Conformal geometry is in the core of pure mathematics. It is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces – discrete surface Ri...

In earlier work, carrying out numerical simulations of the Ricci flows of families of rotationally symmetric geometries on S3, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a “critical” initial geometry– one which is at the transition point between initial geometries (on S3) whose volume-normalized Ricci flows develop a sin...

Ricci flow, since the debut in the famous original work [4] by R. Hamilton, has been one of the major driving forces for the development of Geometric Analysis in the past decades. Its astonishing power is best demonstrated by the breakthrough in solving Poincaré Conjecture and Geometrization Program. For this amazing story, we refer to [1], [7], [10] and the references therein. Meanwhile, Kähle...

In this note, we establish the first variation formula of the adjusted log entropy functional Ya introduced by Ye in [14]. As a direct consequence, we also obtain the monotonicity of Ya along the Ricci flow. Various entropy functionals play crucial role in the singularity analysis of Ricci flow. Let (M, g(t)) be a smooth family of Riemannian metrics on a closed manifold M and suppose g(t) is a ...

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.

In this paper, we study the moduli spaces of noncollapsed Ricci flow solutions with bounded energy and scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study isoperimetric constant control, Kähler Ricci flow and moduli space of gradient shrinking solitons.

We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.

In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on Sn+1, without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman’s hope for a “canonically defined Ricci flow through singularities”.

We will give an elementary proof of a result of R. Hamilton for Ricci flow on compact surfaces. Let M be a compact surface. We will prove the global existence of solution of the Ricci flow ∂gij/∂t = (r − R)gij on M where R is the scalar curvature and r = R

We show that the analogue of Hamilton’s Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston’s circle packing on surfaces. As a consequence, a new proof of Thurston’s existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.