arXiv research

Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.

The paper proves estimates and theorems for Kähler manifolds.

The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.

Upper bound on Stiefel manifold's injectivity radius found.

Characterizes bi-Perron numbers with specific Galois conjugates.

Study dispersive estimates for Schrödinger and wave equations on a cone with specific metric.

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,α}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein m…

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…

We bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and either its conjugate radius or the length of its shortest closed geodesic.

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.

The paper studies conjugate points on Lie groups with specific metrics.

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to i…

In this paper, we prove that, for any integer $n\ge 2,$ there exists an $ε_{n} \ge 0$ so that if $M$ is an n-dimensional complete manifold with sectional curvature $K_{M}\ge 1$ and if $M$ has conjugate radius bigger than $\fracπ{2}$ and contains a geodesic loop of length $2(π-ε_{n}),$ then $M$ is diffeomorphic to the…

A well-known Lemma in Riemannian geometry by Klingenberg says that if $x_0$ is a minimum point of the distance function $d(p,\cdot)$ to $p$ in the cut locus $C_p$ of $p$, then either there is a minimal geodesic from $p$ to $x_0$ along which they are conjugate, or there is a geodesic loop at $p$ that smoothly goes throu…

We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower…

We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$ via conjugate constructio…

A pseudo-Anosov surface automorphism $φ$ has associated to it an algebraic unit $λ_φ$ called the dilatation of $φ$. It is known that in many cases $λ_φ$ appears as the spectral radius of a Perron-Frobenius matrix preserving a symplectic form $L$. We investigate what algebraic units could potentially appear as dilatatio…

The paper counts conjugacy classes of pseudo-Anosov homeomorphisms in Teichmüller space.

The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.

We consider the Lie group PSL(2) (the group of orientation preserving isometries of the hyperbolic plane) and a left-invariant Riemannian metric on this group with two equal eigenvalues that correspond to space-like eigenvectors (with respect to the Killing form). For such metrics we find a parametrization of geodesics…

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

The study examines the growth of conjugacy classes in negatively curved manifolds.

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and po…

Positive injectivity radius for manifolds with Lie structure at infinity.

Proves upper bound on filling radius for manifolds with positive scalar curvature.

Compact theorem for minimal surfaces with lower injectivity radius.

Injectivity radius on Stiefel manifold is π.

Lower bound on boundary injectivity radius for specific tubes.

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.

Study gives bounds on filling radius for Riemannian manifolds.

A condition for a statistical manifold to have an equiaffine structure is studied. The facts that dual flatness and conjugate symmetry of a statistical manifold are sufficient conditions for a statistical manifold to have an equiaffine structure were obtained in [2] and [3]. In this paper, a fact that a statistical man…

The paper tackles efficient computation of optimal transport by approximating conjugates with amortized optimization.

Estimates geodesics on surfaces without conjugate points.

Study finds the covering radius of RM(4,8) is 26.

Clarifies definition of polarized canonical radius in Kahler Ricci flow.

New method speeds up inference for non-conjugate Gaussian processes.

We introduce a multiple conjugation biquandle, and show that it is the universal algebra to define a semi-arc coloring invariant for handlebody-links. A multiple conjugation biquandle is a generalization of a multiple conjugation quandle. We extend the notion of $n$-parallel biquandle operations for any integer $n$, an…

The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.

The paper studies Jacobi fields and conjugate points in projective sprays.

Paper describes how to extend multiple conjugation quandles using maps.

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

For a surface group, a new bound is given for conjugator length function.

The study examines Hopfian properties of conjugation quandles and their underlying groups.

A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a random translation surface. We compute an upper bound on the expected value of the covering radius of a translation surface in any stratum H_1(kappa). The covering radius of a translation surface is the …

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}πR^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $μ_2(B(p,R)…

We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.