A bound for the order of the fundamental group of a complete noncompact Ricci shrinker
For complete shrinking (gradient) Ricci solitons, we observe a quantification of Wylie's result that the fundamental group is finite.
math.DG↗
arXiv subjects
Peng Lu
Publications and source records attributed to Peng Lu.
Uniqueness of asymptotic cones of complete noncompact shrinking gradient Ricci solitons with Ricci curvature decay
We discuss an elementary consequence of the works of (1) Brett Kotschwar and Lu Wang and (2) Ovidiu Munteanu and Jiaping Wang.
math.DG↗
A rigidity theorem for codimension one shrinking gradient Ricci solitons in $\mathbb R^{n+1}$
We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci curvature.
math.DG↗
Finding More Relevance: Propagating Similarity on Markov Random Field for Image Retrieval
To effectively retrieve objects from large corpus with high accuracy is a challenge task. In this paper, we propose a method that propagates visual feature level similarities on a Markov random field (MRF) to obtain a high level correspondence in image space for image pairs. The proposed correspondence between image pair reflects not only the similarity of low-level visual features but also the relations built through other images in the database and it can be easily integrated into the existing bag-of-visual-words(BoW) based systems to reduce the missing rate. We evaluate our method on the standard Oxford-5K, Oxford-105K and Paris-6K dataset. The experiment results show that the proposed method significantly improves the retrieval accuracy on three datasets and exceeds the current state-of-the-art retrieval performance.
cs.CV↗
Rigidity of complete entire self-shrinking solutions to Kahler-Ricci flow
We show that every complete entire self-shrinking solution on complex Euclidean space to the Kahler-Ricci flow must be generated from a quadratic potential.
math.DG↗
A Note on Conformal Ricci Flow
In this note we study conformal Ricci flow introduced by Arthur Fischer. We use DeTurck's trick to rewrite conformal Ricci flow as a strong parabolic-elliptic partial differential equations. Then we prove short time existences for conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We show that Yamabe constant is monotonically increasing along conformal Ricci flow on compact manifolds. We also show that conformal Ricci flow is the gradient flow for the ADM mass on asymptotically flat manifolds.
math.DG↗
The linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation
We show that the linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation. Similar result holds for shrinkers. We also present an interpolation between Perelman's and Cao--Hamilton's Harnacks on a steady soliton.
math.DG↗
Mild Ricci curvature restrictions for steady gradient Ricci solitons
We show for a complete noncompact steady Ricci soliton that there exists a sequence {x_i} of points tending to infinity such that |Rc|(x_i) limits to zero.
math.DG↗
Weak Ricci curvature bounds for Ricci shrinkers
We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.
math.DG↗
Some elementary consequences of Perelman's canonical neighborhood theorem
In this purely expository note, we recall a few known properties of 3-dimensional singularity models. These properties are direct consequences of Perelman's canonical neighborhood theorem for 3-dimensional Ricci flow and compactness theorem for 3-dimensional kappa-solutions.
math.DG↗
A lower bound for the scalar curvature of certain steady gradient Ricci solitons
In this very short note we prove a lower bound for the scalar curvature of certain steady gradient Ricci solitons.
math.DG↗
A lower bound for the scalar curvature of noncompact nonflat Ricci shrinkers
We show that recent work of Ni and Wilking yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kähler--Ricci shrinkers by Feldman, Ilmanen, and Knopf exhibit that this result is sharp.
math.DG↗
A necessary and sufficient condition for Ricci shrinkers to have positive AVR
In this short note we observe that a recent result of C.-W. Chen meshes well with earlier work of H.-D. Cao and D.-T. Zhou, O. Munteanu, J. Carrillo and L. Ni, and S.-J. Zhang to give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci solitons to have positive asymptotic volume ratio.
math.DG↗
Local curvature bound in Ricci flow
Some modification of the old version.In this note we give a proof of a result which is related to Perelman's theorem in Section 10.3 of the paper "The entropy formula for the Ricci flow and its geometric applications".
math.DG↗
Convergence of fundamental solutions of linear parabolic equations under Cheeger-Gromov convergence
In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral estimate of fundamental solutions.
math.DG↗
Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds
In this work we derive local gradient and Laplacian estimates of the Aronson-Bénilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.
math.DG↗
Evolution of Yamabe constant under Ricci flow
In this note under a crucial technical assumption we derive a differential equality of the Yamabe constant $\mathcal{Y} (g (t))$ where $g (t)$ is a solution of the Ricci flow on a closed manifold.
math.DG↗
A note on uniformization of Riemann surfaces by Ricci flow
In this note we clarify that the Rcci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.
math.DG↗