arXiv research

The identity map of certain Einstein manifolds is stable in both energy and bienergy.

The paper proves quaternion projective space is unstable.

We give here a self contained and elementary introduction to the Conley-Zehnder index for a path of symplectic matrices. We start from the definition of the index as the degree of a map into the circle for a path starting at the identity and ending at a matrix for which 1 is not an eigenvalue. We prove some properties …

Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding $L$-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic…

The paper studies harmonic identity maps on Riemannian manifolds.

New energy identity found for biharmonic maps into spheres.

In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres $\mathbb{S}^n$. First, we propose a new approach to isoperimetric inequalities based on energy index. Using this approach we show that for any positive $k$, the $k$-th non-zero eigenvalu…

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on boundary(M) has the homotopy type of a finite aspherical CW-complex. This answers, for…

New statistical manifolds derived from identity map biharmonicity.

Paper proves energy identity and no-neck property for special harmonic maps.

New Bailey pairs derived for tetrahedron index, linking knot invariants.

The paper extends energy identities and neck existence for ε-harmonic maps.

We study the estimation of the parametric components of single and multiple index volatility models. Using the first- and second-order Stein's identities, we develop methods that are applicable for the estimation of the variance index in the high-dimensional setting requiring finite moment condition, which allows for h…

In this paper, we study the blow-up phenomena on the $α_k$-harmonic map sequences with bounded uniformly $α_k$-energy, denoted by $\{u_{α_k}: α_k>1 \quad \mbox{and} \quad α_k\searrow 1\}$, from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold is of a positive l…

We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in $L^p$ for $p>\frac43$. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic ma…

We study general representations of the free group on two generators into $SL(2,C)$, and the connection with generalized Markoff maps, following Bowditch. We show that Bowditch's Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations a…

Theta graph diffeomorphism shows nontrivial mapping class of 4-sphere.

We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.

We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.

The paper bounds the energy index of harmonic Gauss maps on surfaces.

Researchers construct an index map for contact manifolds using K-theory.

Local constancy of index for certain gradient mappings proved.

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

A new diffusion method approximates Schrödinger bridge with improved convergence.

Paper constructs a transfer map for codimension 2 submanifolds in higher index theory.

Paper proves existence of Dirac-harmonic maps with trivial index.

The study counts critical points of Steklov eigenfunctions on manifolds.

The use of bundle gerbes and bundle gerbe modules is considered as a replacement for the usual theory of Clifford modules on manifolds that fail to be spin. It is shown that both sides of the Atiyah-Singer index formula for coupled Dirac operators can be given natural interpretations using this language and that the re…

We construct a closed Riemannian manifold $(N,h)$ and a sequence of $α$-harmonic maps from $S^2$ into $N$ with uniformly bounded energy such that the energy identity for this sequence is not true.

The paper proves an energy identity for harmonic maps near singularities.

We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated to Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of …

We consider estimating the parametric components of semi-parametric multiple index models in a high-dimensional and non-Gaussian setting. Such models form a rich class of non-linear models with applications to signal processing, machine learning and statistics. Our estimators leverage the score function based first and…

We study a new set of coupled field equations motivated by the non-linear supersymmetric sigma model of quantum field theory. These equations couple a map into a Riemannian manifold controlled by a harmonic map like action with a spinor field along that map. We study the solutions which we call Dirac-harmonic maps from…

Defines a map connecting 3d-index and skein module.

Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.

In this note, we study an invariant associated to the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally…

Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla u_n\|_{L^2(M)}+\|τ(u_n)\|_{L^{p}(M)}\big)\leq Λ\quad \text{for some}\,\,p>1, \eeno where $τ(u_n)$ is the tension field of $u_n$, then there hold the so called ene…

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …

Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…

The normal map of curves is analyzed as a vector field on a cylinder.

The Piola identity $\operatorname{div} \operatorname{cof} \nabla f=0$ is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: on…

New framework models high-Hopf-index hopfions using generalized fold maps.

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

Paper derives second variational formula for statistical manifold mappings.

Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps, derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese…

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

We prove existence results for Dirac-harmonic maps using index theoretical tools. They are mainly interesting if the source manifold has dimension 1 or 2 modulo 8. Our solutions are uncoupled in the sense that the underlying map between the source and target manifolds is a harmonic map.