Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth…
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48 results for “linear volume growth”
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.
problem Proving uniqueness of asymptotic limits for noncollapsed Ricci flat manifolds with linear volume growth.
method Relating uniqueness to the existence of a harmonic function asymptotic to a Busemann function, proving uniqueness via a monotone quantity.
result Proves uniqueness of the asymptotic limit and establishes a polynomial convergence rate.
On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvaturemath.DG
The study proves manifolds with specific curvature and volume properties always split off a line at infinity.
problem Understanding the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature.
method Analyzing properties of Busemann functions and constructing examples.
result Manifolds with the specified properties always split off a line at infinity, with bounded diameter of level sets of Busemann functions.
In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…
Proves effective linear volume growth for 3-manifolds with positive scalar curvature.
problem Volume growth of three-manifolds with positive scalar curvature.
method Utilizes the technique of μ-bubbles and almost-splitting theorem.
result Proves effective linear volume growth for 3-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature.
We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.
Study volume growth and asymptotic cones of nonnegative Ricci curvature manifolds.
problem Whether the volume growth order of manifolds is greater than or equal to the dimension of their asymptotic cones.
method Analyzing asymptotic cones and volume growth conditions, extending Sormani's results.
result Existence of asymptotic cones with upper box dimension at most equal to the volume growth order.
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this resul…
Affirmative answer to splitting question for open manifolds with nonnegative Ricci curvature and linear volume growth.
problem Understanding the structure of universal covers of open manifolds with nonnegative Ricci curvature and linear volume growth.
method Proving the universal cover splits off an isometric -factor.
result If an open manifold with nonnegative Ricci curvature has linear volume growth, its universal cover is isometric to a metric product .
Two rigidity theorems for manifolds with nonnegative Ricci curvature and specific volume growth.
problem Characterizing manifolds with nonnegative Ricci curvature and specific volume growth properties.
method Rigidity theorems based on volume growth and existence of harmonic functions.
result Conditions for the Riemannian universal cover to have Euclidean volume growth and existence of nonconstant linear growth harmonic functions.
Study of manifolds with nonnegative Ricci curvature and slow relative volume growth.
problem Understanding fundamental groups of manifolds with specific volume growth.
method Defined a function RV(s) to describe volume growth and studied fundamental groups with slow relative volume growth.
result If RV(s) grows sublinearly, fundamental groups are almost abelian or finite.
We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".
The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Seco…
Study proves uniqueness of asymptotic limits for specific manifolds.
problem Proving uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth.
method Established using natural curvature and cross section assumptions.
result Uniqueness and exponential convergence rate for complete noncollapsed Ricci-flat manifolds with linear volume growth.
Study reveals fundamental group properties of manifolds with specific curvature and growth.
problem Understanding the fundamental groups of manifolds with nonnegative Ricci curvature and linear volume growth.
method Analysis of covering spaces and rigidity results for RCD spaces.
result Fundamental groups of manifolds contain subgroups of finite index or are finite.
We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give …
The paper studies 3D manifolds with positive scalar curvature and volume growth.
problem Understanding the geometry of 3D manifolds with positive scalar curvature.
method Analyzes volume and geometric properties of 3D complete manifolds with positive scalar curvature, considering different curvature conditions.
result Volume growth estimates for 3D manifolds with positive scalar curvature, answering Gromov's question affirmatively.
The Almost Rigidity of Manifolds with Lower Bounds on Ricci Curvature and Minimal Volume Growthmath.DG
We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…
Volume estimates and classification theorem for constant weighted mean curvature hypersurfacesmath.DG
In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces . We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of pro…
The study finds a limit on the volume growth of certain 3-manifolds.
problem Volume growth of noncompact 3-manifolds with specific curvature properties.
method Analyzes 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and positive scalar curvature.
result Optimal asymptotic volume ratio for manifolds with finite first Betti number and linear volume growth.
Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the norm square of the pull-back of the target volume form by such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic ave…
Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.
problem Estimating volume growth for gradient Ricci solitons.
method Survey and prove new volume growth estimates.
result New volume growth estimates for expanding gradient Ricci solitons.
The study provides volume growth estimates for specific types of manifolds.
problem Estimating volume growth for Ricci solitons and quasi-Einstein manifolds.
method Similar to classical results, the study proves volume growth estimates for gradient Ricci solitons and quasi-Einstein manifolds.
result Sharp volume growth estimates for gradient shrinking Ricci solitons and upper bound volume growth estimates for quasi-Einstein manifolds.
