arXiv research

Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.

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 …

Smooth Busemann functions found in harmonic Finsler spaces.

Corrected proof for 3D harmonic manifolds with minimal horospheres.

Study shows volume density in central harmonic spaces can vary arbitrarily.

Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.

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.

The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.

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…

The study proves properties of intersections of horospheres in harmonic spaces.

Paper extends foliation results in higher dimensions for Schwarzschild spaces.

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.

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…

Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.

Proves positive mass theorem for 3-manifolds with a boundary.

Solves Dirichlet problem for harmonic maps to give geodesic insights.

Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with $C^1$ boundary conditions at infinity between asymptotically hyperbolic manifolds.

The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.

Derives new monotone quantities for p-harmonic functions on asymptotically flat 3-manifolds.

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.

Study fractional perimeter asymptotics on Riemannian manifolds as $s$ approaches 0.

In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmid's Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact Kähler manifolds with carefully controlled asymptotics near the compactifying divisor; such a m…

The existence and nonexistence of $λ$-harmonic functions in unbounded domains of $\mathbb{H}^n$ are investigated. We prove that if the $(n-1)/2$ Hausdorff measure of the asymptotic boundary of a domain $Ω$ is zero, then there is no bounded $λ$-harmonic function of $Ω$ for $λ\in [0,λ_1(\mathbb{H}^n)]$, where $λ_1(\mathb…

New type of harmonic Finsler manifolds created.

The metric is quite singular at infinity and it is not complete. Using these expansions, we have a more precise description of the asymptotic behavior of quasi-harmonic functions and of eigenfunctions of drift-Laplacian at infinity.

Study of harmonic maps with extreme Kerr-like singularities.

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…

Study on harmonic maps from surfaces to homogeneous spaces, focusing on bubble formation and geometric constraints.

We study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor $D$-modules. As an application, we show the hard Lefschetz theorem for algebraic semis…

Solves wave equation on non-flat harmonic manifolds using Abel transform and Fourier analysis.

Let $E$ be a holomorphic vector bundle. Let $θ$ be a Higgs field, that is a holomorphic section of $End(E)\otimesΩ^{1,0}_X$ satisfying $θ^2=0$. Let $h$ be a pluriharmonic metric of the Higgs bundle $(E,θ)$. The tuple $(E,θ,h)$ is called a harmonic bundle. Let $X$ be a complex manifold, and $D$ be a normal crossing divi…

Study polynomial growth harmonic functions on infinite penny graphs.

Proves density and mass theorems for specific initial data sets.

New estimates for Hitchin's equations at high energy.

Study harmonic surfaces in 3D space, proving superposition principle.

Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that eve…

Continuous metrics on R^3 with specific properties have non-negative harmonic mass.

We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance to a fixed point in M. We are, in particular, interested in finding optimal (or…

We study the asymptotic behaviour of tame harmonic bundles. First of all, we prove a local freeness of the prolongation by an increasing order. Then we obtain the polarized mixed twistor structure. As one of the applications, we obtain the norm estimate of holomorphic or flat sections by weight filtrations of the monod…

Harmonic maps from complex plane to hyperbolic space constructed using heat flow.

The paper studies geometric properties of $Φ_{(3)}$-harmonic maps and proves Liouville type results.

Study on harmonic functions in spaces with collapsing behaviors.

The paper explores isometric models and Busemann functions for Funk and Hilbert discs.