arXiv research

New spherical curve deformations solve a conjecture.

Study spherical Fourier transform on hypergeometric type harmonic manifolds.

In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and having harmonic pseudo-spherical Gauss map. Then we give a characterization the…

Defines spherical type surfaces via support function and classifies them.

The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.

Study spherically symmetric Finsler metrics with specific curvature properties.

Let $A$ and $A'$ be two Artin groups of spherical type, and let $A_1,\dots,A_p$ (resp. $A'_1,\dots,A'_q$) be the irreducible components of $A$ (resp. $A'$). We show that $A$ and $A'$ are commensurable if and only if $p=q$ and, up to permutation of the indices, $A_i$ and $A'_i$ are commensurable for every $i$. We prove …

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for bra…

The paper classifies Poincaré complexes as topological manifolds.

Study $Θ$-invariants for spherical 3-manifolds via $\mathbb{Z}π$-homology equivalences.

Uniform proof of property R_infinity for specific Artin-Tits groups.

The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…

The study extends knot theory to knotoids using two approaches.

Simple geodesics on spherical tetrahedra identified for specific angles.

Study spherical curves with curvature dependent on distance to a great circle.

The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.

Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.

This paper deals with some simple results about spherical functions of type $δ$, namely new integral formulas, new results about behavior at infinity and some facts about the related $C_σ$ functions.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

Smooth minimizers found for Willmore energy surfaces.

In this paper, we give the general form of spherically symmetric Finsler metrics in $R^n$ and surprisedly find that many well-known Finsler metrics belong to this class. Then we explicitly express projective metrics of this type. The necessary and sufficient conditions that projective Finsler metrics with spherical sym…

Artin groups of type $D_n$ have special cycles and complexes with interesting properties.

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

The paper extends geometric inequalities for nearly spherical sets in various space forms.

The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.

Proves local existence and extension principle for Einstein Yang--Mills system with spherical symmetry.

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit sys…

New flow for capillary surfaces converges to spherical caps.

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natu…

The paper proves stability of inequalities for nearly spherical sets in various spaces.

Artin groups of types $F_4$ and $H_4$ are not commensurable with $D_4$.

Introduces a reduction system for Artin-Tits groups, improving algorithms and proving periodicity results.

Study on Seifert fibered spherical 3-orbifolds, determining their unique fibrations.

The paper defines functions from spherical curves and chord diagrams, proving invariance under specific Reidemeister moves.

In this paper, we classify the spherically symmetric Berwald metrics in $\mathbb{R}^n$. For the spherically symmetric Landsberg metrics, we prove that there do not exist any non-Berwald metrics among the regular case. The partial differential equation systems which can respectively characterize the spherically symmetri…

The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in $S^3$. In recent years, the realization problem for C, T, O, and I-type spherical manifolds has been solved, leaving the D-type manifolds (also known as the prism manifolds) as the only remaining case. Every…

Study proves rigid spectral properties of planets with metric discontinuities.

In this paper, we investigate the spherically symmetric Finsler metrics with isotropic S-curvature and obtain a characterized equation. As an application, we prove that these metrics with Douglas type must be Randers metrics or Berwald metrics. This result leads to two classification theorems.

We study the isometry groups of compact spherical orientable $3$-orbifolds $S^3/G$, where $G$ is a finite subgroup of $\mathrm{SO}(4)$, by determining their isomorphism type. Moreover, we prove that the inclusion of $\mbox{Isom}(S^3/G)$ into $\mbox{Diff}(S^3/G)$ induces an isomorphism of the $π_0$ groups, thus proving …

Conformal qc geometry of spherical qc manifolds are investigated. We construct the qc Yamabe operators on qc manifolds, which are covariant under the conformal qc transformations. A qc manifold is scalar positive, negative or vanishing if and only if its qc Yamabe invariant is positive, negative or zero, respectively. …

The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.

Rigidity shown for spherical product Ricci solitons.

New method finds large curved subcomplexes, proving conjectures for specific groups.

We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irr…

We provide a direct proof for the positivity of Chen-Nester-Tung quasi-local energy with analytic reference in spherical symmetry. A hoop-type theorem for this energy is also established. Finally, the relation between Chen-Nester-Tung and Brown-York quasi-local energies will be discussed.

We define a new finite type invariant for stably homeomorphic class of curves on compact oriented surfaces without boundaries and extend to a regular homotopy invariant for spherical curves.

We prove boundedness and polynomial decay statements for solutions to the spin $\pm1$ Teukolsky-type equation projected to the $\ell=1$ spherical harmonic on Reissner-Nordström spacetime. The equation is verified by a gauge-invariant quantity which we identify and which involves the electromagnetic and curvature tensor…

New rigidity found for 3D warped product domains.