Regularity of weak solutions of a complex Monge-Amp\`ere equation

We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.Comment: 9 pages; removed a sentence from the introduction, other minor change

On Volume Growth of Gradient Steady Ricci Solitons

In this paper we study volume growth of gradient steady Ricci solitons. We show that if the potential function satisfies a uniform condition, then the soliton has at most Euclidean volume growth.Comment: 8 page

Stability of K\"ahler-Ricci flow in the space of K\"ahler metrics

In this paper, we prove that on a Fano manifold $M$ which admits a K\"ahler-Ricci soliton (\om,X), if the initial K\"ahler metric \om_{\vphi_0} is close to \om in some weak sense, then the weak K\"ahler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if \vphi_0 is also $K_X$-invariant, then the weak modified K\"ahler-Ricci flow converges exponentially to a unique K\"ahler-Ricci soliton nearby. Especially, if the Futaki invariant vanishes, we may delete the $K_X$-invariant assumption. The methods based on the metric geometry of the space of the K\"ahler metrics are potentially applicable to other stability problem of geometric flow near a critical metric.Comment: 28 pages, 1 figure

Notes on Perelman's papers

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".Comment: 216 pages, minor corrections mad

Graph Convolutional Neural Networks for Web-Scale Recommender Systems

Recent advancements in deep neural networks for graph-structured data have led to state-of-the-art performance on recommender system benchmarks. However, making these methods practical and scalable to web-scale recommendation tasks with billions of items and hundreds of millions of users remains a challenge. Here we describe a large-scale deep recommendation engine that we developed and deployed at Pinterest. We develop a data-efficient Graph Convolutional Network (GCN) algorithm PinSage, which combines efficient random walks and graph convolutions to generate embeddings of nodes (i.e., items) that incorporate both graph structure as well as node feature information. Compared to prior GCN approaches, we develop a novel method based on highly efficient random walks to structure the convolutions and design a novel training strategy that relies on harder-and-harder training examples to improve robustness and convergence of the model. We also develop an efficient MapReduce model inference algorithm to generate embeddings using a trained model. We deploy PinSage at Pinterest and train it on 7.5 billion examples on a graph with 3 billion nodes representing pins and boards, and 18 billion edges. According to offline metrics, user studies and A/B tests, PinSage generates higher-quality recommendations than comparable deep learning and graph-based alternatives. To our knowledge, this is the largest application of deep graph embeddings to date and paves the way for a new generation of web-scale recommender systems based on graph convolutional architectures.Comment: KDD 201

Normalized Ricci flow on Riemann surfaces and determinants of Laplacian

In this note we give a simple proof of the fact that the determinant of Laplace operator in smooth metric over compact Riemann surfaces of arbitrary genus $g$ monotonously grows under the normalized Ricci flow. Together with results of Hamilton that under the action of the normalized Ricci flow the smooth metric tends asymptotically to metric of constant curvature for $g\geq 1$, this leads to a simple proof of Osgood-Phillips-Sarnak theorem stating that that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of Laplace operator is maximal on metric of constant curvatute.Comment: a reference to paper math.DG/9904048 by W.Mueller and K.Wendland where the main theorem of this paper was proved a few years earlier is adde

Does SN 1987A contain a rapidly vibrating neutron star

If the recently reported 0.5 ms-period pulsed optical signal from the direction of Supernova 1987A originated in a young neutron star, its interpretation as a rotational period has difficulties. The surface magnetic field would have to be much lower than expected, and the high rotation rate may rule out preferred nuclear equations of state. It is pointed out here that a remnant radial vibration of a neutron star, excited in the supernova event, may survive for several years with about the observed (gravitationally redshifted) period. Heavy ions at the low-density stellar surface, periodically shocked by the vibration, may efficiently produce narrow pulses of optical cyclotron radiation in a surface field of about a trillion gauss

Shakedown analysis for rolling and sliding contact problems

There is a range of problems where repeated rolling or sliding contact occurs. For such problems shakedown and limit analyses provides significant advantages over other forms of analysis when a global understanding of deformation behaviour is required. In this paper, a recently developed numerical method. Ponter and Engelhardt (2000) and Chen and Ponter (2001), for 3-D shakedown analyses is used to solve the rolling and sliding point contact problem previously considered by Ponter, Hearle and Johnson (1985) for a moving Herzian contact, with friction, over a half space. The method is an upper bound programming method, the Linear Matching Method, which provides a sequence of reducing upper bounds that converges to the least upper bound associated with a finite element mesh and may be implemented within a standard commercial finite element code. The solutions given in Ponter, Hearle and Johnson (1985) for circular contacts are reproduced and extended to the case when the frictional contact stresses are at an angle to the direction of travel. Solutions for the case where the contact region is elliptic are also given

String Bracket and Flat Connections

Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras \Psi: \H_{2\ast} (L M) \to {\o}(\Mc), where \H_{2\ast} (LM) is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and \Mc is the Maurer-Cartan moduli space of the graded differential Lie algebra \Omega^\ast (M, \adp), the differential forms with values in the associated adjoint bundle of $P$. For a 2-dimensional manifold $M$, our Lie algebra map reduces to that constructed by Goldman in \cite{G2}. We treat different Lie algebra structures on \H_{2\ast}(LM) depending on the choice of the linear reductive Lie group $G$ in our discussion.Comment: 28 pages. This is the final versio

Non-ancient solution of the Ricci flow

For any complete noncompact K$\ddot{a}$hler manifold with nonnegative and bounded holomorphic bisectional curvature,we provide the necessary and sufficient condition for non-ancient solution to the Ricci flow in this paper.Comment: seven pages, latex fil