Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
Study harmonic function growth on curved spaces, proving inequalities.
problem Understanding growth rates of harmonic functions on curved manifolds.
method Applied a double-sided Price inequality to estimate growth rates.
result Effective estimates for harmonic function growth rates on curved manifolds.
We prove some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear boundary condition
Study on 3-manifolds with nonnegative scalar curvature and positive harmonic functions.
problem Characterizing 3-manifolds with nonnegative scalar curvature.
method Exhaustions by level sets of harmonic functions and refined average gradient estimates.
result Contractible 3-manifolds are diffeomorphic to R^3, and handlebodies have genus at most 1.
In this paper, we will give a horizontal gradient estimate of positive solutions of Δbu=−λu on complete noncompact pseudo-Hermitian manifolds. As a consequence, we recapture the Liouville theorem of positive pseudo-harmonic functions on Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.
New methods using spacetime harmonic functions solve geometric inequalities.
problem Geometric inequalities involving mass in spacetime.
method Utilizing spacetime harmonic functions and other elliptic equations.
result Novel concept of total mass and proof of positive mass theorem.
This paper proves a conjecture about unique positive harmonic functions in a ball.
problem Proving the uniqueness of positive harmonic functions in a unit ball for specific parameters.
method Analyzing a partial differential equation to show the solution is constant.
result Guo-Wang's conjecture is proven for the specified parameters.
The paper studies Harnack inequalities on Finsler metric measure spaces.
problem Analyzing Harnack inequalities on Finsler metric measure spaces.
method Using weighted Ricci curvature and distortion conditions, the authors derive an elliptic p-Harnack inequality.
result The paper establishes an elliptic p-Harnack inequality and derives Hölder continuity and gradient estimates for positive harmonic functions.
We show that if p:M→N is a normal Riemannian covering, with N closed, and M has exponential volume growth, then there are non-constant, positive harmonic functions on M. This was conjectured by Lyons and Sullivan in \cite{LS}.
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. The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.
problem Proving Liouville-type theorems for positive harmonic functions on specific types of manifolds.
method Employing the P-function method and a closed conformal vector field inherent to such manifolds.
result Confirms some cases of Wang's conjecture and provides a partial verification of Wang's conjecture on warped product manifolds.
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.
Exponential rate of convergence for harmonic heat flow maps.
problem Analyzing the convergence rate of harmonic heat flow maps.
method Proving exponential convergence rate for harmonic heat flow maps.
result Exponential convergence rate of the harmonic heat flow.
The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.
problem Understanding harmonic functions on 3-manifolds with specific ends.
method Derives monotonic properties of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends.
result Rigidity characterization of spatial Schwarzschild manifolds with two ends.
Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.
For positive p-harmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension n, p and the radius of the ball on which the function is defined. Our approach is based on a careful application of…
Gradient estimate for harmonic functions with boundary condition proved.
problem Proving gradient estimates for harmonic functions with boundary conditions.
method Using weighted f-harmonic functions and infinite dimensional Bakry-Emery Ricci tensor. result Gradient estimates for positive f-harmonic functions with Dirichlet boundary condition. The paper proves a new inequality for 3-manifolds with noncompact boundaries.
problem Proving positivity of a convex combination of ADM masses on 3-manifolds with noncompact boundaries.
method Obtained an integral inequality for asymptotically linear harmonic functions.
result Positivity of a convex combination of ADM masses under a positivity condition on scalar curvatures and boundary mean curvatures.
For any positive natural number r∈N+ we construct new explicit proper r-harmonic functions on the celebrated 3-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $\H^2\times\rn$ and $\s^2\times\rn$.
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…
Treebolic space HT(q,p) is a key example of a strip complex in the sense of Bendikov, Saloff-Coste, Salvatori, and Woess [Adv. Math. 226 (2011), 992-1055]. It is an analog of the Sol geometry, namely, it is a horocylic product of the hyperbolic upper half plane with a "stretching" parameter q and the homogeneous tree T…
The study examines rigidity and stability of gradient estimates on surfaces and manifolds.
problem Rigidity and stability of gradient estimates for positive harmonic functions and solutions to heat equations.
method Sharp gradient estimates for positive harmonic functions and solutions to heat equations on surfaces and manifolds with nonnegative curvature.
result Obtained rigidity and stability results for gradient estimates.
