The paper studies steady solitons with curvature decay and proves their smoothness.
problem Analyzing the properties of steady solitons with curvature decay.
method Bootstrap regularity in harmonic coordinates using the soliton equation.
result Steady gradient Ricci solitons are asymptotically cylindrical under certain curvature decay conditions.
The center of mass in General Relativity is hard to define due to coordinate freedom.
problem Defining the center of mass in General Relativity rigorously and consistently.
method Analyzing the challenges in Newtonian Gravity and using Bartnik's asymptotic harmonic coordinates.
result Examples of initial data sets in General Relativity that do not satisfy center of mass definitions.
The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.
problem Investigating constant harmonic mean curvature surfaces in Schwarzschild spaces.
method Volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces.
result These surfaces form a foliation of the space outside a large ball.
Study on conformal harmonic coordinates on manifolds, proving existence and properties.
problem Existence and properties of conformal harmonic coordinates on Riemannian manifolds.
method Solutions to the conformal Laplace equation, proving up to boundary regularity results, elliptic regularity, and unique continuation results.
result Proves conformal harmonic coordinates are a close conformal analogue of harmonic coordinates.
Refined asymptotics of scalar-flat ALE four-manifolds
problem Asymptotic behavior of scalar-flat ALE four-manifolds
method Constructing preferred coordinates at infinity
result Identifying homogeneous ∣x∣−2 term in metric expansion Formula calculates mass using cube faces and edges.
problem Measuring mass of 3-manifolds.
method Cube faces and edges, mean curvature, dihedral angle, geodesic curvature, angle defect.
result Mass formula connects to Gromov's theory and Gauss-Bonnet theorem.
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or 1-quasiregular mapping between two manifolds with Cr metric tensors (r>1) is a Cr+1 conformal (local) diffeomorphism. …
Proves harmonic coordinates for weak immersions in even dimensions.
problem Existence of harmonic coordinates for weak immersions in Sobolev spaces.
method Analyzes weak immersions in critical Sobolev spaces and uses smallness conditions on the second fundamental form.
result Global harmonic coordinates exist for weak immersions in even dimensions under certain conditions.
Harmonic basis vector fields on surfaces
problem Parameterizing surfaces with harmonic vector fields
method Introducing harmonic basis vector fields and deriving conditions for their existence
result Classifying parameterizations of surfaces with harmonic basis vector fields
Paper studies Laplace operator estimates in harmonic map heat flows.
problem Estimating Laplace operator in harmonic map heat flows outside singularities.
method Investigates estimates using spherical coordinates for T2 and T3 boundary conditions. result Provides higher-order estimates for the Ericksen--Leslie system.
We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers--Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we …
In this paper we pursue the work initiated in \cite{Bahuaud, BahuaudGicquaud}: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to -1 controls the Hölder regularity of the compactified metric. To thi…
Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.
problem Classifying polynomial growth solutions to drift-harmonic equations on specific types of manifolds.
method Inductive argument that alternates between constructing and asymptotically controlling drift-harmonic functions.
result All drift-harmonic functions with polynomial growth asymptotically separate variables and dimensions of spaces are computed.
Researchers create non-degenerate harmonic functions on n-dimensional space.
problem Creating non-degenerate Z2-harmonic functions on Rn. method Using a variant of ellipsoidal coordinates, the construction is explicit and involves Lawlor's necks in Cn. result First known family of non-degenerate Z2-harmonic 1-forms with compact branching sets. Classifies instantons on a specific gravitational instanton and computes partition functions.
problem Classifying finite energy harmonic 2-forms and anti-self-dual Yang-Mills instantons.
method Analyzes U(1)-bundles and computes instantons explicitly. result Unique anti-self-dual Yang-Mills instantons exist and are described explicitly.
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 …
This paper explores the harmonic mean of implied volatility and its relation to local volatility.
problem Understanding the relationship between implied volatility and local volatility.
method Investigates the harmonic mean of a positive function for any fixed maturity, linking it to Fukasawa's invertible map.
result The short-dated implied volatility approaches the arithmetic mean of the local volatility in a new coordinate system.
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
This paper describes the behavior of sequences of solutions to the Kapustin-Witten equations with Nahm pole asymptotics on the product of the half-line with a compact, oriented, Riemannian 3-manifold. These sequences have sub-sequences that either converge to another solution after acting term-wise by an automorphism o…
Corrected proof for 3D harmonic manifolds with minimal horospheres.
problem Proving 3D harmonic manifolds with minimal horospheres are either flat or hyperbolic.
method Provided a corrected proof for the classification of 3D harmonic manifolds.
result Classification of 3D harmonic manifolds: flat or hyperbolic.
