arXiv research

Paper excludes the lowest energy level as an accumulation point for harmonic maps into analytic manifolds.

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some coni…

New mathematical tools for studying knots and links.

New findings on isospectral tori and harmonic maps between flat tori.

Study proves Lojasiewicz inequalities for harmonic maps near simple bubble trees.

Researchers found uncountable harmonic self-maps in complex projective spaces.

We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result…

The paper studies harmonic 1-forms on specific metric measure spaces.

Compactifies character varieties for group actions.

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

We prove a sharp integral gradient estimate for harmonic functions on noncompact Kähler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds achieving this estimate.

The paper identifies magnetic ground states and their role in determining the conformal class of a surface.

The study finds infinitely many p-harmonic maps between spheres for specific p and m.

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 …

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $ψ:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $φ:\s^3\to \s^3$. We show $φ$ to be unstable and estimate its biharmonic index and nullity. Resolving the s…

Researchers resolve string theory ambiguities and define a new metric for massless spectrum.

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for…

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…

We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spect…

We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.

We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero square-integrable harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on (1-forms on M)/Ker(d)?

Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1)$. It is well known that the bottom of spectrum $λ_{0}$ of its unverversal covering satisfies $λ_{0}\leq(n-1) ^{2}/4$. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.

We show that for a suitable class of ``Dirac-like'' operators there holds a Gluing Theorem for connected sums. More precisely, if $M_1$ and $M_2$ are closed Riemannian manifolds of dimension $n\ge 3$ together with such operators, then the connected sum $M_1 # M_2$ can be given a Riemannian metric such that the spectrum…

We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions. As an interesting byproduct of our study, w…

Stability of Morse index for harmonic maps on degenerating surfaces analyzed.

We consider the energy-critical half-wave maps equation $\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterizat…

The study shows how energy density of harmonic maps dominates in $n$-Fuchsian fibers, leading to unique minimal surfaces.

Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\Z_2^k$,…

Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our constructio…

The paper describes spectra of operators on rational homogeneous varieties.

Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.

In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be estimated by the dimension of a certain space of harmonic functions. Applying this t…

Generalizes Gelfand's spectrum to monogenic spinor fields on compact Riemannian manifolds.

Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove…

In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function $f$ has decay $f=O(r^{-1-\varepsilon})$ for some $\varepsilon >0,$ where $r$ is the distance function to a fixed point, then t…

The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.

While the harmonic function solution performs well in many semi-supervised learning (SSL) tasks, it is known to scale poorly with the number of samples. Recent successful and scalable methods, such as the eigenfunction method focus on efficiently approximating the whole spectrum of the graph Laplacian constructed from …

Let H=$Δ+V$ be a Schrödinger on a complete non-compact manifold. It is known since the work of Fischer-Colbrie and Schoen that the finiteness of the negative spectrum of $H$ implies the existence of a function $φ$ solution of $Hφ=0$ outside a compact set. This has consequences for minimal surfaces and for the finitenes…

The Dirac operator d+delta on the Hodge complex of a Riemannian manifold is regarded as an annihilation operator A. On a weighted space L_mu^2 Omega, [A,A*] acts as multiplication by a positive constant on excited states if and only if the logarithm of the measure density of mu satisfies a pair of equations. The equati…

We study both function theoretic and spectral properties of the weighted Laplacian $Δ_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-Émery curvature $Ric_f$ bounded from below by a constant. In particular, we establish a gradient estimate for positive $f-$harmonic functions and a sharp upper…

The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…

Study convex functions on manifolds without focal points, deriving new spectral properties.

This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${\rm Osc}_1$ up to inner automorphisms of ${\rm Osc}_1$. F…

Variant of mSSA improves time series prediction error.

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-Émery Ricci curvature. Among other things, we derive a gradient estimate for positive $f$-harmonic functions and obtain as a consequence the strong Liouville property under…

In this paper, we consider solutions and spectral functions of M-theory from Milne spaces with extra free dimensions. Conformal deformations to the metric associated with the real hyperbolic space forms are derived. For the three-dimensional case, the orbifold identifications $SL(2,{\mathbb Z}+i{\mathbb Z})/\{\pm Id\}$…