Study shows volume density in central harmonic spaces can vary arbitrarily.
problem Volume density in central harmonic spaces can vary arbitrarily.
method Analyzes asymptotics of volume density function in central harmonic manifolds.
result Volume density in central harmonic spaces can be specified arbitrarily and does not determine geometry.
In this paper, we study volume growth, Liouville theorem and the local gradient estimate for f-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.
Harmonic functions on Calabi-Yau manifolds with maximal volume growth are studied.
problem Characterizing harmonic functions on Calabi-Yau manifolds with maximal volume growth.
method Proved a Liouville type theorem for harmonic 1-forms, using a new local L2 estimate of the exterior derivative. result Subquadratic harmonic functions on Calabi-Yau manifolds with maximal volume growth are the real parts of holomorphic functions.
The paper introduces a new complex analytic invariant called the pointed harmonic volume and its relation to the Johnson homomorphism.
problem Exploring new complex analytic invariants related to the complex structure of Riemann surfaces.
method Defining and computing the pointed harmonic volume as a natural extension of Chen's iterated integrals.
result Established a relationship between the harmonic volume and the first extended Johnson homomorphism.
Harmonic manifolds of hypergeometric type have entropy bounds related to real hyperbolic spaces.
problem Bounding the volume entropy of harmonic manifolds of hypergeometric type.
method Normalized Ricci curvature and entropy analysis.
result Upper and lower bounds for volume entropy of harmonic manifolds of hypergeometric type.
New harmonic Hadamard manifolds defined via hypergeometric equations.
problem Characterizing harmonic Hadamard manifolds using hypergeometric equations.
method Defining harmonic Hadamard manifolds of hypergeometric type and using spherical Fourier transform.
result Characterization of harmonic Hadamard manifolds being of hypergeometric type.
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.
We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.
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…
Method extends eigenfunction construction to non-symmetric spaces.
problem Constructing eigenfunctions on harmonic manifolds.
method Applying Sullivan's method to non-compact harmonic manifolds.
result Eigenfunctions constructed for non-symmetric spaces.
The study proves properties of intersections of horospheres in harmonic spaces.
problem Properties of intersections of horospheres in harmonic spaces.
method Constructing volume preserving mappings using Busemann functions.
result Upper bound of the volume of intersection of horospheres is independent of Busemann function differences.
Covering spaces with exponential growth have non-constant positive harmonic functions.
problem Existence of non-constant positive harmonic functions on covering spaces with exponential growth.
method Normal Riemannian covering, exponential volume growth, Lyons and Sullivan conjecture.
result Existence of non-constant positive harmonic functions on M. Gradient estimates for special harmonic functions on manifolds.
problem Estimating gradients of (p,V)-harmonic functions on Riemannian manifolds. method Using Moser iteration method, volume comparison theorem, and Sobolev embedding theorem.
result Explicit global gradient estimates for positive entire (p,V)-harmonic functions. Novel method uses image descriptors to harmonize MRI brain volumes across centers.
problem Inconsistencies in MRI brain volume measurements across different centers and scanners.
method Trained a Relevance Vector Machine (RVM) model using image descriptors to harmonize brain volumes.
result Decreases scanner and center variability while preserving measurements for longitudinal studies.
We give a explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.
Study of harmonic functions on infinite penny graphs.
problem Characterizing harmonic functions on infinite penny graphs.
method Proving volume doubling and Poincaré inequalities, analyzing polynomial growth harmonic functions.
result Finite dimensional property of ancient solutions of the heat equation.
The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed on…
We show that noncompact simply connected harmonic manifolds with volume density Θp(r)=sinhn−1r is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density Θp(r)=sinh2n−1rcoshr is isometric to the complex hyperbolic space. A similar re…
In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with …
Study dynamics of Lp-multipliers on harmonic manifolds with exponential volume growth.
problem Characterize the behavior of Lp-multipliers on harmonic manifolds of purely exponential volume growth. method Analyzing the dynamics of Lp-multipliers on non-compact harmonic manifolds, using Fourier transformation and properties of radial functions. result Show that Lp-multipliers acting nicely on smooth functions with compact support for p≤2 cannot be chaotic. In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h. We prove the following equivalences for asymptotically harmonic manifolds X under the additional assumpti…
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of …
Study relates Gromov norm to harmonic norm on non-positively curved manifolds.
problem Relating norms on homology classes to cohomology.
method Relates Gromov norm to harmonic norm, using volume and geometric quantities.
result Obtains double-sided bounds on norms.
Entropy derived from Colding's volume on Ricci-flat manifolds.
problem Deriving Perelman's entropy from Colding's monotonic volume.
method Applying Colding's monotonic volume to Perelman's N-space for harmonic functions on Ricci-flat manifolds.
result Entropy is the limit of Colding's monotonic volume.
