Geometric Rigidity in Nonlocal Models: The Ordered Mean Curvature Problem with Integrable Kernels
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We address the "ordered curvature" problem for nonlocal mean curvature defined by integrable, radially symmetric kernels. Analogous to the classical Li-Nirenberg conjecture, we investigate whether surfaces with nonlocal curvature monotonic in a given direction possess symmetry. We prove that if a bounded, connected set satisfies a monotone nonlocal curvature condition, it must be symmetric with respect to a hyperplane orthogonal to that direction. Our proof utilizes a variational argument to establish pairwise equality at boundary points and a generalized Alexandrov moving plane method adapted for nonlocal operators.
