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.

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.

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…

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

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…

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…

In this note we show that a compact asymptotically harmonic manifold without focal points is either flat or a rank one locally symmetric space.

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…

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

Paper extends foliation results in higher dimensions for Schwarzschild spaces.

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

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 $L^2$ harmonic theories.

New type of harmonic Finsler manifolds created.

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.

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

Solves Dirichlet problem for harmonic maps to give geodesic insights.

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

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…

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

Study on solitons in specific geometric manifolds, proving manifold properties and presenting examples.

Study examines harmonic functions in sub-Riemannian and RCD settings.

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…

Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.

Motivated by Witten's spinor proof of the positive mass theorem, we analyze asymptotically constant harmonic spinors on complete asymptotically flat nonspin manifolds with nonnegative scalar curvature.

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…

Study shows mass-capacity inequality for specific geometric manifolds.

We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

Derives monotonic quantities for $p$-harmonic functions on manifolds.

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…

We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the map at infinity. In particular, the results can be applied to F-harmonic maps from some pinched mani…

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

We prove the existence of harmonic spinor fields in axisymmetric Riemannian 3-manifolds having nonnegative scalar curvature and asymptotic to the usual constant time hypersurface of Melvin's magnetic universe. Such a spinor can be used in the proof of the uniqueness of the magnetized Schwarzschild solution.

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

We prove that, in general, given a $p$-harmonic map $F:M\to N$ and a convex function $H:N\to\mathbb{R}$, the composition $H\circ F$ is not $p$-subharmonic. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. The key of the proof is an asymptotic…

We present a method in nonlinear elliptic systems to study curvature decays on asymptotically locally Euclidean (ALE) manifolds. In particular, we show that scalar flat Kahler and harmonic ALE metrics of real dimension n are of order n-2.

The paper proves a new inequality for 3-manifolds with noncompact boundaries.

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

We study a second order differential equation corresponding to rotationally symmetric $F$-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.

On a Riemannian surface, the energy of a map into a Riemannian manifold is a conformal invariant functional, and its critical points are the harmonic maps. Our main result is a generalization of this theorem when the starting manifold is even dimensional. We then build a conformal invariant functional for the maps betw…

The paper proves density and positive mass theorems for incomplete manifolds.

The study bounds harmonic functions on manifolds with nonnegative Ricci curvature.