Maximal Volume Growth Research Articles

Subquadratic harmonic functions on Calabi‐Yau manifolds with maximal volume growth

AbstractOn a complete Calabi‐Yau manifold with maximal volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon‐Hein. We prove this result by proving a Liouville‐type theorem for harmonic 1‐forms, which follows from a new local estimate of the exterior derivative.

Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian manifolds

Abstract We show that there exists a nonconstant CR-holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative pseudohermitian bisectional curvature with the CR maximal volume growth property. This is the very first step toward the CR analogue of the Yau uniformization conjecture which states that any complete noncompact Sasakian manifold of positive pseudohermitian bisectional curvature is CR biholomorphic to the standard Heisenberg group. More precisely, we first construct CR-holomorphic functions with controlled growth in a sequence of exhaustion domains in Sasakian manifolds by applying the Cheeger–Colding theory. Secondly, via the CR analogue of a tangent cone at infinity and the three-circle theorem, we are able to take the subsequence to obtain a nonconstant CR-holomorphic function of polynomial growth.

Kähler manifolds with almost nonnegative curvature

In this paper, we construct local and global solutions to the K\"ahler-Ricci flow from a non-collapsed K\"ahler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed K\"ahler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete K\"ahler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and maximal volume growth.

Weakly PIC1 Manifolds with Maximal Volume Growth

In this article we study the local estimates of Ricci flow when the curvature of the initial metric is of almost PIC1 and use it to study complete manifolds with weakly PIC1 and maximal volume growth.

Compactification of certain Kähler manifolds with nonnegative Ricci curvature

We prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete non-compact Ricci-flat Kähler manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, such an affine variety degenerates in two steps to the unique metric tangent cone at infinity.

Empowering Forest Owners with Simple Volume Equations for Poplar Plantations in the Órbigo River Basin (NW Spain)

Hybrid poplar plantations are becoming increasingly important as a source of income for farmers in northwestern Spain, as rural depopulation and farmers aging prevent landowners from planting other labor-intensive crops. However, plantation owners, usually elderly and without formal forestry background, lack of simple tools to estimate the size and volume of their plantations by themselves. Therefore, farmers are usually forced to rely on the estimates made by the timber companies that are buying their trees. With the objective of providing a simple, but empowering, tool for these forest owners, simple equations based only on diameter were developed to estimate individual tree volume for the Órbigo River basin. To do so, height and diameter growth were measured for 10 years (2009–2019) in 404 trees growing in three poplar plantations in Leon province. An average growth per tree of 1.66 cm year−1 in diameter, 1.52 m year−1 in height, and 0.03 m3 year−1 in volume was estimated, which translated into annual volume increment of 13.02 m3 ha−1 year−1. However, annual volume increment was different among plots due to their fertility, with two plots reaching maximum volume growth around 11 years since planting and another at 13 years, encompassing the typical productivity range in plantations in this region. Such data allowed developing simple but representative linear, polynomial and power equations to estimate volume explaining 93%–98% of the observed variability. Such equations can be easily implemented in any cellphone with a calculator, allowing forest owners to accurately estimate their timber existences by using only a regular measuring tape to measure tree diameter. However, models for height were less successful, explaining only 75%–76% of observed variance. Our approach to generate simplified volume equations has shown to be viable for poplar, but it could be applied to any species for which several volume equations are available.

Growth–density relationship in mixed stands – Results from long-term experimental plots

Forest management mainly controls wood growth through the regulation of stand density. Knowledge of the growth–density relationship is based on numerous studies in pure stands. Currently, in times of more climate plasticity and more nature-oriented forests, silviculture with mixed stands is demanded of forest policy. It is well known that the competition mechanisms in mixed stands are different to those of pure stands. Hence, the question arises for forest managers whether the regulation of density must be different, too.In an attempt to answer this question, we investigated 481 stand surveys from 124 long-term experimental plots of European beech stands, mixed with conifers and also broadleaf trees. With generalized additive models, we estimated the influence of the density on stand volume growth and how it changed with stand development and shade tolerance of the admixed species. The density of the mixed stands was given on a relative scale. 100% density took into account the maximum species-specific density plus a mixed stand specific addition.We found that mixed stands have a constant growth level in the medium- to high-density phase (70–100%). Although we have tried to take into account the higher density in mixed stands, 10% of the stands outperformed the 100% benchmark and exhibited a growth gain in this over-dense phase (>100%). Finally, the growth-density response resembles a cubic function with a saddle or optimum between medium- to high-density phase and an increase of growth in the low-density and the over-density phase. The inclusion of stand development led to the oblation of the growth–density response in younger ages. The admixture of shade-intolerant species with beech caused a more intensified cubic course of the relationship.Finally, we could show that a short-term decrease of the density also has an effect on growth. In terms of growth, weak thinnings are better than severe thinnings. Thinnings from above have a significant positive effect on growth, especially for longer observation periods of up to 10 years.We concluded that mixed stands need an adapted thinning compared to pure stands. Regular, weak thinnings from above with a very high stand density enables maximum volume growth gains in European beech mixed stands. We could show that the growth-density relationship is conceptually different to pure stands, because mixing effect causes density section above 100%. However, the study emphasizes the need for a systematic experimental setup for a better understanding of intraspecific interactions at different densities.