Study dynamics of -multipliers on harmonic manifolds with exponential volume growth.
problem Characterize the behavior of -multipliers on harmonic manifolds of purely exponential volume growth.
method Analyzing the dynamics of -multipliers on non-compact harmonic manifolds, using Fourier transformation and properties of radial functions.
result Show that -multipliers acting nicely on smooth functions with compact support for cannot be chaotic.
Paper proves volume growth estimate for steady gradient Ricci solitons.
problem Estimating the volume growth of steady gradient Ricci solitons.
method Proved a volume growth estimate using Nash entropy.
result Volume growth rate is no smaller than $r^{ rac{n+1}{2}}$.
The study shows that the visible range from a point on harmonic manifolds follows an exponential distribution.
problem Understanding the visible range from a point on harmonic manifolds.
method Analyzing Poisson Boolean models on harmonic manifolds, focusing on the geometric mechanism of tube volumes around geodesic segments.
result The visible range from a point on harmonic manifolds follows an exponential distribution.
In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fund…
Study on volume growth of horospheres in specific Heintze groups.
problem Volume growth of horospheres in diagonalizable Heintze groups.
method Analysis of volume growth in Heintze groups with diagonalizable structure.
result Two distinct isometry and quasi-isometry classes of horospheres with explicit volume growth calculations.
Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cu…
Graphs with stronger curvature grow faster.
problem Understanding volume growth on graphs with various curvatures.
method Examined inner-outer and Ricci-Ollivier curvatures to relate them to volume growth.
result Graphs with stronger inner-outer curvature growth have faster volume growth.
Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.
problem Understanding if manifolds with specific curvature bounds and volume growth must be of finite topological type.
method Constructs a family of dimensional open manifolds with positive Ricci curvature and sectional curvature bounds.
result Volume growth can be arbitrarily close to quadratic, and Betti numbers are infinite.
The paper examines properties and rigidity of self-expanders in Euclidean space.
problem Characterizing and estimating properties of self-expanders in Euclidean space.
method Analyzing mean curvature flow, volume growths, and stability of self-expanders.
result Proves the uniqueness of certain self-expanders in 3D space.
In this paper, it is shown that for any closed orientable -manifold with positive simplicial volume, the growth of the Seifert volume of its finite covers is faster than the linear rate. In particular, each closed orientable -manifold with positive simplicial volume has virtually positive Seifert volume. The resu…
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.
We make some improvements to our previous results. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. These results al…
Study volume growth in Milnor fibers using real Lagrangians.
problem Volume growth in Milnor fibers of Brieskorn polynomials.
method Investigate real Lagrangians, use Smith inequality for involutions in wrapped Floer homology.
result Uniform lower bound of volume growth for a class of Brieskorn polynomials.
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 growt…
In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have s…
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finit…
Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.
problem Estimating heat kernels and Green's functions on specific Riemannian manifolds.
method Analyzing noncompact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Sharp Moser-Trudinger inequalities on manifolds with Euclidean volume growth.
The study proves that certain manifolds can have metrics with specific volume growth.
problem Determining if manifolds with positive scalar curvature can have metrics with a given volume growth.
method Using Gromov-Lawson and Grimaldi-Pansu constructions, the study proves the existence of metrics with the desired volume growth on specific manifolds.
result The study positively answers the question for manifolds that are infinite connected sums of closed manifolds with positive scalar curvature.
Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.
problem Estimating the essential spectrum of minimal submanifolds.
method Using volume growth to bound the bottom of the essential spectrum.
result Improved essential spectrum estimate for minimal submanifolds.
New proof linking scalar curvature to volume growth on 3-manifolds.
problem Relating scalar curvature to volume growth on 3-manifolds.
method Theory of μ-bubbles and almost splitting theorem.
result New proof of recent result by Munteanu--Wang.
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
The paper explores curvature-free effects in manifolds with volume growth and ends-counting.
problem Investigating curvature-free effects in manifolds with volume growth and ends-counting.
method Establishing two main theorems about volume growth and ends-counting.
result Proves the existence of smooth bounded mean-concave exhaustion and escaping geodesic lines.
Upper bound on geodesic ball volume in Riemannian manifolds.
problem Bounding geodesic ball volume in Riemannian manifolds.
method Techniques to provide an upper bound on geodesic ball volume.
result Upper bound on geodesic ball volume in Euclidean space.