YAU'S UNIFORMIZATION CONJECTURE FOR MANIFOLDS WITH NON-MAXIMAL VOLUME GROWTH

The well-known Yau＇s uniformization conjecture states that any complete noncompact Kahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed, by G. Liu in [23]. In the first part, we will give a survey on thc progress. In the second part, we will consider Yau＇s conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number Cn1 is an essential condition to solve Yau＇s conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, Cn1 is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau＇s uniformization conjecture on Kahler manifolds with minimal volume growth.

Using machine learning algorithms to estimate stand volume growth of Larix and Quercus forests based on national-scale Forest Inventory data in China

Estimating the volume growth of forest ecosystems accurately is important for understanding carbon sequestration and achieving carbon neutrality goals.However,the key environmental factors affecting volume growth differ across various scales and plant functional types.This study was,therefore,conducted to estimate the volume growth of Larix and Quercus forests based on national-scale forestry inventory data in China and its influencing factors using random forest algorithms.The results showed that the model performances of volume growth in natural forests(R^(2)=0.65 for Larix and 0.66 for Quercus,respectively)were better than those in planted forests(R^(2)=0.44 for Larix and 0.40 for Quercus,respectively).In both natural and planted forests,the stand age showed a strong relative importance for volume growth(8.6%–66.2%),while the edaphic and climatic variables had a limited relative importance(<6.0%).The relationship between stand age and volume growth was unimodal in natural forests and linear increase in planted Quercus forests.And the specific locations(i.e.,altitude and aspect)of sampling plots exhibited high relative importance for volume growth in planted forests(4.1%–18.2%).Altitude positively affected volume growth in planted Larix forests but controlled volume growth negatively in planted Quercus forests.Similarly,the effects of other environmental factors on volume growth also differed in both stand origins(planted versus natural)and plant functional types(Larix versus Quercus).These results highlighted that the stand age was the most important predictor for volume growth and there were diverse effects of environmental factors on volume growth among stand origins and plant functional types.Our findings will provide a good framework for site-specific recommendations regarding the management practices necessary to maintain the volume growth in China's forest ecosystems.

Open Manifolds with Nonnegative Ricci Curvature and Large Volume Growth

In this paper, we prove that if M is an open manifold with nonnegativeRicci curvature and large volume growth, positive critical radius, then sup Cp = ∞.As an application, we give a theorem which supports strongly Petersen's conjecture.

VOLUME GROWTH ESTIMATES OF MANIFOLDS WITH NONNEGATIVE CURVATURE OUTSIDE A COMPACT SET

In this article, using the properties of Busemann functions, the authors prove that the order of volume growth of Kahler manifolds with certain nonnegative holomorphic bisectional curvature and sectional curvature is at least half of the real dimension. The authors also give a brief proof of a generalized Yau＇s theorem.

Prostate Volume Growth Rate Changes over Time:Results from Men 18 to 92 Years Old in a Longitudinal Community-based Study

Previous investigations have shown that changes in total prostate volume（TPV） are highly variable among aging men,and a considerable proportion of aging men have a stable or decreasing prostate size.Although there is an abundance of literature describing prostatic enlargement in association with benign prostatic hyperplasia,less is known about the appropriate age cut-off points for TPV growth rate.In this community-based cohort study,TPV was examined once a year in men who had consecutive health checkup,during a follow-up of 4 years.A total of 5058 men（age 18–92 years old） were included.We applied multiple regression analyses to estimate the correlation between TPV growth rate and age.Overall,3232（63.9%） men had prostate growth,and 1826（36.1%） had a stable or decreased TPV during the study period.The TPV growth rate was correlated negatively with baseline TPV（r= –0.32,P〈0.001）.Among 2620 men with baseline TPV 〈15 cm^3,the TPV growth rate increased with age（β=0.98,95% CI：0.77%–1.18%） only up to 53 years old.Among 2188 men with baseline TPV of 15–33.6 cm3,the TPV growth rate increased with age（β=0.84,95% CI,0.66%–1.01%） only up to 61 years old after adjusting for factors of hypertension,obesity,baseline TPV,diabetes mellitus and dyslipidemia.In this longitudinal study,the TPV growth rate increased negatively with baseline TPV,only extending to a certain age and not beyond.Further research is needed to identify the mechanism underlying such differences in prostate growth.

Open Manifolds with Nonnegative Ricci Curvature and Large Volume Growth

in this paper,we prove that a complete n-dimensional Riemannian manifold with nonnegative kth-Ricci curvature, large volume growth has finite topological type provided that lim r→∞{（vol[B（p.r]/ωnrn-αM）rk（n-1）/k＋1（1-α/2）}≤for some COllstant ε〉0 We also prove that a conlplete Riemannian manifold with nonnegative kth-Ricci curvature and undler some pinching conditions is diffeomorphic to R^n.

Volume growth for gradient shrinking solitons of Ricci-harmonic flow

In this paper,we derive an estimate on the potential functions of complete noncompact gradient shrinking solitons of Ricci-harmonic flow,and show that complete noncompact gradient shrinking Ricci-harmonic solitons have Euclidean volume growth at most.

Closed Geodesics and Volume Growth of Open Manifolds with Sectional Curvature Bounded from Below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.

A New Proof of Calabi-Yau's Theorem

We give a new proof of Calabi-Yau's theorem on the volume growth of Rie- mannian manifolds with non-negative Ricci curvature.

THE VOLUME AND TOPOLOGY OF A COMPLETE RIEMANNIAN MANIFOLD

It is conjectured that the manifold with nonnegative Ricci curvature and weaked bounded geometry is of finite topological type, if The paper partially solves this conjecture. In the same time, the paper also discusses the volume growth of a manifold with asymptotically nonnegative Ricci curvature.

Some Results on Space-Like Self-Shrinkers

We study space-like self-shrinkers of dimension n in pseudo-Euclidean space Rm^m＋n with index m. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove rigidity results under minor growth conditions in terms of the mean curvature or the image of Gauss maps.

Infinity Behavior of Bounded Subharmonic Functions on Ricci Non-negative Manifolds

In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau’s Liouville theorem on bounded harmonic functions.

Ricci Curvature and Fundamental Group

Abstract By refined volume estimates in terms of Ricci curvature, the two results due to J. Milnor （1968） are generalized.

On the Isometric Minimal Immersion into a Euclidean Space

We study the volume growth of the geodesic balls of a minimal submanifold in a Euclidean space.A necessary condition for the isometric minimal immersion into a Euclidean space is obtained. A classification of non-positively curved minimal hypersurfaces in a Euclidean space is given.

Sharp heat kernel bounds and entropy in metric measure spaces

We establish the sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying the RCD(0, N)(equivalently, RCD~*(0, N), condition with N∈N\ {1} and having the maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the(minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in the Riemannian manifolds, and some of them appear more explicit and sharper than the ones in metric measure spaces obtained recently by Jiang et al.(2016).