We study the class of holomorphic and isometric submersions between finite-type Teichm "uller spaces. We prove that, with potential exceptions coming from low-genus phenomena, any such map is a forgetful map @math obtained by filling in punctures. This generalizes a classical result of Royden and Earle-Kra asserting that biholomorphisms between finite-type Teichm "uller spaces arise from mapping classes. As a key step in the argument, we prove that any @math -linear embedding @math between spaces of integrable quadratic differentials is, up to scale, pull-back by a holomorphic map. We accomplish this step by adapting methods developed by Markovic to study isometries of infinite-type Teichm "uller spaces.
We mention also a result of Antonakoudis-Aramayona-Souto @cite_9 stating that any holomorphic map @math between moduli spaces is forgetful, as long as @math and @math . One can see this is as a parallel of our result, with our metric constraint replaced by an equivariance condition.
@cite_9
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We prove that every non-constant holomorphic mapeM& em&⊂g,p& sub&→ eM& em&⊂& g',p'& sub& between moduli spaces of Riemann surfaces is a forgetful map, provided that g ≥ 6 and g' ≤ 2g-2. The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in the moduli space of Deligne-Mumford stable pointed curves which includes the space of canonical divisors as an open subset. The theory leads to geometric combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus g curves are of pure codimension g in the moduli spaces of stable pointed curves. In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors. As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases). We give a log-geometric description of the space of twisted canonical divisors constructed by Farkas--Pandharipande. In particular, we introduce the notion of a principal rubber @math -log-canonical divisor, and we study its moduli space. It is a proper Deligne--Mumford stack admitting a perfect obstruction theory whose virtual fundamental cycle is of dimension @math . In the so-called strictly meromorphic case with @math , the moduli space is of the expected dimension and the push-forward of its virtual fundamental cycle to the moduli space of stable curves equals the weighted fundamental class of the moduli space of twisted canonical divisors. Conjecturally, it yields a formula of Pixton generalizing the double ramification cycle in the moduli space of stable curves. It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner–Nordstrom data for the Einstein–Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously. Under additional assumptions, it is shown that the Hawking mass blows up identically along this bifurcate null hypersurface, and thus the metric cannot be extended twice differentiably; in fact, it cannot be extended in a weaker sense characterized at the level of the Christoffel symbols. The proof combines estimates obtained in previous work with an elementary Cauchy stability argument. There are no restrictions on the size of the support of the scalar field, and the result applies to both the future and past boundary of spacetime. In particular, it follows that for an open set in the moduli space of solutions around Reissner–Nordstrom, there is no spacelike component of either the future or the past singularity.