arXiv research

Sharp inequalities on Siegel domains and complex hyperbolic spaces established.

Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.

The paper derives and proves the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.

In this paper we put together some tools from differential topology and analysis in order to study second order semi-linear partial differential equations on a Riemannian manifold $M$. We look for solutions that are constants along orbits of a given group action. Using some results obtained by Helgason in [J DIFFER GEO…

In this work, we show an injectivity result and support theorems for integral moments of a m-tensor field on a simple, real analytic, Riemannian manifold. Integral moments of m-tensor field were first introduced by Sharafutdinov. At first we generalize a Helgason type support theorem proven by Krishnan and Stefanov in …

In this paper, we address the problem of determining a function in terms of its orbital integrals on Lorentzian symmetric spaces. It has been solved by S. Helgason for even-dimensional isotropic Lorentzian symmetric spaces via a limit formula involving the Laplace-Beltrami operator. The result has been extended by J. O…

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines of ${\R}^n$ with the weight $t^k$: $ (I^k\!f)(x,ξ)=\int_{-\infty}^\infty t^kłf(x+tξ),ξ^m\r\,dt. $ We give the range characterization for the operator $f\mapsto(I^0\!f,I^1\!f,\dots, I^m\!f)$ on the Schwartz space of rank $m$ smo…

Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown th…

For a one-parameter family of simple metrics of constant curvature ($4κ$ for $κ\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also t…

Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, …

This paper establishes inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.

Researchers found new dimensions for exceptional Lie group realizations.

New research finds 145 infinite families of CS spheres are standard.

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

New theory proves infinite homology 3-spheres in homology 4-spheres.

Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.

Reduces weak reducing pairs to spheres in 3-sphere Heegaard surfaces.

The study shows how to construct $d$-spheres from $(d-1)$-spheres and $d$-balls without additional vertices.

New proof for sphere recognition algorithm.

Proves stability of convex spheres with similar geodesic lengths.

Soliton spheres are immersed 2-spheres in the conformal 4-sphere S^4=HP^1 that allow rational, conformal parametrizations f:CP^1->HP^1 obtained via twistor projection and dualization from rational curves in CP^{2n+1}. Soliton spheres can be characterized as the case of equality in the quaternionic Pluecker estimate. A …

Study on sphere immersions and their stability indices.

The paper constructs biharmonic maps between spheres using polynomial maps.

Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.

Author provides an alternate proof of the free ribbon lemma.

Paper studies inscribed sphere and lines through centers of Apollonius spheres in n dimensions.

Sharp convergence theorem for sphere submanifolds proved.

In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mea…

We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds $M^{2n}$, where $n=7$ or $8$, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…

In this paper we show that for m>n the set of cobordism classes of maps from m-sphere to n-sphere is trivial. The determination of the cobordism homotopy groups of spheres admits applications to the covers for spheres.

Study finds infinite families of Sasaki-Einstein metrics on spheres.

We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum…

We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in the unit sphere S(x). In the Morse case these graphs are either spheres, the empty…

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

Standard $S^4$ proved to be diffeomorphic to a curious homotopy sphere.

We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the $3$-Sphere and the $4$-Sphere' (arXiv:1706.01405), using the free mathematical software Sage.

New actions found on exotic spheres using group theory.

Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.

New bounds and examples for sphere unknotting numbers.

We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.

Survey of Dupin hypersurfaces in Lie sphere geometry.

Paper studies spaces of flattenings of simplicial spheres and their homotopy type.

The paper constructs homotopy 4-spheres using pochette surgery.

New compact mean convex hypersurfaces found for positive λ.

For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers. It also gives rise to a complete invariant for homology bordism classe…

Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.

Spheres in curve complexes are almost simply connected.