Minimal surfaces, equidistribution vs scarring

Antoine Song, University of California, Berkeley

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(The colloquium password will be distributed to Princeton University and IAS members. We ask that you do not share this password. If you would like to be included in the colloquium and are not a member of either institution, please email the organizer Casey Kelleher ( with an email requesting to participate which introduces yourself, your current affiliation and stage in your career.)

Given a closed Riemannian manifold, a classical result states that the eigenfunctions of the Laplacian equidistribute on average in the sense that the normalized sum of the L^2-densities converges to the uniform probability measure. On the other hand, subsequences of eigenfunctions can exhibit the opposite behavior known as scarring.

Minimal hypersurfaces are geometric non-linear analogues of eigenfunctions. The understanding of their existence theory was substantially improved recently. These developments lead to natural questions about the spatial distribution of minimal hypersurfaces in closed manifolds.

In this talk I will survey previous results and discuss some joint works with F.C. Marques and A. Neves, and with X. Zhou, related respectively to generic equidistribution on average, and to generic scarring for minimal hypersurfaces. It turns out that for closed Riemannian 3-manifolds, scarring is a common phenomenon. The proofs partly use some Weyl law type asymptotics associated to minimal hypersurfaces.