arXiv research

The paper examines metrics on foliated manifolds that have special geometric properties.

Combines Gaussian process and Geometric Harmonics for better uncertainty estimation.

We study integral geometric properties of non-compact harmonic spaces.

The paper characterizes harmonic maps and studies their properties on Riemannian manifolds.

Harmonic gauge simplifies geometric analysis of Riemannian metrics.

Constructs harmonic maps between special geometric shapes.

The abstract discusses p-harmonic forms and their geometric properties, proving new theorems about Lp-cohomology.

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

Study of Type IIA flow on symplectic Lie algebras for geometric structures.

The paper proves a geometric capacitary inequality for sub-static manifolds with harmonic potentials.

Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of …

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

A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to …

Study spherical Fourier transform on hypergeometric type harmonic manifolds.

We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of an…

The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.

Analyzes harmonic and biharmonic maps from gradient Ricci solitons.

New methods using spacetime harmonic functions solve geometric inequalities.

The paper proves geometric rigidity using harmonic twisted spinors and scalar curvature comparison.

We study several geometric and analytic aspects of Dirac-harmonic maps with curvature term from closed Riemannian surfaces.

We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.

The method of geometric harmonics is adapted to the situation of incomplete data by means of the iterated geometric harmonics (IGH) scheme. The method is tested on natural and synthetic data sets with 50--500 data points and dimensionality of 400--10,000. Experiments suggest that the algorithm converges to a near optim…

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses a geometric flow a…

Develops a surrogate model for predicting system responses using GDMaps and geometric harmonics.

In this paper, we introduce a new concept so called harmonic complex structure by using harmonic theory for vector bundle-valued differential forms. It is a new structure intermediates between complex structure and Kähler structure. From differential geometric viewpoint, it is a natural generalization of Kähler structu…

We study geometric quantization of the harmonic oscillator in terms of a singular real polarization given by fibres of the energy momentum map.

In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fun…

Study on harmonic maps in special geometric spaces.

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic…

Uniform small energy regularity for fractional geometric problems proved.

In this paper we generalize harmonic maps and morphisms to the \emph{degenerate semi-Riemannian category}, in the case when the manifolds $M$ and $N$ are \emph{stationary} and the map $φ:M\to N$ is \emph{radical-preserving}. We characterize geometrically the notion of \emph{(generalized) horizontal (weak) conformality}…

Study shows how maps from certain geometric spaces behave near their edges.

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…

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to …

The study shows that the visible range from a point on harmonic manifolds follows an exponential distribution.

We study a geometrical condition (PHWC) which is weaker than horizontal weak conformality. In particular, we show that harmonic maps satisfying this condition, which will be called {\em pseudoharmonic morphisms}, include harmonic morphisms and can be described as pulling back certain germs to certain other germs. Final…

Develops methods to construct harmonic and wave maps into variable-curvature surfaces.

Our aim in this paper is to investigate some geometrical properties of Berger Spheres i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields. We determine all vector fields which are critical points for the energy functional restricted to vector fields. We also see that do not exist any v…

We find geometric conditions on a four-dimensional almost Hermitian manifold under which the almost complex structure is a harmonic map or a minimal isometric imbedding of the manifold into its twistor space.

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

Study on harmonic functions in spaces with collapsing behaviors.

Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ be an associate fiber bundle. Our interested is to study harmonic sections of the projection $π_{E}$ of $E$ into $M$. Our first purpose is to give a stochastic characterization of harmonic section from $M$ into $E$ and a geometric characterization of harmonic se…

Study finds bound on energy of minimal spheres on complex manifolds.

We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional …

We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced …

We study Dirac-harmonic maps from surfaces to manifolds with torsion, which is motivated from the superstring action considered in theoretical physics. We discuss analytic and geometric properties of such maps and outline an existence result for uncoupled solutions.

Study pseudo-harmonic maps on Weyl manifolds.