Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far.

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In this article we introduce higher order Bergman functions for bounded complete Reinhardt domains in a variety with possi- bly isolated singularities. These Bergman functions are invariant under biholomorhic maps. We use Bergman functions to deter- mine all the biholomorhic maps between two such domains. As a result, we can construct an infinite family of numerical invari- ants from the Bergman functions for such  domains in An variety (x, y, z) ∈ C3 : xy = zn+1 . These infinite family of numerical invariants are actually a complete set of invariants for either the set of all bounded strictly pseudoconvex complete Reinhardt domain in An variety or the set of all bounded pseudoconvex complete Reinhardt domains with real analytic boundaries in An variety. In particular the moduli space of these domains in An variety is constructed explicitly as the image of this complete family of numerical invariants. It is well known that An variety is the quo- tient of cyclic group of order n + 1 on C2. We prove that the moduli space of bounded complete Reinhardt domains in An vari- ety coincides with the moduli space of the corresponding bounded complete Reinhardt domains in C2. Since our complete family of numerical invariants are computable, we have solved the biholo- morphically equivalent problem for large family of domains in C2.

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang–Yau quasi-local mass for a surface in spacetime introduced in Wang and Yau (Phys Rev Lett 102(2):021101, 2009 ; Commun Math Phys 288(3):919–942, 2009) is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the surface into the Minkowski space. A critical point of the quasi-local energy is an isometric embedding satisfying the Euler–Lagrange equation. In this article, we prove results regarding both local and global minimizing properties of critical points of the Wang–Yau quasi-local energy. In particular, under a condition on the mean curvature vector we show a critical point minimizes the quasi-local energy locally. The same condition also implies that the critical point is globally minimizing among all axially symmetric embedding provided the image of the associated isometric embedding lies in a totally geodesic Euclidean 3-space.

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