Maximal Volume Growth Research Articles

Large time behavior of the heat kernel

In this paper, we study the large time behavior of the heat kernel on complete Riemannian manifolds with nonnegative Ricci curvature, which was studied by P. Li with additional maximum volume growth assumption. Following Y. Ding’s original strategy, by blowing down the metric, using Cheeger and Colding’s theory about limit spaces of Gromov-Hausdorff convergence, combining with the Gaussian upper bound of heat kernel on limit spaces, we succeed in reducing the limit behavior of the heat kernel on manifold to the values of heat kernels on tangent cones at infinity of manifold with renormalized measure. As one application, we get the consistent large time limit of heat kernel in more general context, which generalizes the former result of P. Li. Furthermore, by choosing different sequences to blow down the suitable metric, we show the first example manifold whose heat kernel has inconsistent limit behavior, which answers an open question posed by P. Li negatively.

On the Perelman’s reduced entropy and Ricci flat manifolds with maximal volume growth

In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman's reduced volume of Ricci flow. We show that if $(M^n,g)$ is an noncompact complete Ricci flat manifold with maximal volume growth satisfying $|Rm|(x)\to 0$ as $d(x)=d_g(x,p)\to \infty$, then $M^n$ has the quadratic curvature decay. Some applications to this result are also presented.

Morphological selection of Euclidean and fractal patterns of nonequilibrium growth of ice in supercooled water

A morphology diagram of Euclidean and fractal patterns of the nonequilibrium growth of ice in supercooled water has been constructed based on the experimental dependence of the growth rate of ice crystals of different shapes on the initial supercooling of water. An empirical morphological principle follows from this diagram: with an increase in water supercooling, the phase implementing the maximum volume growth rate of ice crystals is selected.

Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

We consider the class of minimal surfaces given by the graphical strips $${{\mathcal S}}$$ in the Heisenberg group $${{\mathbb {H}}^1}$$ and we prove that for points p along the center of $${{\mathbb {H}}^1}$$ the quantity $${\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}}$$ is monotone increasing. Here, Q is the homogeneous dimension of $${{\mathbb {H}}^1}$$ . We also prove that these minimal surfaces have maximum volume growth at infinity.

Comparing volume growth in pure and mixed stands of Pinus sylvestris and Quercus pyrenaica

• In Mediterranean forestry, it is important to improve knowledge about mixed stands dynamics, including their productivity. Previous studies have focused on the interactions between different species (competition, reduction of competition and facilitation) depending on site, species composition and structure. • At the centre of this research are the possible differences between pure and mixed stands of Pinus sylvestris and Quercus pyrenaica in terms of density-growth relationships and volume growth per species. • Using data from the second and third Spanish National Forest Inventory (606 plots), volume increment models for these species were fitted. Both species displayed a similar density-growth pattern for pure and mixed stands, with a maximum volume growth at maximum density. Volume increment per occupied area was also found to be greater in mixed stands as opposed to pure stands, suggesting a species interaction with reduced levels of competition in the former. However, the total volume growth was generally lower in mixed stands since the growth rate of oak is much lower. • The results highlight the expedience of favouring P. sylvestris-Q. pyrenaica mixed stands with higher proportions of pine trees in order to gain the benefits of a more complex forest whilst retaining an acceptable level of wood production.

On the Complex Structure of Kähler Manifolds with Nonnegative Curvature

We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative and bounded holomorphic bi-sectional curvature and maximal volume growth is biholomorphic to complex Euclidean space Cn. We also show that the volume growth condition can be removed if we assume the Kähler manifold has average quadratic scalar curvature decay and positive curvature operator.

Entropy and reduced distance for Ricci expanders

Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij/∂t = − 2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W+ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals μ+ and v+. The forward reduced volume θ+ is monotone in general and constant exactly on expanders. A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/tn/2 (Hamilton) and -λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like tn/2(maximal volume growth) then W+, θ+ and -λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture.

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete non-compact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have maximal volume growth, then M is biholomorphic to C 2. This gives a partial affirmative answer to the well-known conjecture of Yau [41] on uniformization theorem. During the proof, we also verify an interesting gap phenomenon, predicted by Yau in [42], which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.

Optimal stand density: a solution

The search for a stand density that maximizes total volume growth has continued since the beginning of forestry without producing a definite answer. One of the reasons is that the effect of density on growth is not always separated from those of tree size and age. Such a separation is not easy when the relationship between density and growth is expressed as a graph (Langsaeter's curve). This study develops a simple model that accounts for each main growth predictor individually. It allows one to calculate the density that maximizes volume growth at any given moment (current annual increment of volume). Just as the maximum Reineke's index, this density optimum does not change with age. Fitting the model to long-term data confirms the obvious: because a complete crown closure intercepts more light than a broken canopy, the densest stands produce maximum volume growth. Thus, the current optimal density is equal to maximum density. What is less obvious is that maximum current stand volume growth does not sum up to maximum stand volume. In addition to density, stand volume depends on average tree size, which is larger in less dense stands. The high current density that maximizes volume growth also minimizes diameter increment, which eventually reduces average diameter and stand volume. When density is kept at a stationary (over time) level by thinning, maximum volume is produced by stands with substantially lower density than the current optimum. even higher volume can probably be obtained when density changes with age. The challenge is to find this optimum path of density.

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 $$ \lim _{{r \to \infty }} \frac{{r^{2} }} {{V{\left( r \right)}}}{\int_{B{\left( r \right)}} {\Delta hdv{\kern 1pt} = {\kern 1pt} {\kern 1pt} 0} } $$ 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.

Complete manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity

In this paper, we study complete Riemannian manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity. We show that M is of finite topological type if it has either maximal volume growth or minimal volume growth.

Sharp bounds for the Green’s function and the heat kernel

The main purpose of this note is two-fold. The first is to give simple proofs to some recent theorems of Colding-Minicozzi in [C-M] on the asymptotic behavior of the Green’s function on manifolds with non-negative Ricci curvature and maximal volume growth. The second is to give sharp upper and lower estimates for the heat kernel on this class of manifolds. In fact, after integrating these estimates with respect to the time variable, they yield sharp pointwise bounds for the Green’s function. The interested reader should refer to [C-M] for a detailed history of this subject. Throughout this paper, we assume that M is an n-dimensional complete noncompact manifold with non-negative Ricci curvature. With the exception of Theorem 2.1 and Corollary 2.3, we will assume that n ≥ 3. Let us denote the distance between x, y ∈M by d(x, y). If Bx(r) denotes the geodesic ball of radius r centered at x, then we denote Vx(r) and Ax(r) to be the volume and the area of Bx(r) and ∂Bx(r), respectively. In all the theorems, with the exception of Theorem 1.2, we will assume that M has maximal volume growth. This means that for some fixed point p ∈M ,

On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth

On etudie le comportement asymptotique de varietes de Riemann. On construit des coordonnees a l'infini sous certaines conditions intrinseques

?-flat manifolds and Riemannian submersions

In this paper, we show that a certain rigidity condition (∑-flatness) for open nonnegatively curved manifoldsM is preserved by Riemannian submersions. The result can be applied to quotients ofM by groups of isometries. ∑-flat metrics are also used to derive a splitting theorem for distance tubes of maximal volume growth.

On Yau’s uniformization conjecture

Let $M^n$ be a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $\mathbb{C}^n$. This confirms Yau's uniformization conjecture when M has maximal volume growth.