arXiv research

Study bounds the Morse index of a special torus to 1.

We develop an analog of harmonic replacement in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron. The technique, as introduced by Jost and further developed by Colding and Minicozzi, involves taking a map $v\colonΣ\to M$ defined on a surface $Σ$ and replacing its values on…

The paper proves existence and instability of weak $r$-harmonic maps.

Let $\varphi\in C^0 \cap W^{1,2}(Σ, X)$ where $Σ$ is a compact Riemann surface, $X$ is a compact locally CAT(1) space, and $W^{1,2}(Σ,X)$ is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map $u:Σ\to X$ homotopic to $\varphi$ or there exists a co…

We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences …

Harmonic maps intersect all minimal surfaces with bounded curvature.

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…

We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lie in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Co…

Closed Riemannian 4 or 5-manifolds contain branched immersed closed minimal surfaces.

We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-…

Localized curvature bounds ensure harmonic maps are constant.

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown that the theo…

The study extends classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.

Theorem proves spectral rigidity of warped product metrics.

In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory …

Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature map translations. This is not the case for rotations. Global rotation equivariance is typic…

We derive a positive mass theorem for asymptotically flat manifolds with boundary whose mean curvature satisfies a sharp estimate involving the conformal Green's function. The theorem also holds if the conformal Green's function is replaced by the standard Green's function for the Laplacian operator. As an application,…

A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifo…

B.-Y. Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpreta…

We consider a conformally invariant version of the Calderón problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions $\geq …

É.Ghys proved that the linking numbers of modular knots and the "missing" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\rm SL}_2\mathbb{Z}$. In this paper, we replace ${\rm SL}_2\mathbb{Z}=Γ_{2,3}$ by the triangle group $Γ_{p,q}$ for any coprime …

Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.

One purpose of this article is to draw attention to the seminal work of J. Mealy in 1989 on calibrations in semi-riemannian geometry where split SLAG geometry was first introduced. The natural setting is provided by doing geometry with the complex numbers C replaced by the double numbers D, where i with i^2 = -1 is rep…

We explore the potential of a popular distributional semantics vector space model, word2vec, for capturing meaningful relationships in ecological (complex polyphonic) music. More precisely, the skip-gram version of word2vec is used to model slices of music from a large corpus spanning eight musical genres. In this newl…

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

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.

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.

Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.

J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, We study k-harmonic immersion into a sphere, and get the rerationship between radious and "k"…

Extends p-harmonic map theory for new properties.

$\infty$-Harmonic maps are a generalization of $\infty$-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic $\infty$-harmonic maps from and into a sphere, quadratic $\infty$-harmonic maps between E…

Dirac-harmonic maps are uncoupled under certain conditions.

Recent advances in conditional image generation tasks, such as image-to-image translation and image inpainting, are largely accounted to the success of conditional GAN models, which are often optimized by the joint use of the GAN loss with the reconstruction loss. However, we reveal that this training recipe shared by …

The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.

Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details)…

Analyzes harmonic and biharmonic maps from gradient Ricci solitons.

Study heat flow for half-harmonic maps and harmonic maps with free boundary.

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

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