# Project Details

### Description

1. Overview

This proposal is to continue the PI’s research in global harmonic analysis and asymptotic geometry, with emphasis on growth and zeros of eigenfunctions and their relations to the dynamics of the geodesic flow. The PI proposes a number of projects on the growth of Lp and Kakeya-Nikodym norms of eigenfunctions as the eigenvalue tends to infinity and on nodal sets of Laplace eigenfunctions in the real and complex domains. A second emphasis is on the use of Bergman kernels to study asymptotic geometry in the setting of K¨ahler metrics on a complex manifold.

Sogge and the PI have a continuing project to determine the geometry of compact Riemannian manifolds (M, g) which carry sequences of eigenfunctions that extremize universal Lp bounds. Recently, we proved that for real analytic surfaces, existence of extremal sequences for high Lp norms implies existence of a ‘pole’ of M through which all geodesics are closed. In higher dimensions the result is weaker and one project is to determine if poles must exist when dimM ≥ 3.

The PI and Sogge also introduced a phase space Kakeya-Nikodym norm and are trying to relate it to L4 norms. Using recent quantum ergodic restriction theorems (joint work with H. Christianson and J. Toth) the PI and Junehyuk Jung proved that the number of nodal domains of Dirichlet and Neumann eigenfunctions on non-positively curved surfaces with concave boundary increase to infinity with the eigenvalue. This raises many further questions on critical points and nodal domains that the PI plans to explore. The PI also studies the ‘complex geometry’ of nodal sets using tools adapted from Bergman kernels on positive line bundles over K¨ahler manifolds, e.g. a recent proof that there is an optimal number of intersections of geodesics and nodal hypersurfaces when the geodesic flow is ergodic.

In K¨ahler geometry, the PI has an ongoing project with Yanir Rubinstein on the IVP (initial value problem) for geodesics in the space of K¨ahler metrics. The PI and Rubinstein proposed a conjectured solution, which led to a conjecture on the off-diagonal of the Bergman kernel. Recent

results on this conjecture by M. Christ and J. Sj¨ostrand have interesting implications for geodesics.

The PI also has a continuing project with S. Klevtsov and others on integration over spaces of Bergman metrics as an approximation to integration over all metrics on a surface.

1.1. Intellectual merit of the proposed activity. The inverse spectral problem, ‘Can you hear the shape of a drum’, is a classical unsolved problem that remains actively studied today. The PI’s result on analytic domains with one symmetry is the strongest positive known to date. Eigenfunctions of Schr¨odinger operators are the stationary states of quantum mechanics, and their sizes and shapes are important in both physics and chemistry. The PI is working with several physicists (Frank Ferrari and Semyon Klevtsov, who is now working with Paul Wiegmann) on applications of Bergman kernels to problems in physics (integration over spaces of metrics on surfaces).

1.2. Broader impacts resulting from the proposed activity. Each year, the PI co-organizes on average one or two workshops or programs which are aimed at graduate students and post-docs as well as senior researchers. In the near future, the PI is co-organizing a quarter long program at the Simons Center on ‘Large N limits in K¨ahler geometry” with R. Berman, S. Klevtsov and P. Wiegmann. He is also giving a mini-course on inverse spectral theory at Luminy (Marseille) in April, 2015. In 2013 the PI taught a summer sc

This proposal is to continue the PI’s research in global harmonic analysis and asymptotic geometry, with emphasis on growth and zeros of eigenfunctions and their relations to the dynamics of the geodesic flow. The PI proposes a number of projects on the growth of Lp and Kakeya-Nikodym norms of eigenfunctions as the eigenvalue tends to infinity and on nodal sets of Laplace eigenfunctions in the real and complex domains. A second emphasis is on the use of Bergman kernels to study asymptotic geometry in the setting of K¨ahler metrics on a complex manifold.

Sogge and the PI have a continuing project to determine the geometry of compact Riemannian manifolds (M, g) which carry sequences of eigenfunctions that extremize universal Lp bounds. Recently, we proved that for real analytic surfaces, existence of extremal sequences for high Lp norms implies existence of a ‘pole’ of M through which all geodesics are closed. In higher dimensions the result is weaker and one project is to determine if poles must exist when dimM ≥ 3.

The PI and Sogge also introduced a phase space Kakeya-Nikodym norm and are trying to relate it to L4 norms. Using recent quantum ergodic restriction theorems (joint work with H. Christianson and J. Toth) the PI and Junehyuk Jung proved that the number of nodal domains of Dirichlet and Neumann eigenfunctions on non-positively curved surfaces with concave boundary increase to infinity with the eigenvalue. This raises many further questions on critical points and nodal domains that the PI plans to explore. The PI also studies the ‘complex geometry’ of nodal sets using tools adapted from Bergman kernels on positive line bundles over K¨ahler manifolds, e.g. a recent proof that there is an optimal number of intersections of geodesics and nodal hypersurfaces when the geodesic flow is ergodic.

In K¨ahler geometry, the PI has an ongoing project with Yanir Rubinstein on the IVP (initial value problem) for geodesics in the space of K¨ahler metrics. The PI and Rubinstein proposed a conjectured solution, which led to a conjecture on the off-diagonal of the Bergman kernel. Recent

results on this conjecture by M. Christ and J. Sj¨ostrand have interesting implications for geodesics.

The PI also has a continuing project with S. Klevtsov and others on integration over spaces of Bergman metrics as an approximation to integration over all metrics on a surface.

1.1. Intellectual merit of the proposed activity. The inverse spectral problem, ‘Can you hear the shape of a drum’, is a classical unsolved problem that remains actively studied today. The PI’s result on analytic domains with one symmetry is the strongest positive known to date. Eigenfunctions of Schr¨odinger operators are the stationary states of quantum mechanics, and their sizes and shapes are important in both physics and chemistry. The PI is working with several physicists (Frank Ferrari and Semyon Klevtsov, who is now working with Paul Wiegmann) on applications of Bergman kernels to problems in physics (integration over spaces of metrics on surfaces).

1.2. Broader impacts resulting from the proposed activity. Each year, the PI co-organizes on average one or two workshops or programs which are aimed at graduate students and post-docs as well as senior researchers. In the near future, the PI is co-organizing a quarter long program at the Simons Center on ‘Large N limits in K¨ahler geometry” with R. Berman, S. Klevtsov and P. Wiegmann. He is also giving a mini-course on inverse spectral theory at Luminy (Marseille) in April, 2015. In 2013 the PI taught a summer sc

Status | Finished |
---|---|

Effective start/end date | 9/1/15 → 8/31/19 |

### Funding

- National Science Foundation (DMS‐1506591-001)