arXiv research

Recalls intrinsically harmonic forms and open problems.

Criterion for flat circle bundles using intrinsically harmonic forms.

Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of …

The paper studies harmonic graphs in the Heisenberg group and their properties.

We go further on the study of harmonicity for almost contact metric structures already initiated by Vergara-Diaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric st…

Study sesqui-harmonic map flow from Riemannian surfaces

Develops a framework for generating harmonic maps from a unit ball to a sphere.

The study of harmonicity for almost contact metric structures was initiated by Vergara-Díaz and Wood and continued by González-Dávila and the present author. By using the intrinsic torsion and some restriction on the type of almost contact metric structure, González-Dávila and the present author have characterised harm…

We discuss the construction of Sp(2)Sp(1)-structures whose fundamental form is closed. In particular, we find 10 new examples of 8-dimensional nilmanifolds that admit an invariant closed 4-form with stabiliser Sp(2)Sp(1). Our constructions entail the notion of SO(4)-structures on 7-manifolds. We present a thorough inve…

The n-dimensional torus is uniquely characterized by specific harmonic forms.

Bayesian framework for sphere regression using Gaussian fields.

Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.

Sharp Veronese rigidity theorem for submanifolds of unit ball.

We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delau…

For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold M, where G-structures are considered as sections of the quotient bundle O(M)/G. Then, we deduce the corresponding first and second variation formulae and the characterising condi…

Proves harmonic coordinates for weak immersions in even dimensions.

Multimodal learning aims to discover the relationship between multiple modalities. It has become an important research topic due to extensive multimodal applications such as cross-modal retrieval. This paper attempts to address the modality heterogeneity problem based on Gaussian process latent variable models (GPLVMs)…

We propose a novel framework for combining datasets via alignment of their intrinsic geometry. This alignment can be used to fuse data originating from disparate modalities, or to correct batch effects while preserving intrinsic data structure. Importantly, we do not assume any pointwise correspondence between datasets…

New method uses spherical harmonics to simplify learning single-index models.

In this article, we study the regularity of minimizing and stationary $p$-harmonic maps between Riemannian manifolds. The aim is obtaining Minkowski-type volume estimates on the singular set $S(f)=\{x \ \ s.t. \ \ f \text{ is not continuous at } x\}$, as opposed to the weaker and non quantitative Hausdorff dimension bo…

Let $M$ be a closed oriented manifold of dimension $n$ and $ω$ a closed 1-form on it. We discuss the question whether there exists a Riemannian metric for which $ω$ is co-closed. For closed 1-forms with nondegenerate zeros the question was answered completely by Calabi in 1969. The goal of this paper is to give an answ…

Estimates the index of the Laplace operator on planar domains with Robin boundary condition.

We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We generalize Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians:…

This paper contains the technical foundations from stochastic differential geometry for the construction of geometrically intrinsic nonlinear recursive filters. A diffusion X on a manifold N is run for a time interval T, with a random initial condition. There is a single observation consisting of a nonlinear function o…

New Ricci curvature means derived from plane curvatures.

We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent …

3D spheres with certain properties approach the round sphere.

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…

Study topological properties of foliations induced by closed 1-forms on orbifolds.

Let (M,I, ω, Ω) be a nearly Kaehler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form Ωand the Hermitian form ωwhich satisfies $dω=3λ\ReΩ, d\ImΩ=-2λω^2$, for a non-zero real constant λ. We develop an analogue of Kaehler relations on M, proving several useful identities for various intrinsic Laplacians on M. Wh…

The structure equations for a surface are introduced and two required results based on the Codazzi equations are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem of Bonnet is obtained. A tranformation formula for the connection forms is developed. It is proved that the an…

The paper classifies flows of SU(2)-structures on 4-manifolds.

Let $M$ be a submanifold of a Riemannian manifold $(N,g)$. $M$ induces a subbundle $O(M,N)$ of adapted frames over $M$ of the bundle of orthonormal frames $O(N)$. Riemannian metric $g$ induces natural metric on $O(N)$. We study the geometry of a submanifold $O(M,N)$ in $O(N)$. We characterize the horizontal distributio…

Stability of biharmonic maps in critical dimension proven.

Enhanced probabilistic sampling on manifolds using Double Diffusion Maps and Geometric Harmonics.

Abstract theory of flows of geometric structures on manifolds.

Study cohomology classes related to harmonic maps on submersions.

Study on harmonic maps on weighted Riemannian foliations.

The paper examines triviality of Ricci-Bourguignon harmonic solitons.

Study cohomology classes related to $n$-harmonic morphisms and $F$-harmonic maps.

The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

New algorithm reduces unfairness in bandit problems by balancing exploration and exploitation.

f-Harmonic maps were first introduced and studied by Lichnerowicz in \cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. We prove that a map betwe…

Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.

We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $p\to \infty$. Infinity harmoncity appears in many familiar contexts. For example,…

Study examines maximal domains of radial harmonic functions across different curvature types.

New p-harmonic and harmonic morphisms found on Lie groups.