalls that for the conventional surface diffusion flow, there is a well-developed theory of global existence and asymptotic stability for closed hypersurfaces near spheres, including convergence to a stationary sphere (e.g., Escher–Mayer–Simonett and subsequent works).

By contrast, the authors note that the corresponding stability problems for the exponential surface diffusion flow, driven by V = Δ_Γ f(−κ) with f(r) = e^r, have not been resolved. This highlights a gap between the linearized (conventional) and fully nonlinear exponential curvature dependence in the stability analysis near spherical equilibria.