In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold (M,g) corresponding to e…
The paper explores α−harmonic maps and their stability, proving key properties and conditions.
problem Existence and stability of α−harmonic maps between Riemannian manifolds. method Analysis of α−energy functional, construction of α−harmonic maps, and stability conditions. result Conditions for the stability of α−harmonic maps and their instability from compact manifolds. An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurfac…
Continuous metrics on R^3 with specific properties have non-negative harmonic mass.
problem Proving non-negativity of mass for continuous metrics.
method Defining harmonic mass and using properties of approximating smooth metrics.
result The harmonic mass of continuous metrics is non-negative.
We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, w…
The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.
problem Geometric stability of the positive mass theorem and related conjectures.
method Analysis of harmonic maps to flat model spaces under integral curvature bounds.
result Upgrading harmonic maps to diffeomorphisms when conditions are met, proving quantitative closeness to model spaces.
It is known that the L2-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the L2-norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular…
Sharp gradient estimates extended to surfaces with lower Ricci curvature.
problem Rigidity of Cheng-Yau gradient estimates on surfaces with lower Ricci curvature.
method Extending Cheng-Yau gradient estimates to surfaces with lower Ricci curvature bound and higher-dimensional Riemannian manifolds.
result Pointwise Cheng-Yau gradient estimates for higher-dimensional Riemannian manifolds and monotonicity formulas for positive harmonic functions.
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 paper connects mass, harmonic functions, and capacity in asymptotically flat 3-manifolds.
problem Connections among ADM mass, harmonic functions, and capacity in asymptotically flat 3-manifolds.
method New formulae for ADM mass via harmonic functions, monotone quantities, and geometric inequalities.
result The mass-to-capacity ratio is bounded below by 1 - sqrt(normalized Willmore functional of the boundary).
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.
The paper provides gradient estimates for a specific equation on Riemannian manifolds.
problem Gradient estimates for positive solutions to a specific equation on Riemannian manifolds.
method Obtained gradient bounds for positive solutions without depending on the solution's bounds or the Laplacian of the distance function.
result Gradient bound of a positive solution does not depend on the solution's bounds or the Laplacian of the distance function.
The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.
problem Understanding exceptional sets for radial limits of superharmonic functions on curved manifolds.
method Analysis of radial geodesic rays, Poisson integrals, Green potentials, and Riesz decomposition.
result Sharp bounds on Hausdorff dimensions of exceptional sets for superharmonic functions.
The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.
problem Proving dihedral rigidity for cubic initial data sets in 3D spacetime.
method By studying the level sets of spacetime harmonic functions and extending previous work on dihedral rigidity for prisms in hyperbolic space.
result The paper proves dihedral rigidity for cubes in 3D spacetime, extending previous results.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.
problem Functional and geometric inequalities on Finsler measure spaces.
method Local uniform Poincaré and Sobolev inequalities, mean value inequality, Harnack inequalities, and gradient estimates.
result Global gradient estimates for positive harmonic functions on Finsler measure spaces.
New proof removes decay assumptions for spacetime positive mass theorem.
problem Proving the rigidity of the spacetime positive mass theorem without additional decay assumptions.
method Uses spacetime harmonic functions and Liouville's theorem, and an alternative proof based on Killing development.
result Removes additional decay assumptions for the spacetime positive mass theorem.
Proves positive mass theorem for 3-manifolds with a boundary.
problem Proving the positive mass theorem for specific 3-manifolds.
method Uses harmonic level set approach.
result Validates the positive mass theorem for new class of manifolds.
The paper studies harmonic 1-forms on specific metric measure spaces.
problem Analyzing harmonic 1-forms on non-compact smooth metric measure spaces.
method Establishing splitting and vanishing theorems for Lfp harmonic 1-forms under curvature conditions. result Two new theorems for Lfp harmonic 1-forms are proven. In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional c…
The paper proves a discrete positive mass theorem for graphs.
problem Formulating and proving a discrete positive mass theorem for graphs.
method Introducing asymptotically flat graphs, defining ADM mass, and using discrete harmonic functions.
result An asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph.
Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.
problem Rigidity of harmonic maps between manifolds with curvature constraints.
method Proves an extension of Eells-Sampson theorem under positive sectional curvature upper bounds.
result Recover Hamilton's rigidity result for positive Ricci curvature.
We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem: Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on…
Paper bounds mass of 3D Einstein data using spacetime harmonic functions.
problem Calculating the mass of 3D asymptotically flat initial data for the Einstein equations.
method Uses spacetime harmonic functions to give a lower bound for the ADM mass.
result New proof of spacetime positive mass theorem and rigidity statement.