Smooth Solutions and Discrete Imaginary Mass of the Klein-Gordon Equation in the de Sitter Background

Using methods in the theory of semisimple Lie algebras, we can obtain all smooth solutions of the Klein-Gordon equation on the 4-dimensional de Sitter spacetime (dS^4). The mass of a Klein-Gordon scalar on dS^4 is related to an eigenvalue of the Casimir operator of so(1,4). Thus it is discrete, or quantized. Furthermore, the mass m of a Klein-Gordon scalar on dS^4 is imaginary: m^2 being proportional to -N(N+3), with N >= 0 an integer.Comment: 23 pages, 4 figure

An obstacle problem for a class of Monge-Amp\`ere type functionals

In this paper we study an obstacle problem for Monge-Amp\`ere type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu's equation

Relative Algebro-Geometric stabilities of Toric Manifolds

In this paper, we study the relative Chow and $K$-stability of toric manifolds. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds. The reduction of relative Chow stability on toric manifolds will be investigated by the Hibert-Mumford criterion in two ways. One is to consider the criterion for the maximal torus action and its weight polytope. Then we obtain a reduction by the strategy of Ono \cite{Ono13}, which fits into the relative GIT stability detected by Sz\'ekelyhidi. The other way is to use the criterion for $\mathbb{C}^{\times}$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas \cite{D02, RT07}. In the end, we determine the relative K-stability of all toric Fano threefolds and present a counter-example for relatively $K$-stable manifold, but which is asymptotically relatively Chow unstable.Comment: 24 pages, 2 tables: v2 has minor changes and several added reference