A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Perelman's Ricci flow emerges in quantum gravity, linking math and physics.
In recent years, there has seen much interest and increased research activities on Perelman's paper. Section one and two of this paper aim to establish Perelman's local non-collapsing result for the Ricci flow. This will provide a positive lower bound on the injectivity radius for the Ricci flow under blow-up analysis.…
New spinorial functional connects Perelman's W- and F-functionals.
Entropy derived from Colding's volume on Ricci-flat manifolds.
Defines Perelman's functionals on manifolds with non-isolated conical singularities.
We show a very simple and general total second variation formula for Perelman's -functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of Kähler structures. We deduce a quite simple and general total second variation formula for P…
We will simplify the earlier proofs of Perelman's collapsing theorem of 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's semi-convex analysis of distance functions to construct the desired local Seifert fibration structure on collapsed 3-manifolds. The verification of Perelma…
We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibratio…
The study examines Perelman singular manifolds and their properties.
In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow and, in particular, we study them from the point of view of Perelman's functionals. In a first part, we study Perelman's and functionals of cones and characterize their finiteness in terms of the -functional of…
We show a quite simple second variation formula for Perelman's -functional along the modified Kähler-Ricci flow over Fano manifolds.
The paper proves the monotonicity of a modified Perelman's W-entropy for mean curvature flow.
In this paper we present a major application of the l-function and the reduced volume of Perelman, namely their application to the analysis of the asymptotical limits of kappa solutions of the Ricci flow.
This is a short note in which we show how to calculate the value of Perelman's nu-functional for a variety of metrics. In particular we complete the calculation of values for the known 4-dimensional Einstein and shrinking Ricci soliton metrics.
The main purpose of this paper is to present a number of analytic and geometric properties of the -function and the reduced volume of Perelman, including in particular the monotonicity, the upper bound and the rigidities of the 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.
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
In this thesis we give a review on Ricci flow, an overview on Poincare conjecture, maximum principle, Li-Yau-Perelman estimate, Two functional F and W of Perelman, Reduced volume and reduced length and k-non collapsing estimate
In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a Kähler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is stable along the Kähler-Ricci flow. The Chinese version of this note has appeared in…
In this paper, we develop the theory of Perelman's -functional on manifolds with isolated conical singularities. In particular, we show that the infimum of -functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvatur…
The article proves a new entropy formula for surfaces with boundaries.
This paper characterizes mu-cscK metrics using Perelman's W-entropy.
In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prov…
By means of a Kaluza-Klein type argument we show that the Perelman's F-functional is the Einstein-Hilbert action in a space with extra ``phantom'' dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his first famous paper. As a consequence the…
In this short note we observe that the concavity of Perelman's -functional over a neighborhood of a Kähler-Ricci soliton inside the space of Kähler potentials is a direct consequence of author's solution of the variational stability problem for Kähler-Ricci solitons. Independently, we provide a rather simp…
Ancient Ricci flows with asymptotic solitons have uniform bounds and inequalities.
We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by c…
Proves curvature bounds for close to 1 Perelman's reduced volume.
In this note, we construct families of functionals of the type of -functional and -functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using t…
In this paper, we prove that on a compact manifold with isolated conical singularity the spectrum of the Schrödinger operator consists of discrete eigenvalues with finite multiplicities, if the scalar curvature satisfies a certain condition near the singularity. Moreover, we obtain an asymptotic behavior fo…
In this paper, we consider the generalized lambda constant and the existence of ground states of the generalized Perelman's W-functional from a variational formulation. One result is concerned with the estimation of the generalized constant. The other results are about the existence of ground states of generalized …
In this paper, we prove the characterization of the -super Perelman Ricci flows by various functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton type dim…
Classifies ancient and expanding Ricci flows with specific groups.
We calculate Perelman's invariant for compact complex surfaces and a few other smooth four-manifolds. We also prove some results concerning the dependence of Perelman's invariant on the smooth structure.
Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.
New topological quantum gravity theories linked to Ricci flow.
Unified approach to analysis on Ricci nonnegative manifolds and flows using optimal transport.
Entropy study on synthetic spaces with curvature bounds.
New estimates show all stable Einstein manifolds are linear stable with respect to Perelman's ν-entropy.
We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Müller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result …
In this article, we introduce a new method (based on Perelman's lambda-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (A) a Ricci-flat metric is a local maximizer of lambda in a C^2,alpha-sense iff its Lichne…
This is essentially a note on Section 7 of Perelman's first paper on Ricci flow. We list some basic properties of the index form for Perelman's -length, which are analogous to the ones in Riemannian case (with fixed metric), and observe that Morse's index theorem for Perelman's -length holds…
The paper models star dynamics using Ricci flow and Perelman entropy, revealing chaotic behavior.
The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau, the gradient estimate of Ni, the monotonicity of the Perelman's entropy and the volum…
Using Perelman's results on Kahler Ricci flow, we prove that the K energy is bounded from below if and only if the F functional is bounded from below in the canonical Kahler class.
The paper extends Perelman's theorems on Ricci flow entropy.
New proof for curvature and diameter estimates on Fano manifolds.
As an application of his entropy formula, Perelman proved that every compact shrinking breather is a shrinking gradient Ricci soliton. We give a proof for the complete noncompact case by using Perelman's -geometry. Our proof follows the argument in Lu and Zheng of constructing an ancient solution, and remo…