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In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.
We establish curvature estimates and a convexity result for mean convex properly embedded $[varphi,vec{e}_{3}]$-minimal surfaces in $mathbb{R}^3$, i.e., $varphi$-minimal surfaces when $varphi$ depends only on the third coordinate of $mathbb{R}^3$. Le
In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions
We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabais theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to sphe
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and Kahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures
In the following article we discuss Delaunay triangulations for a point cloud on an embedded surface in $mathbb{R}^3$. We give sufficient conditions on the point cloud to show that the diagonal switch algorithm finds an embedded Delaunay triangulation.