Let G = (V, E) be a finite connected weighted graph, and assume 1 ≤ α ≤ p ≤ q. In this paper, we consider the following p-th Yamabe type equationon G, where ∆ p is the p-th discrete graph Laplacian, h ≤ 0 and f > 0 are real functions defined on all vertices of G. Instead of the approach in [12], we adopt a new approach, and prove that the above equation always has a positive solution u > 0 for some constant λ ∈ R. In particular, when q = p our result generalizes the main theorem in [12] from the case of α ≥ p > 1 to the case of 1 ≤ α ≤ p. It's interesting that our new approach can also work in the case of α ≥ p > 1. critical points of the Hilbert-Einstein function S on the space of Riemannian metrics on M restricted to [g], the conformal class of g, S(g) = M s g dµ g V ol(M, g) 2/pm .

For triangulated surfaces and any p > 1, we introduce the combinatorial p-th Calabi flow which precisely equals the combinatorial Calabi flows first introduced in H. Ge's thesis [9] (or see H. Ge [13]) when p = 2. The difficulties for the generalizations come from the nonlinearity of the p-th flow equation when p = 2. Adopting different approaches, we show that the solution to the combinatorial p-th Calabi flow exists for all time and converges if and only if there exists a circle packing metric of constant (zero resp.) curvature in Euclidean (hyperbolic resp.) background geometry. Our results generalize the work of H.

Let G = (V, E) be a connected infinite and locally finite weighted graph, ∆ p be the p-th discrete graph Laplacian. In this paper, we consider the p-th Yamabe type equationwhere h and g are known, 2 < α ≤ p. The prototype of this equation comes from the smooth Yamabe equation on an open manifold. We prove that the above equation has at least one positive solution on G.where ∆ p is p-th discrete graph Laplacian. Now, we recall the main result in [7].

We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which generalizes Li-Rubinstein's curvature estimate for one divisor case.

No abstract