Notes on Perelman's papers

@article{Kleiner2006NotesOP,
  title={Notes on Perelman's papers},
  author={Bruce Kleiner and John Lott},
  journal={arXiv: Differential Geometry},
  year={2006}
}
These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds". 

PERELMAN'S PROOF OF THE POINCAR´ E CONJECTURE: A NONLINEAR PDE PERSPECTIVE

We discuss some of the key ideas of Perelman's proof of Poincare's conjecture via the Hamilton program of using the Ricci flow, from the perspec- tive of the modern theory of nonlinear partial

Hyperbolic thermostat and Hamilton's Harnack inequality for the Ricci flow

In this paper, we will recover Hamilton's Harnack inequality for the Ricci flow from the view point of Hyperbolic thermostat.

Perelman's reduced volume and a gap theorem for the Ricci flow

In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean

Hyperbolic thermostat and Hamilton's Harnack inequality for the Ricci flow

In this paper, we will recover the Hamilton’s Harnack inequality for the Ricci flow from the view point of “Hyperbolic thermostat”.

Some Elementary Consequences of Perelman’s Canonical Neighborhood Theorem

In this purely expository note, we deduce a few known direct consequences of Perelman’s canonical neighborhood theorem for 3-dimensional Ricci flow and compactness theorem for 3-dimensional

A Simple Proof On Poincar\'e Conjecture

We give a simple proof on the Poincar\'e's conjecture which states that every compact smooth $3-$manifold which is homotopically equivalent to $S^3$ is diffeomorphic to $S^3$.

Relative volume comparison of Ricci flow

In this paper we derive a relative volume comparison of Ricci flow under a certain local curvature condition. It is a refinement of Perelman’s no local collapsing theorem in Perelman (2002).

Relative volume comparison of Ricci Flow and its applications

In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of K\"ahler-Ricci flow on a minimal manifold. This new

Optimal transport and Perelman’s reduced volume

We show that a certain entropy-like function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.

Perelman, Poincare, and the Ricci Flow

In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and
...

References

SHOWING 1-10 OF 71 REFERENCES

A note on Perelman’s LYH inequality

We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known

A NOTE ON PERELMAN’S LYH TYPE INEQUALITY

We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known

On the l-Function and the Reduced volume of Perelman II

The main purpose of this paper is to present a number of analytic and geometric properties of the l-function and the reduced volume of Perelman, including in particular the monotonicity, the upper

Perelman???s Stability Theorem

We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curv > k Gromov-Hausdorff converging to a compact Alexandrov

Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds

Abstract.In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called $$\bar{\lambda}$$. We show here that, for completely elementary reasons, this invariant simply equals the

Scalar curvature and the existence of geometric structures on 3-manifolds, II

This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences

RECENT DEVELOPMENTS ON THE RICCI FLOW

This article reports recent developments of the research on Hamilton's Ricci flow and its applications.

Lectures on the Ricci Flow

1. Introduction 2. Riemannian geometry background 3. The maximum principle 4. Comments on existence theory for parabolic PDE 5. Existence theory for the Ricci flow 6. Ricci flow as a gradient flow 7.

Volume collapsed three-manifolds with a lower curvature bound

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.

Algorithmic Topology and Classification of 3-Manifolds

Simple and special polyhedra.- Complexity theory of 3-manifolds.- Haken theory of normal surfaces.- Applications of the theory of normal surfaces.- Algorithmic recognition of S3.- Classification of
...