In this article, we consider Bergman kernels with respect to modules at boundary points, and obtain a log-subharmonicity property of the Bergman kernels, which implies a concavity property related to the Bergman kernels. As applications, we reprove the sharp effectiveness result related to a conjecture posed by Jonsson–Mustaţă and the effectiveness result of strong openness property of the modules at boundary points.
We construct parabolic analogues of (global) eigenvarieties, of patched eigenvarieties and of (local) trianguline varieties, that we call, respectively, Bernstein eigenvarieties, patched Bernstein eigenvarieties, and Bernstein paraboline varieties. We study the geometry of these rigid analytic spaces, in particular (generalising results of Breuil–Hellmann–Schraen) we show that their local geometry can be described by certain algebraic schemes related to the generalised Grothendieck–Springer resolution. We deduce several local–global compatibility results, including a classicality result (with no trianguline assumption at p), and new cases towards the locally analytic socle conjecture of Breuil in the non-trianguline case.
We study the three dimensional quantum many-body dynamics with repulsive Coulomb interaction in the mean-field setting. The Euler–Poisson equation is its limit as the particle number tends to infinity and Planck’s constant tends to zero. By a new scheme combining the hierarchy method and the modulated energy method, we establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities as well as kinetic and potential energies before the 1st blow up time of the limiting Euler–Poisson equation.
Using matrix model, Mironov and Morozov recently gave a formula which represents Kontsevich–Witten tau function as a linear expansion of Schur Q-polynomials. In this paper, we will show directly that the Q-polynomial expansion in this formula satisfies the Virasoro constraints, and consequently obtains a proof of this formula without using matrix model. We also give a proof for Alexandrov’s conjecture that Kontsevich–Witten tau function is a hypergeometric tau function of the BKP hierarchy after re-scaling.
It was conjectured by Escobar (J Funct Anal 165:101–116, 1999) that for an n-dimensional ($n\ge 3$) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by $c>0$, the first nonzero Steklov eigenvalue is greater than or equal to c with equality holding only on isometrically Euclidean balls with radius 1/c. In this paper, we confirm this conjecture in the case of nonnegative sectional curvature. The proof is based on a combination of Qiu–Xia’s weighted Reilly-type formula with a special choice of the weight function depending on the distance function to the boundary, as well as a generalized Pohozaev-type identity.