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.
We relate Kostant's theorem on the cohomology of a flag manifold G/B with the geometry of the Bruhat-Poisson structure. We express Kostant's harmonic forms in terms of the moment maps (for the torus action) and the Liouville volume forms for the symplectic structures on the Schubert cells induced by the Bruhat-Poisso…
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.
Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in t…
A complete Riemannian manifold without conjugate points is called asymptotically harmonic if the mean curvature of its horospheres is a universal constant. Examples of asymptotically harmonic manifolds include flat spaces and rank one locally symmetric spaces of noncompact type. In this paper we show that this list exh…
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.
The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.
problem Characterizing and understanding harmonic Finsler manifolds.
method Investigation of various types of harmonic Finsler manifolds, characterizations via mean curvature and Laplacian, and construction techniques.
result Certain harmonic Finsler manifolds are of Einstein type and examples of non-Riemannian Finsler harmonic manifolds are provided.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
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 classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek-Ricci space. Independently, Y. Nikola…
Explicit mass bound for 3D asymptotically flat manifolds using harmonic functions.
problem Finding an explicit lower bound for the mass of 3D asymptotically flat Riemannian manifolds.
method Using linear growth harmonic functions and scalar curvature, a new proof of the positive mass theorem is achieved.
result Achieved a new proof of the positive mass theorem in dimension three.
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.
Paper extends foliation results in higher dimensions for Schwarzschild spaces.
problem Existence of foliations by constant harmonic mean curvature hypersurfaces in asymptotically Schwarzschild manifolds.
method Generalization to higher dimensions, proving existence under arbitrary dimensionality.
result Existence of foliations by constant harmonic mean curvature hypersurfaces in asymptotically Schwarzschild manifolds of arbitrary dimension.
Study precise asymptotic behavior of functions in singular metric spaces.
problem Singular metric spaces with incomplete geometry.
method Expansions of quasi-harmonic and eigenfunctions.
result More precise description of asymptotic behavior at infinity.
In this note we show that a compact asymptotically harmonic manifold without focal points is either flat or a rank one locally symmetric space.
Study examines harmonic functions in sub-Riemannian and RCD settings.
problem Characterizing harmonic functions in sub-Riemannian and RCD settings.
method Analyzes weak and strong asymptotically mean value harmonic functions.
result Weakly amv-harmonic functions are equivalent to harmonicity in Carnot groups.
Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h. In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds X with mild curvature boundedness c…
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 improves regularity estimates for harmonic maps into ellipsoids.
problem Independence of regularity estimates on harmonic maps with varying target dimensions.
method Analyzes harmonic maps into ellipsoids, uses Palais-Smale sequences, and critical metrics.
result Enhanced regularity estimates for Laplace harmonic eigenmaps.
Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.
problem Existence of harmonic maps from product of hyperbolic planes to hyperbolic space.
method Existence result for asymptotic Dirichlet problem.
result Existence of harmonic maps with given boundary data.
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.
Solves Dirichlet problem for harmonic maps to give geodesic insights.
problem Asymptotic Dirichlet problem for harmonic maps.
method Holographic characterization using conformal geodesics.
result Characterizes conformal geodesics on the boundary.
We prove that for any open Riemann surface M and any non constant harmonic function h:M→R, there exists a complete conformal minimal immersion X:M→R3 whose third coordinate function coincides with h. As a consequence, complete minimal surfaces with arbitrary conformal structure and wh…
Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with C1 boundary conditions at infinity between asymptotically hyperbolic manifolds.
Derives new monotone quantities for p-harmonic functions on asymptotically flat 3-manifolds.
problem Estimating the mass of 3-manifolds with non-negative scalar curvature and minimal boundary.
method Derives monotone quantities for p-harmonic functions and applies them to derive a sharp mass-capacity estimate.
result Derives a sharp mass-capacity estimate relating the ADM mass of a 3-manifold to the p-capacity of its boundary.
Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a flat manifold, provided M is asymptotically harmonic of constant h=0.
Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0.
New insights into manifold properties using Seiberg-Witten and L2 harmonic theories.
problem Characterizing properties of 4-manifolds with specific geometric conditions.
method Combining Seiberg-Witten theory on compact manifolds and L2 harmonic theory on non-compact manifolds, with a new argument for asymptotic properties. result Found a pair of homeomorphic 4-manifolds with distinct geometric properties under Riemannian metrics.