## View abstract

### 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. document.getElementById('cloakf2bf3500c689114e8e2e1534ca3eb090').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyf2bf3500c689114e8e2e1534ca3eb090 = 'jt&#111;th' + '&#64;'; addyf2bf3500c689114e8e2e1534ca3eb090 = addyf2bf3500c689114e8e2e1534ca3eb090 + 'm&#97;th' + '&#46;' + 'mcg&#105;ll'; var addy_textf2bf3500c689114e8e2e1534ca3eb090 = 'jt&#111;th' + '&#64;' + 'm&#97;th' + '&#46;' + 'mcg&#105;ll';document.getElementById('cloakf2bf3500c689114e8e2e1534ca3eb090').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyf2bf3500c689114e8e2e1534ca3eb090 + '\'>'+addy_textf2bf3500c689114e8e2e1534ca3eb090+'<\/a>'; .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).