The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension o…
This paper classifies critical metrics on compact manifolds with boundary and harmonic Weyl tensor.
problem Classifying critical metrics on compact manifolds with boundary and harmonic Weyl tensor.
method Complete classification through geometric conditions and equivalence of assumptions.
result Critical metrics with harmonic Weyl tensor on simply connected compact manifolds with boundary are isometric to geodesic balls in simply connected space forms.
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.
H. Hotelling proved that in the n-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric spaces computing the volumes of the tubes explicitly in these …
Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.
problem Generalizing Schwarz Lemma for a specific type of harmonic maps.
method Conditions on eigenvalues and Ricci curvature are used to prove the lemma.
result Schwarz Lemma for VT harmonic maps proved with distance and volume decreasing properties.
The study bounds harmonic functions on manifolds with nonnegative Ricci curvature.
problem Bounding harmonic functions on manifolds with specific curvature properties.
method Analyzing the asymptotic volume ratio and eigenvalue counting function.
result Sharp upper bounds for harmonic functions with polynomial growth.
Study spherical Fourier transform on hypergeometric type harmonic manifolds.
problem Spherical Fourier transform on harmonic Hadamard manifolds.
method Representation of spherical functions by Gauss hypergeometric functions.
result Inversion formula, convolution rule, and Plancherel theorem are derived.
Every connected, weighted graph with non-negative curvature has exactly two ends.
problem Characterizing the structure of connected, weighted graphs with non-negative curvature.
method Extremal Lipschitz extensions, variational principle, study of harmonic functions.
result Every salami has exactly two ends and no vertices with positive curvature.
Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.
problem Generalizing monotonicity formulas for harmonic functions in mRCD(0,N) spaces. method New estimates for harmonic functions and a functional version of the outer volume cone theorem.
result Proven rigidity and almost rigidity statements for harmonic functions in mRCD(0,N) spaces. 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.
We compute the space of L2 harmonic forms (outside the middle degrees) on negatively curved Kaehler manifolds of finite volume.
The Fourier transform on harmonic manifolds with exponential volume growth is studied.
problem Investigating the Fourier transform on harmonic manifolds with purely exponential volume growth.
method Definition and analysis of the Fourier transform, proof of inversion formula and Plancherel theorem.
result Established a Fourier inversion formula and Plancherel theorem for the class of harmonic manifolds.
This note gives a correction to the proof of the main result of "Harmonic representatives for cuspidal cohomology classes" by J. Dodziuk, J. McGowan and Peter Perry, an article that appeared in Serge Lang memorial volume.
Bounding characteristic numbers of Riemannian manifolds via volume.
problem Bounding characteristic numbers of Riemannian manifolds.
method Using Chern-Weil theory and connections constructed from harmonic metric tensors with bounded Hölder norms.
result Characteristic numbers are bounded proportionally to the volume of Riemannian manifolds.
The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci provided examples showing that in the noncompact case the conjecture is wrong. H…
Compact RCD spaces are proven to be smooth manifolds under specific harmonicity conditions.
problem Characterizing compact RCD spaces as harmonic manifolds.
method Analyzing heat kernel and geodesic ball volumes for harmonicity.
result Compact RCD spaces are isometric to smooth manifolds under given conditions.
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
Study bounds Neumann and Steklov eigenvalues on manifolds and submanifolds.
problem Bounding Neumann and Steklov eigenvalues on manifolds and submanifolds.
method Using conformal and extrinsic volumes, the paper derives upper bounds for eigenvalues.
result Upper bounds for harmonic mean of Neumann and Steklov eigenvalues.
We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.
A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifold…
Study growth rates of harmonic functions on curved surfaces.
problem Understanding the growth rates of harmonic functions on curved surfaces.
method Gradient estimate and frequency analysis on complete surfaces and manifolds with non-negative curvature.
result Existence and properties of nonconstant polynomial growth harmonic functions on manifolds with maximal volume growth.
Paper proves almost rigidity theorem and applies it to study RCD(0,N) spaces.
problem Understanding the structure of noncompact RCD(0,N) spaces with linear volume growth.
method Developed an almost rigidity theorem and applied it to study RCD(0, N) spaces.
result Obtained sublinear growth of diameter of geodesic spheres and non-existence of harmonic functions with polynomial growth.
Harmonic maps are described using Jacobi elliptic functions.
problem None explicitly stated; focuses on existing work.
method Use of Jacobi elliptic functions to describe harmonic maps.
result Harmonic maps can be described using Jacobi elliptic functions.
The study extends classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.
problem Extending classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.
method Investigated the restricted mean-value property on Riemannian manifolds, focusing on non-tangential boundary behavior.
result Extended a classical result of Fenton to non-positively curved Harmonic manifolds of purely exponential volume growth.