Aspectos metodológicos para generar diagramas de manejo de la densidad de rodales con base en el índice de Reineke

La densidad indica el grado de aglutinamiento de los árboles en un rodal, sus cambios afectan el rendimiento maderable y el tamaño de los individuos. El modelo de Reineke y su índice de densidad del rodal (IDR) se usan en la construcción de diagramas para manejar la densidad (DMD), el IDR define el límite superior del autoaclareo y el punto mínimo de ocupación total del sitio. A pesar de su importancia, en la construcción de un DMD es común que exista desconocimiento o poca claridad sobre aspectos metodológicos relevantes. El objetivo es presentar elementos teóricos básicos que el responsable del manejo forestal debe considerar para construir un DMD. Bajo el enfoque clásico de modelación de la función de Reineke, se abordan conceptos sobre densidad, el tipo y características de la fuente de información dasométrica por utilizar, la selección de sitios en alta densidad, el ajuste estadístico de la función, criterios metodológicos para desplazar y delimitar la línea de autoaclareo, y juicios para determinar las zonas de crecimiento de Langsater. Se resalta que la línea de autoaclareo puede delimitarse a partir de sitios de muestreo para inventario maderable, y los de densidad alta se seleccionan aplicando el enfoque del IDR máximo (IDRmáx); el método de mínimos cuadrados ordinarios lineales para ajustar la función de Reineke es robusto y corrige la heterocedasticidad. Para el crecimiento máximo en volumen, los rodales es factible manejarlos entre 35 y 65 % con respecto al IDRmáx. Los niveles óptimos de densidad deben determinarse por especie.

Chern–Ricci flows on noncompact complex manifolds

In this work, we obtain existence criteria for Chern-Ricci flows on noncompact manifolds. We generalize a result by Tossati-Wienkove on Chern-Ricci flows to noncompact manifolds and at the same time generalize a result for Kahler-Ricci flows by Lott-Zhang to Chern-Ricci flows. Using the existence results, we prove that any complete noncollapsed Kahler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kahler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume. Combining this result with the result of Chau-Tam, we give another proof that a complete noncompact Kahler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to the complex Euclidean space. This last result has already been proved by Gang Liu recently using other methods. This last result is partial confirmation of a uniformization conjecture of Yau.

The growth rate of harmonic functions

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than three, we prove that quantitative strong unique continuation yields the existence of nonconstant polynomial growth harmonic functions. Also the uniform bound of frequency for linear growth harmonic functions on such manifolds is obtained, and this confirms a special case of Colding-Minicozzi conjecture on frequency.

Degenerations of Cn and Calabi–Yau metrics

We construct infinitely many complete Calabi–Yau metrics on Cn for n≥3, with maximal volume growth and singular tangent cones at infinity. In addition, we construct Calabi–Yau metrics in neighborhoods of certain isolated singularities whose tangent cones have singular cross section, generalizing work of Hein and Naber.

Weighted Littlewood–Paley inequalities for heat flows in RCD spaces

We establish inequalities on vertical Littlewood–Paley square functions for heat flows in the weighted L2 space over metric measure spaces satisfying the RCD⁎(0,N) condition with N∈[1,∞) and the maximum volume growth assumption. In the noncompact setting, the later assumption can be removed by showing that the volume of the ball growths at least linearly. The estimates are sharp on the growth of the 2-heat weight and the 2-Muckenhoupt weight considered. The p-Muckenhoupt weight and the p-heat weight are also compared for all p∈(1,∞).

