On Isosystolic Inequalities for T^n, RP^n, and M^3

If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M, we show that the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g) are related by the following isosystolic inequality: Sys(M,g)^n \leq n! Vol(M,g). The inequality can be regarded as a generalization of Burago and Hebda's inequality for closed essential surfaces and as a refinement of Guth's inequality for closed n-manifolds whose Z/2Z-cohomology has the maximal cup-length. We also establish the same inequality in the context of possibly non-compact manifolds under a similar cohomological condition. The inequality applies to (i) T^n and all other compact euclidean space forms, (ii) RP^n and many other spherical space forms including the Poincar\'e dodecahedral space, and (iii) most closed essential 3-manifolds including all closed aspherical 3-manifolds.Comment: 34 pages, 0 figures. v2 contains expository revisions and some additional reference

Systematic Features of Axisymmetric Neutrino-Driven Core-Collapse Supernova Models in Multiple Progenitors

We present an overview of two-dimensional (2D) core-collapse supernova simulations employing neutrino transport scheme by the isotropic diffusion source approximation. We study 101 solar-metallicity, 247 ultra metal-poor, and 30 zero-metal progenitors covering zero-age main sequence mass from $10.8 M_{\odot}$ to $75.0 M_{\odot}$. Using the 378 progenitors in total, we systematically investigate how the differences in the structures of these multiple progenitors impact the hydrodynamics evolution. By following a long-term evolution over 1.0 s after bounce, most of the computed models exhibit neutrino-driven revival of the stalled bounce shock at $\sim$ 200 - 800 ms postbounce, leading to the possibility of explosion. Pushing the boundaries of expectations in previous one-dimensional (1D) studies, our results confirm that the compactness parameter $\xi$ that characterizes the structure of the progenitors is also a key in 2D to diagnose the properties of neutrino-driven explosions. Models with high $\xi$ undergo high ram pressure from the accreting matter onto the stalled shock, which affects the subsequent evolution of the shock expansion and the mass of the protoneutron star under the influence of neutrino-driven convection and the standing accretion-shock instability. We show that the accretion luminosity becomes higher for models with high $\xi$, which makes the growth rate of the diagnostic explosion energy higher and the synthesized nickel mass bigger. We find that these explosion characteristics tend to show a monotonic increase as a function of the compactness parameter $\xi$.Comment: 15 pages, 22 figures, 277 progenitors added, accepted to PAS