A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

Abstract

We survey Mirzakhani’s work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of nonexperts.

Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-ε$

“…where A ⊂ M g is any Borel subset, 1 A : M g → {0, 1} is its characteristic function, and where dX is short for dvol WP (X). One may see the book [Wol10] for recent developments on Weil-Petersson geometry, and see the recent survey [Wri20] for works of Mirzakhani including her coworkers on random surfaces in the Weil-Petersson model.…”

“…It is unknown whether Theorem 1 also holds with 3 16 replaced by 1 4 (e.g. see [Wri20,Problem 10.4]). We remark here that it is even unknown whether there exists a sequence of hyperbolic surfaces {X gn } with genus g n going to infinity such that λ 1 (X gn ) tends to 1 4 (e.g.…”

Bootstrap bounds on closed Einstein manifolds

“…This builds on Mirzakhani's result that generically λ 1 > 0.0024 as g → ∞ [29]. It is conjectured that this 3/16 can be replaced by 1/4 [30]. There is a similar result for the eigenvalues of random covers of a surface [31].…”

A high-genus asymptotic expansion of Weil-Petersson volume polynomials

“…are bounded by C n (K) log(g + n) K , thanks to the estimate on the approximating polynomials coefficients (7) • prove the point (2) of Theorem 2 later, that is to say the remainder estimate, using Proposition 14.…”

“…In this article, we provide an expansion in decreasing powers of g of the volume V g,n (x) for any x. This expansion was encountered by the authors in an ongoing project, aiming at proving that random hyperbolic surfaces have an optimal spectral gap (see Wright's conjecture [7,Problem 10.3]).…”