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Session S33 - Spectral Geometry

Wednesday, July 21, 16:30 ~ 16:50 UTC-3

Lower bounds for eigenfunction restrictions in lacunary regions

John Toth

McGill, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it..a

Let $(M,g)$ be a compact, real-analytic Riemannian manifold and $u_h \in C^{\omega}(M)$ be a sequence of $L^2$-normalized Laplace eigenfunctions with $(-h^2 \Delta_g - 1) u_h = 0$. We assume that this sequence has a localized defect measure $d\mu$ in the sense that $$ \text{supp} \, \pi_* d\mu = K, \quad M \setminus K \neq \emptyset.$$ Using Carleman estimates in the lacunary region $M \setminus K,$ we show that for any separating hypersurface $H \subset (M\setminus K)$ sufficiently close to $\partial K,$ there exist constants $h_0(H), C_H>0$ such that for $h \in (0, h_0(H)],$ $$ \int_{H} |u_h|^2 d\sigma_H \geq e^{- C_H /h}.$$ Consequently, In the terminology of Toth and Zelditch, all such hypersufaces are good for the eigenfunction sequence $\{ u_h \}.$ This is joint work with Yaiza Canzani.

Joint work with Yaiza Canzani (University of North Carolina, USA).

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