A new complete Calabi\u2013Yau metric on {mathbb {C}}^3

Motivated by the study of collapsing Calabi–Yau 3-folds with a Lefschetz K3 fibration, we construct a complete Calabi–Yau metric on {mathbb {C}}^3 with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi–Yau metric has singular tangent cone at infinity {mathbb {C}}^2/{mathbb {Z}}_2 times {mathbb {C}}, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein’s Ph.D. thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms.

Ertragskundliche Aspekte und Strukturdynamik der Buchen-Lichtwuchsdurchforstung

Effects of heavy selective thinnings in beech stands with regard to stand productivity and structural diversity Based on thinning trials in beech stands initiated in the early 1970s, the effects of selective thinnings in favour of 100 to 120 future crop trees were analysed with regard to volume productivity and the stands' structural diversity. The results show that these kinds of thinnings reduce volume productivity by approximately 6% compared to the maximum volume growth. The reduction in volume growth is mostly caused by the fact that the strongly released crop trees have reduced their height growth for the benefit of a lateral crown expansion, while basal area growth was not affected. Calculations indicate that silvicultural systems working with only 50 future crop trees per hectare may decrease the maximum volume production by about 12%. Furthermore, the results show that the intense crown releases caused by selection thinnings increase the horizontal and vertical diversity of beech stands, since the trees in the understorey profit from better light conditions. The indices calculated for diameter and height diversity of the selection thinning plots approach the values being typical for single tree selection forests (“Plenterwälder”). Already after three to four consecutive selection thinnings, the beech stands' diameter distributions displayed an exponentionally decreasing shape typical for single tree selection forests close to equilibrium. Growth simulations revealed that it might even be possible to directly transform beech stands by selection thinnings into single tree selection structures. However, presumably negative effects on timber quality to be expected under single tree selection structures should be considered as a critical aspect with regard to theoretically possible transformation.

On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

Abstract Suppose ( M n , g ) {(M^{n},g)} is a Riemannian manifold with nonnegative Ricci curvature, and let h d ⁢ ( M ) {h_{d}(M)} be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that h d ⁢ ( M ) {h_{d}(M)} is finite. Later on, there are many researches which give better estimates of h d ⁢ ( M ) {h_{d}(M)} . In this paper, we study the behavior of h d ⁢ ( M ) {h_{d}(M)} when d is large. More precisely, suppose ( M n , g ) {(M^{n},g)} has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of h d ⁢ ( M ) {h_{d}(M)} in terms of the growth order d, the dimension n and the asymptotic volume ratio α = lim R → ∞ ⁡ Vol ⁢ ( B p ⁢ ( R ) ) R n {\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}} . When α = ω n {\alpha=\omega_{n}} , i.e., ( M n , g ) {(M^{n},g)} is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of h d ⁢ ( ℝ n ) {h_{d}(\mathbb{R}^{n})} .

Yau's uniformization conjecture for manifolds with non-maximal volume growth

The well-known Yau's uniformization conjecture states that any complete noncompact Kähler 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 the 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 C1n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, C1n 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 Kähler manifolds with minimal volume growth.

Heat kernel estimates on connected sums of parabolic manifolds

We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li–Yau estimate. The key result is the on-diagonal upper bound of the heat kernel at a central point. Contrary to the non-parabolic case (which was settled in [15]), the on-diagonal behavior of the heat kernel in our case is determined by the end with the maximal volume growth function. As examples, we give explicit heat kernel bounds on the connected sums R2#R2 and R1#R2 where R1=R+×S1.

Sharp heat kernel bounds and entropy in metric measure spaces

We establish sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying $\RCD(0,N)$ ( equivalently, $\RCD^\ast(0,N)$) condition with $N\in \mathbb{N}\setminus\{1\}$ and having 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 Riemannian manifolds and also in metric measure spaces obtained recently by the author with R. Jiang and H. Zhang in \cite{JLZ2014}.

Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture

We study the uniformization conjecture of Yau by using the GromovHausdorff convergence. As a consequence, we confirm Yau’s finite generation conjecture. More precisely, on a complete noncompact Kahler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course of the proof, we prove if M is a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, then M is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on Kahler manifolds with nonnegative bisectional curvature.

On the volume growth of Kähler manifolds with nonnegative bisectional curvature

Let $M$ be a complete K\"ahler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth, we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni. There are two essential ingredients in the proof: The Cheeger-Colding theory on Gromov-Hausdorff convergence of manifolds; the three circle theorem for holomorphic functions.